Toroidalization of Dominant Morphisms of 3-Folds Steven Dale

Toroidalization of Dominant Morphisms of 3-Folds

Steven Dale Cutkosky

Author address:

Department of Mathematics, University Missouri, Columbia, Mis-souri 65211

E-mail address: [email protected]

Contents

Chapter 1. Introduction 1

Chapter 2. An outline of the proof 5

Chapter 3. Notation 17

Chapter 4. Toroidal morphisms and prepared morphisms 19

Chapter 5. Toroidal ideals 27

Chapter 6. Toroidalization of morphisms from 3-folds to surfaces 29

Chapter 7. Preparation above 2 and 3-points 33

Chapter 8. Preparation 49

Chapter 9. The τ invariant 57

Chapter 10. Super parameters 79

Chapter 11. Good and perfect points 95

Chapter 12. Relations 113

Chapter 13. Well prepared morphisms 119

Chapter 14. Construction of τ -well prepared diagrams 127

Chapter 15. Construction of a τ -very well prepared morphism 169

Chapter 16. Toroidalization 211

Chapter 17. Proofs of the main results 217

Chapter 18. List of technical terms 219

Bibliography 221

v

Abstract

This book contains a proof that a dominant morphism from a 3-fold X to avariety Y can be made toroidal by blowing up in the target and domain. We giveapplications to factorization of birational morphisms of 3-folds.

Received by the editor February 11, 2006.1991 Mathematics Subject Classification. Primary 514E05, 14J30; Secondary 14B25, 14B05.Key words and phrases. Toroidal, Monomial.The first author was supported in part by NSF Grant DMS #0240819.

vi

CHAPTER 1

Introduction

Suppose that f : X → Y is a dominant morphism of algebraic varieties, over afield k of characteristic zero. If X and Y are nonsingular, f : X → Y is toroidalif there are simple normal crossing (SNC) divisors DX on X and DY on Y suchthat f−1(DY ) = DX , and f is locally given by monomials in appropriate etale localparameters on X. The idea of toroidalization is developed in [KKMS]. A precisedefinition of this concept is in [AK], [KKMS] and Definition 4.3 of this paper. Theproblem of toroidalization is to determine, given a dominant morphism f : X → Y ,if there exists a commutative diagram

(1.1)X1

f1→ Y1

Φ ↓ ↓ Ψ

Xf→ Y

such that Φ and Ψ are products of blow ups of nonsingular subvarieties, X1 andY1 are nonsingular, and there exist simple normal crossing divisors DY1 on Y1 andDX1 = f−1(DY1) on X1 such that f1 is toroidal (with respect to DX1 and DY1).This is stated in Problem 6.2.1. of [AKMW].

In this paper we consider a stronger form of toroidalization which we will callstrong toroidalization. Suppose that f : X → Y is a dominant morphism of non-singular projective varieties, DY is a SNC divisor on Y and DX = f−1(DY ) is aSNC divisor on X such that the locus sing(f) where the morphism f is not smoothis contained in DX . The problem of strong toroidalization is to determine if thereexists a commutative diagram (1.1) such that Φ and Ψ are products of blow upsof nonsingular subvarieties which are supported in the preimages of DX and DY

respectively, and make SNCs with the respective preimages of DX and DY , and f1is toroidal with respect to DY1 = Ψ−1(DY ) and DX1 = Φ−1(DX).

Toroidalization, and related concepts, have been considered earlier in differentcontexts, mostly for morphisms of surfaces. Strong torodialization is the strongeststructure theorem which could be true for general morphisms. The concept oftorodialization fails completely in positive characteristic. A simple counter examplein characteristic p > 0 is obtained from the map of curves s = tp(1 + t).

In the case when Y is a curve, toroidalization follows from embedded resolutionof singularities ([H]). When X and Y are surfaces, there are several proofs in print([AkK], [CP], Corollary 6.2.3 [AKMW]). They all make use of special propertiesof the birational geometry of surfaces. An outline of proofs of the above cases canbe found in the introduction to [C3].

In this paper, we prove strong toroidalization for dominant morphisms of 3-folds. Combining this with our theorem in [C3] of strong toroidalization in thecase when X is a 3-fold and Y is a surface, we obtain the following theorem.

1

2 1. INTRODUCTION

Theorem 1.1. Suppose that X is a 3-fold, and f : X → Y is a dominant mor-phism of nonsingular varieties over an algebraically closed field k of characteristic0. Further suppose that there is a simple normal crossings (SNC) divisor DY on Ysuch that DX = f−1(DY ) is a SNC divisor which contains the non smooth locus ofthe map f . Then there exists a commutative diagram of morphisms

X1f1→ Y1

Φ ↓ ↓ Ψ

Xf→ Y

where Φ,Ψ are products of possible blow ups for the preimages of DX and DY re-spectively, and f1 is toroidal with respect to DY1 = Ψ−1(DY ) and DX1 = Φ−1(DX).

A variety over a field k is a quasi projective variety over k. A 3-fold is a 3dimensional variety.

A possible blow up on a nonsingular variety X with toroidal structure DX isthe blow up of a nonsingular subvariety V of DX such that V makes simple normalcrossings with DX . That is, for all q ∈ V , local equations of the components of DX

containing q are part of a minimal set of generators of the ideal of V at q.We also obtain the following strong toroidalization theorem for morphisms of

(possibly singular) varieties.

Theorem 1.2. Suppose that X is a 3-fold and f : X → Y is a dominantmorphism of 3-folds over an algebraically closed field k of characteristic 0. Furthersuppose that there is an equidimensional codimension 1 reduced subscheme DY ofY such that DY contains the singular locus of Y , and DX = f−1(DY ) containsthe non smooth locus of the map f . Then there exists a commutative diagram ofmorphisms

X1f1→ Y1

Φ ↓ ↓ Ψ

Xf→ Y

where Φ,Ψ are products of blow ups of nonsingular curves and points supportedabove DX and DY respectively, DY1 = Ψ−1(DY ) is a simple normal crossingsdivisor on Y1, DX1 = f−1

1 (DY1) is a simple normal crossings divisor on X1 and f1is toroidal with respect to DY1 and DX1 .

This paper is the synthesis of the e-prints [C5], [C7] and [C8]. [C5] provesTheorem 1.3 and proves Theorems 1.1 and 1.2 in the birational case. [C8] provesTheorems 1.1 and 1.2 in the case of a general morphism of 3-folds.

If we relax some of the restrictions in the definition of toroidalization, there areother constructions producing a toroidal morphism f1, which are valid for arbitrarydimensions ofX and Y . In [AK] it is shown that a diagram (1.1) can be constructedwhere Φ is weakened to being a modification (an arbitrary birational morphism).

In [C1], [C2] and [C4], it is shown that a diagram (1.1) can be constructedwhere Φ and Ψ are locally products of blow ups of nonsingular centers and f1is locally toroidal, but the morphisms Φ, Ψ and f1 may not be separated. Thisconstruction is obtained by patching local solutions, at least one of which containsthe center of any given valuation. The morphism f1 is called “locally monomial”.

1. INTRODUCTION 3

The proof of toroidalization of morphisms of 3-folds to surfaces in [C3] breaksup into two parts: a reduction to prepared morphisms and then a proof of toroidal-ization of prepared morphisms from 3-folds to surfaces. This second step is extendedto arbitrary prepared morphisms from n-folds to srufaces in [CK].

The first step in the proof of Theorems 1.1 and 1.2 is the following theorem,proven in Chapters 6 - 8.

Theorem 1.3. Suppose that f : X → Y is a dominant morphism of nonsingularprojective 3-folds over an algebraically closed field k of characteristic zero, withtoroidal structures determined by SNC divisors DY on Y and DX = f−1(DY ) onX such that DX contains the singular locus of f . Then there exists a commutativediagram

X1f1→ Y1

Φ ↓ ↓ Ψ

Xf→ Y

such that Ψ and Φ are products of possible blow ups for the preimages of DY , DX

respectively, such that f1 is prepared for DY1 = Ψ−1(DY ) and DX1 = Φ−1(DX).

A prepared morphism f : X → Y of 3-folds has the property that locally, asuitable projection of Y onto a surface S, when composed with f , is close to beinga toroidal morphism from the 3-fold X to the surface S. This theorem uses thetheorem of toroidalization of morphisms of 3-folds to surfaces in [C3] to simplifythe morphism f .

It has been shown in [AKMW] that weak factorization of birational morphismsholds in characteristic zero, and arbitrary dimension. That is, birational morphismsof complete varieties can be factored by an alternating sequence of blow ups andblow downs of non singular subvarieties. Weak factorization of birational (toric)morphisms of toric varieties, (and of birational toroidal morphisms) has been provenby Danilov [D1] and Ewald [E] (for 3-folds), and by Wlodarczyk [W1], Morelli [Mo]and Abramovich, Matsuki and Rashid [AMR] in general dimensions.

Our Theorem 1.2, when combined with weak factorization for toroidal mor-phisms ([AMR]), gives a new proof of weak factorization of birational morphismsof 3-folds. We point out that our proof uses an analysis of the structure of powerseries of local germs of a mapping, as opposed to the entirely different proof of weakfactorization, using geometric invariant theory, of [AKMW].

Corollary 1.4. Suppose that f : X → Y is a birational morphism of 3-foldswhich are proper over an algebraically closed field k of characteristic zero. Thenthere exists a commutative diagram of morphisms factoring f ,

X1 X3 Xn

· · · X X2 X4 Y

where each arrow is a product of blow ups of points and nonsingular curves.

The problem of strong factorization, as proposed by Abhyankar [Ab2] andHironaka [H], is to factor a birational morphism f : X → Y by constructing adiagram

Z

Xf→ Y

4 1. INTRODUCTION

where Z → X and Z → Y factor as products of blow ups of nonsingular subvarieties.Oda [O] has proposed the analogous problem for (toric) morphisms of toric varieties.

A birational morphism f : S → Y of (nonsingular) surfaces can be directlyfactored by blowing up points (Zariski [Z1] and Abhyankar [Ab1]), but there areexamples showing that a direct factorization is not possible in general for 3-folds(Shannon [Sh] and Sally[S]).

We also obtain as an immediate corollary to Theorem 1.2 the following newresult, which reduces the problem of strong factorization of 3-folds to the case oftoroidal morphisms.

Corollary 1.5. Suppose that the Oda conjecture on strong factorization ofbirational toroidal morphisms of 3-folds is true. Then the Abhyankar, Hironakastrong factorization conjecture of birational morphisms of complete (characteristiczero) 3-folds is true.

Abhyankar’s local factorization conjecture [Ab2], which is “strong factoriza-tion” along a valuation, follows from local monomialization (Theorem A [C1]), toreduce to a locally toroidal morphism, and local factorization for toroidal mor-phisms along a valuation Christensen [Ch] (for 3-folds), and Karu [K] or [CS] ingeneral dimensions.

CHAPTER 2

An outline of the proof

Suppose that f : X → Y is a dominant morphism of nonsingular projective3-folds, over an algebraically closed field k of characteristic zero, and suppose thatDY is a simple normal crossings divisor on Y such that DX = f−1(DY ) is also asimple normal crossings divisor, defining toroidal structures on X and Y . Furthersuppose that the locus of points in X where f is not smooth is contained in DX .We will refer to points where three components of DY (or DX) intersect as 3-point,points where 2-components intersect as 2-points, and the remaining points of DY

(and DX) as 1-points.Permissible parameters are defined in Chapter 4. A set of regular parameters

in OY,q (or OY,q) are permissible parameters at q if the product of the first of theseparameters is a local equation of DY at q.

There is a natural stratification of Y into locally closed subsets such that f isincreasingly complex in higher strata.

For q ∈ Y −DY , we have that f is smooth above q, so that (since f is genericallyfinite) if p ∈ f−1(q), and u, v, w are regular parameters at q, then u, v, w are regularparameters at p.

The non finite locus of f is

NFf = q ∈ Y | dim f−1(q) > 0.

NFf is a closed subset of DY and has codimension ≥ 1 in DY (codimension ≥ 2in Y ). If q ∈ DY − NFf , then f is finite above q, and by Abhyankar’s Lemma([Ab4] and XIII 5.3 [G]), there exist permissible parameters u, v, w at q such thatif p ∈ f−1(q), there exist permissible parameters x, y, z at p such that one of thefollowing forms hold:

q a 1-point, p a 1-point

(2.1) u = xa, v = y, w = z


(2.2) u = xa, v = yb, w = z


(2.3) u = xa, v = yb, w = zc.

If q is a general point of a 1-dimensional component of NFf , then f behaves as adominant morphism of nonsingular surfaces (defined over a non algebraically closedfield) at q. Toroidalization of maps of surfaces over non closed fields of characteristic0 is constructed in [CP] and Corollary 6.2.3 [AKMW]. We conclude that thereexists a neighborhood V of q in Y such that NFf ∩ V is a nonsingular curve, and

5

6 2. AN OUTLINE OF THE PROOF

there exists a commutative diagram

W1f1→ V1

Φ1 ↓ ↓ Ψ1

W = f−1(V )f→ V

such that Φ1, Ψ1 are products of blow ups of nonsingular curves which dominate C,and for q1 ∈ DV1 = Ψ−1

1 (DV ), there exist permissible parameters u, v, w ∈ OV1,q1

such that if p ∈ f−1

1 (q1), there exist formal permissible parameters x, y, z at p suchthat one of the following local forms hold:

q1 is a 1-point, p1 is a 1-point

(2.4) u = xa, v = y, w = z

q1 is a 2-point, p1 is a 1-point

(2.5) u = xa, v = xb(α+ y), w = z

with 0 6= α ∈ k.q1 is a 2-point, p1 is a 2-point

(2.6) u = xayb, v = xcyd, w = z.

The final strata is a finite set of points contained in NFf . The map f isextremely complex above these points and defies a simple analysis.

Our goal is to construct a commutative diagram

X1f1→ Y1

Φ1 ↓ ↓ Ψ1

Xf→ Y

such that Φ1, Ψ1 are products of blow ups of nonsingular curves and points whichare possible blow ups (Chapter 3) such that for all q ∈ DY1 = Ψ−1

1 (DY ), thereare permissible parameters at q such that for all p ∈ f−1

1 (q), u, v, w are toroidalforms at p. That is, there exist permissible parameters x, y, z ∈ OX1,p such thatone of the forms 1 - 6 after Definition 4.3. hold. Such a map f1 is called toroidal(Definition 4.3).

Observe that the forms (2.1) - (2.6) above are toroidal forms.In this paper, we develop a series of algorithms to manipulate local germs of

mappings, expressed in terms of series and polynomials. We are required to blowup in both the domain and target.

The main result of resolution of singularities [H] tells us that we can constructa diagram

X1

↓ f

Xf→ Y

where for all points p ∈ X1 and q = f(p) ∈ Y there are permissible parametersx, y, z at p and u, v, w at q, such that we have an expression

(2.7)u = xaybzcγ1

v = xdyezfγ2

w = xgyhziγ3

2. AN OUTLINE OF THE PROOF 7

where γ1, γ2, γ3 are units at p. The algorithms of resolution give us no informationabout the structure of the units γ1, γ2, γ3. In general, we will have that the monomi-als xaybzc, xdyezf , xgyhzi are algebraically dependent. We may then hope to blowup nonsingular subvarieties of Y (and blow up above X1 to resolve the indetermi-nancy of the resulting rational map), leading to new regular parameters at a pointq′ above q with regular parameters u′, v′, w′ which are obtained by dividing u, v, wby each other. We will eventually obtain an expression such as w′ = γ3 − γ3(p) bythis procedure, which need not be any better than the expressions we started withfor our original map f : X → Y .

In attempting to obtain a sufficiently deep understanding of map germs toprove toroidalization, we must understand information about germs such as (2.7),up to the level of a series expansion of the units γi. In considering this problem,we are led to generalized notions of multiplicity, which can actually increase aftermaking blow ups above X (in the proof of toroidalization of morphisms of 3-folds tosurfaces [C3]). In summary, the problem of toroidalization requires new methods,which are not contained in any proofs of resolution of singularities.

In Definition 4.6 we define a prepared morphism f : X → Y of 3-folds. This isrelated to the notion of a prepared morphism from a 3-fold to a surface (Definition6.5 [C3]). The basic idea of a prepared morphism f : X → Y of 3-folds is thatlocally an appropriate projection of Y onto a surface S, when composed with f ,is (close to) a toroidal morphism from a 3-fold to a surface. If f is prepared, bya simple local calculation involving Jacobian determinants, we have expressions ofthe form of Definition 4.1 and Lemma 4.2 (or 2 (b), 2 (c) of Definition 4.6) at allpoints of X. A typical case, when p is a 1-point and q is a 2-point, is

u = xa, v = xb(α+ y), w = g(x, y) + xcz

where 0 6= α ∈ k and g(x, y) is a series.

Construction of a prepared morphismIn Chapters 6 - 8 we prove Theorem 1.3. It is shown that we can construct

from our given morphism f : X → Y a commutative diagram

X1f1→ Y1

Φ ↓ ↓ Ψ

Xf→ Y

where Φ,Ψ are products of possible blow ups of nonsingular curves and points,and f1 is prepared with respect to the SNC divisors DX1 = Φ−1(DX) and DY1 =Ψ−1(DY ).

The essential ingredient of the proof is our theorem on toroidalization of dom-inant morphisms from 3-folds to surfaces (Theorem 19.11 [C3]).

Let NFf be the non finite locus of f . Above points of DY − NFf , we see by(2.1) - (2.3) that f is prepared.

Above general points of one dimensional components of NFf , we can also easilyshow that f is prepared (but in general is not toroidal).

Let Λ ⊂ DY be the finite set of points above which f is not prepared. Byblowing up 2-curves and 3-point above Y , we can assume that Λ is a finite set of1-points in Y (Lemmas 7.1 and 7.2).


Suppose that q ∈ Λ. We can choose permissible parameters u, v, w at q. LetV be a neighborhood of q in Y such that u, v, w are uniformizing parameters (λ :V → spec(k[u, v, w]) is etale).

Let π : spec(k[u, v, w]) → S = spec(k[u, v]) be the projection. We applyTheorem 6.1 to the map π f to construct a commutative diagram

(2.8)W1 → S1

Φ ↓ ↓ Ψf−1(V ) → S

such that W1 → S1 is toroidal.Now Ψ is a sequence of blow ups of points which induce a sequence of blow ups

of curves Y1 → Y such that there is a factorization W1 → Y1 which is prepared.The curves blown up above Y dominate the curve C with local equations u = v = 0at q. We can choose u, v so that C is a general curve through q on DY . With abit of care, we can construct the diagrams (2.8) so that they patch to a projectivemorphism

Y1f1→ Y1

↓ ↓X

f→ Y

such that f1 is prepared.

The τ-invariantIt may appear that the local forms of a prepared morphism are very simple, and

we can easily modify them to obtain a toroidal form. However, a little explorationwith these local forms will reveal that the notion of being prepared is stable underblow ups of 2-curves, but is in general not stable under blow ups of points and curveswhich are not 2-curves. It also will quickly become apparent that it is absolutelynecessary to blow up subvarieties of Y other than 2-curves to toroidalize. Thisleads to the notion of super parameters (Definition 10.1), which is necessary for allthe blow ups which we consider to preserve the notion of being prepared, and for aglobal invariant τ to behave well under blow ups.

To prove Theorem 1.2, we may assume that f is prepared. This condition ispreserved throughout the proof.

We define the τ -invariant of a point p ∈ DX in Definition 9.1. τf (p) = −∞ iff is toroidal at p, and τf (p) is a positive integer otherwise.

The simplest case is when p is a 1-point and q = f(p) is a 1-point. We havepermissible parameters u, v, w at q and x, y, z at p such that

u = xa, v = y, w =∑i<c

αi(y)xi + xc(z + β)

with β ∈ k. Here u = 0 is a local equation of DY and x = 0 is a local equation ofDX . f is toroidal at p if and only if c = 0. In this case we define τf (p) = −∞.

If c > 0, we define (as follows (9.11))

τf (p) = |(aZ +∑αi 6=0

iZ)/aZ|.

The most complicated case is when p is a 3-point. Since f is prepared, f(p) = qis a 2-point or a 3-point. There are permissible parameters u, v, w in OY,q and x, y, z


in OX,p (xyz = 0 is a local equation of DX , uv = 0 or uvw = 0 is a local equationof DY ) and there is an expression (9.1) of Definition 9.1

(2.9)u = xaybzc

v = xdyezf

w =∑i≥0 αiMi +N

with 0 6= αi ∈ k, Mi = xaiybizci , N = xgyhzi, N 6 |Mi for all i,

rank(a b cd e f

)= 2,det

a b cd e fai bi ci

= 0 for all i,

det

a b cd e fg h i

6= 0.

f is toroidal at p (and τf (p) = −∞) if and only if q is a 3-point and

w = γN

where γ is a unit in OX,p.Otherwise, define (as follows (9.1) of Definition 9.1)

Hp = (a, b, c)Z + (d, e, f)Z +∑i

(ai, bi, ci)Z,

Ap =

(a, b, c)Z + (d, e, f)Z + (a0, b0, c0)Z if q is a 3-point (we have w = γM0

where γ is a unit in OX,p)Z(a, b, c)Z + (d, e, f)Z if q is a 2-point.

Now define

(2.10) τf (p) = |Hp/Ap|.τf is well defined (Lemma 9.2) and upper semi-continuous (Lemma 9.3).We define (after Definition 9.1)

τf (X) = maxτf (p) | p ∈ DX.We have that τf (X) ≥ 1 or τf (X) = −∞. f is toroidal if and only if τf (X) = −∞.

The proof of Theorem 1.2 is by descending induction on τf (X). In our proofof Theorem 1.2 we may thus assume that τ = τf (X) 6= −∞.

Let (Definition 9.4))

GX(f, τ) = p ∈ DX | τf (p) = τ.GX(f, τ) is Zariski closed in X (since τf is upper semi continuous). Let (Definition9.4))

GY (f, τ) = f(GX(f, τ)).GY (f, τ) is Zariski closed in Y since f is proper.

By Theorem 14.4, there exist sequences of blow ups of 2-curves

X1f1→ Y1

↓ ↓X

f→ Y

such that f1 is τ -prepared (Definition 9.4). That is, f1 is prepared, and GY1(f1, τ)contains no 2-curves or 3-points.


From now on, we assume that f : X → Y is τ -prepared and τ > 1. The caseτ = 1 is actually the simplest, but is a little different from the general case τ > 1.

Relations, and τ-quasi-well prepared morphismsA τ -quasi-well prepared morphism f : X → Y is a τ -prepared morphism with

a relation R for f which has certain nice properties.A Relation R (Definition 12.5) consists of a finite number of primitive relations

Ri for f . To each primitive relation Ri is associated a pre-relation Ri on Y .A pre-relation Ri (Definitions 12.1, 12.3) consists of a locally closed subset

U(Ri) of DY consisting of 1-points and a finite number of 2-points. For eachq ∈ U(Ri) we have permissible parameters u = uRi(q)

, v = vRi(q), w = wRi(q)

at qsuch that uRi(q)

, vRi(q)∈ OY,q.

If q is a 2-point, we have a = aRi(q), b = bRi

(q), e = eRi(q) ∈ Z and 0 6= λ =

λRi(q) ∈ k such that e > 1, and gcd(a, b, e) = 1.If q is a 1-point, we have a = aRi

(q), e = eRi(q) ∈ Z and 0 6= λ = λRi

(q) ∈ ksuch that e > 1, and gcd(a, e) = 1.

We can formally represent the relation Ri by the (meromorphic) form

(2.11) we − λuavb = 0

if q is a 2-point,

(2.12) we − λua = 0

if q is a 1-point.We will only allow morphisms Ψ : Y1 → Y in our construction which are

the blow up of an admissible center V for Ri (Definition 12.2). V must be apossible center, and if q ∈ U(Ri), then some subset of our permissible parametersu = uRi(q)

, v = vRi(q), w = wRi(q)

are local equations of V at q.For instance, if q is a 1-point, the possible local equations for V at q are

u = v = w = 0 (the point q) and u = w = 0 (the curve which is the intersectionof the (possibly formal) surface w = 0 with the component E of DY containing q).Of course this curve germ must be algebraic for V to be an admissible center.

Suppose that Ψ : Y1 → Y is the blow up of an admissible center. Then wedefine (after Definition 12.2) the transform R

1

i of the pre-relation Ri on Y1. Forq ∈ U(Ri), we define U(R

1

i ) ∩ Ψ−1(q) to be the points of Ψ−1(q) which lie on thestrict transform of the surface germ wRi(q)

= 0.For instance, suppose that Ψ1 is the blow up of a curve V , and q ∈ V ∩ U(Ri)

is a 1-point. Then we must have that uRi(q)= wRi(q)

= 0 are local equations of Vat q. The only point q1 ∈ Ψ−1(q) which is on the strict transform of wRi(q)

= 0 haspermissible parameters defined by

uRi(q)= u

R1i (q1)

, vRi(q)= v

R1i (q1)

, wRi(q)= u

R1i (q1)

wR

1i (q1)

.

The strict transform of the form weRi(q)

− λuaRi(q)

= 0 is the meromorphic form

weR

1i (q1)

− λua−eR

1i (q1)

= 0.

We have eR

1i (q1)

= e, aR

1i (q1)

= a− e.A primitive relation Ri for f (Definition 12.5) associated to the pre-relation Ri

on Y consists of a locally closed subset T (Ri) of f−1(U(Ri)) ∩GX(f, τ).


Suppose that p ∈ T (Ri), and q = f(p). We must have that u = uRi(q), v =

vRi(q)are toroidal forms at p, and there exists an expression

weRi(q)

= uavbΛ

if q is a 2-point,weRi(q)

= uaΛ

if q is a 1-point, where Λ is a unit in OX,p, with Λ(p) = λ.When we construct a τ -quasi-well prepared morphism, we must further insist

that uRi(q), vRi(q)

, wRi(q)do not satisfy a relation at p with a smaller value of e (3

of Definition 13.1).Now suppose that we have a commutative diagram

X1f1→ Y1

Φ ↓ ↓ Ψ

Xf→ Y

where Ψ is an admissible blow up, f1 is τ -prepared, and for p ∈ DX1 , τf1(p) ≤τf (Φ(p)). Let R

1

i be the transform of Ri on Y1.Suppose that for

p ∈ f−11 (U(R

1

i )) ∩ Φ−1(T (Ri)) ∩GX1(f1, τ),

we have that uR

1i, vR

1i

are toroidal forms at p. Then we can define a primitiverelation R1

i on X1 such that

T (R1

i ) = f−11 (U(R

1

i )) ∩ Φ−1(T (Ri)) ∩GX1(f1, τ).

The resulting relation R1 is called the transform of R (Definition 12.6).A τ -prepared morphism f : X → Y is τ -quasi-well prepared if there is a relation

R for f such that the sets T (Ri) are a partition of GX(f, τ) (Definition 13.1).The basic building block for the construction of a toroidalization will be a

τ -quasi-well prepared diagram

X1f1→ Y1

Φ ↓ ↓ Ψ

Xf→ Y

consisting of a commutative diagram of morphisms such that Ψ is an admissibleblow up, f1 is τ -prepared, for p ∈ DX1 , τf1(p) ≤ τf (Φ(p)), the transform R1 of Ris defined for f1, and f1 is τ -quasi-well prepared for this relation. The necessarydiagrams of this type are constructed in Chapter 14.

To get an idea of how this works in the simplest case, we will construct such adiagram above a germ of a 1-point p ∈ T (Ri) which maps to a 1-point q ∈ U(Ri).Let

u = uRi(q), v = vRi(q)

, w = wRi(q),

and suppose that the admissible center V with local equations u = w = 0 is blownup by Ψ. This case is worked out in detail in the proof of Lemma 14.11.

Points q1 of Ψ−1(q) have local equations u1, v1, w1 which are defined by one ofthe following cases:

q1 is a 1-point, with


(2.13) u = u1, v = v1, w = u1(w1 + α)

for some α ∈ k, or q1 is a 2-point,

(2.14) u = u1w1, v = v1, w = w1.

Now we must have permissible parameters x, y, z at p for u, v, w defined by

u = xa, v = y, w =m∑j=0

aij (y)xij + xn(z + β).

where im < n, aij (y) 6= 0 for all j and β ∈ k. We have that weRi(q) = uaRi

(q)Λ,where Λ is a unit in OX,p such that Λ(p) = λRi

(q). Thus we have that ai0(0) 6= 0,and a 6 | i0 (since eRi

(q) > 1 and gcd(aRi(q), eRi

(q)) = 1). Further, the rationalmap X → Y1 is defined at p, so we take X1 = X in a neighborhood of p.

First suppose that a ≤ i0. Since a 6 | i0, we have a < i0. q1 = f1(p) thus haspermissible parameters u1, v1, w1 satisfying (2.13) with α = 0. Thus q1 ∈ U(R

1

i ).We have

u1 = xa, v1 = y, w1 =m∑j=0

aij (y)xij−a + xn−a(z + β).

We have

weRi

(q)

1 = uaRi

(q)−eRi(q)

1 Λ

where Λ(p) = λRi(q). We compute

τf1(p) = |aZ +∑mj=0(ij − a)Z/aZ|

= |aZ +∑mj=0 ijZ/aZ|

= τf (p) = τ.

Thus p ∈ T (R1i ).

Now suppose that a > i0. We have an expression w = xi0γ, where γ is the unitseries

γ = ai0(y) + ai1(y)xi1−i0 + · · ·+ aim(y)xim−i0 + xn−i0(β + z).

Let x = xγ1i0 . We have expressions (by the type of calculation worked out in

Chapter 9)

u = xaγ−ai0 = xa(P (y, xi1−i0 , . . . , xim−i0) + xn−i0(α+ z))

v = yw = xi0

where x, y, z are regular parameters in OX,p, P is a unit series, and α ∈ k.q1 = f1(p) has permissible parameters u1, v1, w1 satisfying (2.14). Thus q 6∈

U(R1

i ). w1, u1, v1 are permissible parameters at the 2-point q1, and we have

(2.15)w1 = xi0

u1 = xa−i0(P (y, xi1−i0 , . . . , xim−i0) + xn−i0(α+ z))v1 = y.

This is case 2 (b) of Definition 4.6 of a local form of a prepared morphism.


We compute τf1(p) from (9.10) of Definition 9.1. We have

τf1(p) ≤ |(i0Z +∑mj=0(ij − i0 + a− i0)Z)/(i0Z + (a− i0)Z)|

= |(aZ +∑mj=0 ijZ)/(i0Z + aZ)|

< |(aZ +∑mj=0 ijZ)/aZ| = τf (p) = τ

The strict inequality in line 3 above follows from the condition eRi(q) > 1, which

implies that a 6 | i0.The construction of a τ -quasi-well prepared morphism is rather complicated.

This is accomplished in Theorem 15.4. One case where the basic idea is fairlytransparent is when a 1 point p with τf (p) = τ maps to a 1 point q. We start withpermissible parameters u, v, w ∈ OY,q such that there are permissible parametersx, y, z in OX,p such that there is an expression

u = xa, v = y, w =m∑j=0

aij (y)xij + xn(α+ z),

where aij (y) 6= 0 for all j, im < n and α ∈ k. Let φ(u) =∑a|ij aij (v)u

ija . We

define a pre-relation R at q by setting

uRi(q)= u, vRi(q)

= v, wRi(q)= w − φ(u).

Since we are assuming τ > 1, there exists j1 such that a 6 | j1 and

wRi(q)= xj1Λ

where Λ ∈ OX,p is such that x 6 | Λ. Part of the difficulty is to arrange things so thatΛ is actually a unit in OX,p. Assume that this is the case here. Let d = gcd(a, j1).Let e = a

d > 1, a = j1d . We have an expression

weRi(q)

= uaΛe.

Setting

eRi(q) = e, aRi

(q) = a, λRi(q) = Λ(p)e,

we define the primitive relation Ri at p.

τ-very well prepared morphismsA morphism f : X → Y is τ -very well prepared with respect to a relation R if it

is τ -quasi-well prepared, and satisfies some other properties (Definition 13.9). Oneof these properties is that R has a pre-algebraic structure (Definition 13.4). That is,if q ∈ U(Ri) for some pre-relation Ri associated to R, we have that wRi(q)

∈ OY,q.If f is τ -very-well prepared, then there exists a finite, distinguished set of

nonsingular algebraic surfaces Ω(Ri) in Y such that U(R) ∩ Ω(Ri) = U(Ri), andfor q ∈ U(Ri), wRi(q)

= 0 is a local equation of Ω(Ri). There is a SNC divisor Fi onΩ(Ri) such that the intersection graph of Fi is a tree. The irreducible componentsof Fi are the nonsingular curves V = E · Ω(Ri), which are the intersection of acomponent E of DY containing a point of U(Ri) with Ω(Ri). Such V are closed inY . Further, V is a *-permissible center (Definition 13.12). That is, V is a possible


center for DY and there exists a commutative diagram of morphisms

(2.16)X1

f1→ Y1

Φ1 ↓ ↓ Ψ1

Xf→ Y

where Ψ1 is the blow up of V (possibly followed by blow ups of some special 2-pointsq not mapping to GY (f, τ)), such that f1 is τ -quasi-well prepared, τf1(p) ≤ τf (p)for p ∈ DX1 , and f1 is τ -very-well prepared.

A τ -very-well prepared morphism is constructed by constructing an appropriatesequence of τ -quasi-well prepared diagrams.

Making τf (X) decreaseSuppose that f : X → Y is τ -very-well prepared. In Theorem 16.1, we construct

a commutative diagramXn

fn→ Yn↓ ↓X

f→ Y

such that fn is prepared and τfn(Xn) < τ . By induction on τ , we then obtain theproof of Theorem 1.2.

We fix an index i of the relations Ri, with corresponding surface Ω(Ri). Acurve C on Y is good if it is a component of the divisor Fi on Ω(Ri), and if j issuch that C ∩ Ω(Rj) 6= ∅, then C is a component of Fj .

In our construction we begin with i = 1, and blow up a good curve V on Y ,by a morphism (2.16). Let R1 be the transform of R on Xn. Since the intersectiongraph of Fi is a tree, there exists a good curve. Suppose that p ∈ X is a point withτf (p) = τ and q = f(p) ∈ V . Suppose that p1 ∈ Φ−1

1 (p). We assume that p and qare 1-points. this is the simplest but most illustrative case. Set q1 = f1(p1). Let

u = uR1(q), v = vR1(q)

, w = wR1(q).

V has local equations u = w = 0 at q. Thus q1 has regular parameters u1, v, w1

with

(2.17) u = u1w1, w = w1

or

(2.18) u = u1, w = u1(w1 + α)

with α ∈ k. Leta = aR1

(q), e = eR1(q), λ = λR1

(q).The relation R1 is determined at q by a form

(2.19) we − λua.We computed earlier (in the discussion on relations) that in this case p1 = p, andτf1(p1) < τf (p) unless (2.18) holds. In this case, we saw that α = 0 in(2.18), andwe have

aR

11(q1) = a− e, e

R11(q1) = e, λ

R11(q1) = λ,

with aR

11(q1) > 0.

IfT (R1

1) = Φ−11 (T (R1)) ∩ f−1

1 (U(R11)) ∩GX1(f1, τ) = ∅,


then we increase i to 2.Suppose otherwise. We have SNC divisors F 1

i = Ψ−11 (Fi) on the surfaces

Ω(R1

i ) = Ψ−11 (Ω(Ri))). There exists a good curve on Y1 for the SNC divisor F 1

i onthe surface Ω(R

1

1)). We continue to iterate, blowing up good curves. After a finitenumber of blow ups, we obtain a reduction in τ . The easiest case to see this in isabove the 1-point q considered above. If τf1(p1) = τf (p), we blow up the curve C1

with local equations u1 = w1 = 0 at q1. If there is a point in T (R21) above p1, we

must have a reduction aR

21(q2) = a− 2e. Since this number must be positive, after

a finite number of iterations, we must have a decrease in τ .We then continue this algorithm for the preimages of all of the surfaces Ω(Ri).

The algorithm terminates in the construction of a morphism with a drop in τ asdesired.

The final proof of Theorem 1.2 is given in Chapter 17.

The relationship of the birational case to the generically finite caseIf f : X → Y is a birational morphism of projective, nonsingular 3-folds, and

q ∈ Y is such that dim f−1(q) < 2, then the morphism f factors as a product ofblow ups of nonsingular curves above a neighborhood of q [D2]. If f : X → Y isalso prepared, the situation is even simpler. In this case the germ of the nonfinitelocus NFf of f at q is a nonsingular curve C, and f : X → Y factors through theblow up of C in a neighborhood of q (Lemma 4.8).

This statement fails completely if f : X → Y is not birational.We give a simple example of a projective, prepared, dominant morphism of

3-folds f : X → Y , with toroidal structures DY and DX = f−1(DY ) such that fis prepared, and there exists a 1-point q ∈ DY such that the germ of Ff at q issingular.

LetY = spec(k[u, v, w]) = k3,

and q ∈ Y be the origin. Let

X0 = spec(k[x, y, z]) = k3.

Let DY be the plane u = 0.Define f0 : X0 → Y by the inclusion of k-algebras k[u, v, w]→ k[x, y, z] defined

by u = x(x− 2), v = y, w = z. DX0 = f−10 (DY ) is the (disjoint) union of the two

planes x = 0 and x = 2.Now defineX → X0 to be the blow up of the two disjoint, nonsingular curves C1

and C2, where C1 is defined by x = z = 0, and C2 is defined by x− 2 = z− y2 = 0.Let f : X → Y be the composite map. X has a toroidal structure defined byDX = f−1(DY ). It can be shown after a little calculation that f is prepared. Infact, at the 1-point q, the permissible parameters u, v, w realize a form (4.5) and(4.11), or (4.6) and (4.12) of Definition 4.1 at all points of f−1(q), so that theconclusions of Lemma 4.9 holds for u, v, w at q.

However, we see that the nonfinite locus NFf of f is the union of two curvesγ1 and γ2 where γ1 is defined by u = w = 0, and γ2 is defined by u = w − v2 = 0.Ff thus has a singularity at q. Observe that both components of Ff are in factnonsingular at q, which is consistent with Lemma 4.10, which shows that all formalcomponents of the nonfinite locus of a prepared morphism of 3-folds are nonsingularat a 1-point.


We further see that there is no point or curve V in Y such that f : X → Yfactors through the blow up of V .

This contrast, between the extremely simple structure of birational morphismsof 3-folds above 1-points, and the much more complicated structure above 1-pointsin the general case of morphisms of 3-folds, is the major obstruction to extendingthe proofs of the e-print [C5] to the general case.

CHAPTER 3

Notation

Throughout this paper, k will be an algebraically closed field of characteristiczero. A variety is a quasi projective variety over k. A curve, surface or 3-foldis a quasi projective variety over k of respective dimension 1, 2 or 3. If X is avariety, and p ∈ X is a nonsingular point, then regular parameters at p are regularparameters in OX,p. Formal regular parameters at p are regular parameters inOX,p. If X is a variety and V ⊂ X is a subvariety, then IV ⊂ OX will denote theideal sheaf of V . If V and W are subvarieties of a variety X, we denote the schemetheoretic intersection Y = spec(OX/IV + IW ) by Y = V ·W .

Let f : X → Y be a morphism of varieties. We will denote the singular locusof f by sing(f). sing(f) is the closed set of points p ∈ X such that f is not smoothat p. If D is a (reduced) Cartier divisor on Y , then f−1(D) will denote the reduceddivisor f∗(D)red.

Suppose that a, b, c, d ∈ Q. Then we will write (a, b) ≤ (c, d) if a ≤ b and c ≤ d.Suppose that V is an affine variety over k. x1, . . . , xn ∈ Γ(V,OV ) are uni-

formizing parameters on V if the natural morphism V → Spec(k[x1, . . . , xn]) ∼= A3

is etale. Suppose that Λ : V → X is an etale morphism. DefineDX∩V = Λ−1(DX).A toroidal structure on a nonsingular variety X is a simple normal crossing

divisor (SNC divisor) DX on X. We will say that an ideal sheaf I ⊂ OX is toroidalif I is locally generated by monomials in local equations of components of DX .

We will say that a nonsingular curve C which is a subvariety of a nonsingular3-fold X with toroidal structure DX makes simple normal crossings (SNCs) withDX if for all p ∈ C, there exist regular parameters x, y, z at p such that x = y = 0are local equations of C, and xyz = 0 contains the support of DX at p.

Suppose that X is a nonsingular 3-fold with toroidal structure DX . If p ∈ DX

is on the intersection of three components of DX then p is called a 3-point. Ifp ∈ DX is on the intersection of two components of DX (and is not a 3-point) thenp is called a 2-point. If p ∈ DX is not a 2-point or a 3-point, then p is called a1-point. If C is an irreducible component of the intersection of two components ofDX , then C is called a 2-curve. Σ(X) will denote the closed locus of 2-curves onX.

A possible center on a nonsingular 3-fold X with toroidal structure defined bya SNC divisor DX , is a point on DX or a nonsingular curve in DX which makesSNCs withDX . A possible center on a nonsingular surface S with toroidal structuredefined by a SNC divisor DS is a point on DS . We will also call the blow up of apossible center a possible blow up.

Observe that if Φ : X1 → X is the blow up of a possible center, then DX1 =Φ−1(DX) is a SNC divisor on X1. Thus DX1 defines a toroidal structure on X1.All blow ups Φ : X1 → X considered in this paper will be of possible centers, andwe will impose the toroidal structure on X1 defined by DX1 = Φ−1(DX).

17

18 3. NOTATION

We define the nonfinite locus NFf of a proper morphism f : X → Y to be

q ∈ Y | dim f−1(q) > 0.The fundamental locus Fg of a birational map g : X −−− 7→ Y is the complementof the maximal open subset of X on which g is a morphism.

By a general point q of a variety V , we will mean a point q which satisfiesconditions which hold on some nontrivial open subset of V . The exact open condi-tion which we require will usually be clear from context. By a general section of acoherent sheaf F on a projective variety X, we mean the section corresponding toa general point of the k-linear space Γ(X,F).

If X is a variety, k(X) will denote the function field of X. A 0-dimensionalvaluation ν of k(X) is a valuation of k(X) such that k is contained in the valuationring Vν of ν and the residue field of Vν is k. If X is a projective variety whichis birationally equivalent to X, then there exists a unique (closed) point p1 ∈ X1

such that Vν dominates OX1,p1 . p1 is called the center of ν on X1. If p ∈ X is a(closed) point, then there exists a 0-dimensional valuation ν of k(X) such that Vνdominates OX,p (Theorem 37, Section 16, Chapter VI [ZS]). For a1, . . . , an ∈ k(X),ν(a1), . . . , ν(an) are rationally dependent if there exist α1, . . . , αn ∈ Z which arenot all zero, such that α1ν(a1) + · · · + αnν(an) = 0 (in the value group of ν).Otherwise, ν(a1), . . . , ν(an) are rationally independent. The valuations of k(X)form a quasi compact topological space called the Zariski Riemann manifold of X.This is discussed in [Z], [ZS] and the Appendix [Li] to [Z2].

If x1, . . . , xn are indeterminates, and Mi =∏nj=1 x

aij

j are Laurent monomialsfor 1 ≤ i ≤ m, then we will denote rank(M1, . . . ,Mm) = rank(aij).

Suppose that S is a finite set. Then |S| will denote the cardinality of S.

CHAPTER 4

Toroidal morphisms and prepared morphisms

Suppose that X is a nonsingular variety with toroidal structure DX .Suppose that q ∈ X. We say that u, v, w are (formal) permissible parameters

at q (for DX) if u, v, w are regular parameters in OX,q such that u = 0 is a localequation of DX at q if q is a 1-point, uv = 0 is a local equation of DX at q if q isa 2-point and uvw = 0 is a local equation of DX at q if q is a 3-point. u, v, w arealgebraic permissible parameters if we further have that u, v, w ∈ OX,q.

Definition 4.1. Let f : X → Y be a dominant morphism of nonsingular3-folds with toroidal structures DY on Y and DX = f−1(DY ) on X such thatsing(f) ⊂ DX . Suppose that u, v, w are (possibly formal) permissible parametersat q ∈ DY . Then u, v are toroidal forms at p ∈ f−1(q) if there exist permissibleparameters x, y, z in OX,p such that one of the following cases hold.

1. q is a 2-point or a 3-point, p is a 1-point and

(4.1) u = xa, v = xb(α+ y)

where 0 6= α ∈ k.2. q is 2-point or a 3-point, p is a 2-point and

(4.2) u = xayb, v = xcyd

with ad− bc 6= 0.3. q is a 2-point or a 3-point, p is a 2-point and

(4.3) u = (xayb)k, v = (xayb)t(α+ z)

where 0 6= α ∈ k, a, b, k, t > 0 and gcd(a, b) = 1.4. q is a 2-point or a 3-point, p is a 3-point and

(4.4) u = xaybzc, v = xdyezf

where

rank(a b cd e f

)= 2.

5. q is a 1-point, p is a 1-point and

(4.5) u = xa, v = y

6. q is a 1-point, p is a 2-point and

(4.6) u = (xayb)k, v = z

with a, b, k > 0 and gcd(a, b) = 1

Regular parameters x, y, z as in Definition 4.1 will be called permissible param-eters for u, v, w at p.

19

20 4. TOROIDAL MORPHISMS AND PREPARED MORPHISMS

Lemma 4.2. Let notation be as in Definition 4.1. Suppose that q ∈ DY ,p ∈ f−1(q) and u, v, w are permissible parameters at q such that u, v are toroidalforms at p. Then there exist permissible parameters x, y, z for u, v, w at p such thatan expression of Definition 4.1 holds for u and v at p, and one of the followingrespective forms for w holds at p.

1. q is a 2-point or a 3-point, p is a 1-point, u, v satisfy (4.1) and

(4.7) w = g(x, y) + xcz

where g is a series.2. q is 2-point or a 3-point, p is a 2-point, u, v satisfy (4.2) and

(4.8) w = g(x, y) + xeyfz

where g is a series.3. q is a 2-point or a 3-point, p is a 2-point, u, v satisfy (4.3) and

(4.9) w = g(xayb, z) + xcyd

where g is a series and rank(u, xcyd) = 2.4. q is a 2-point or a 3-point, p is a 3-point, u, v satisfy (4.4) and

(4.10) w = g(x, y, z) +N

where g is a series in monomials M in x, y, z such that rank(u, v,M) = 2,and N is a monomial in x, y, z such that rank(u, v,N) = 3.

5. q is a 1-point, p is a 1-point, u, v satisfy (4.5) and

(4.11) w = g(x, y) + xcz

where g is a series.6. q is a 1-point, p is a 2-point, u, v satisfy (4.6) and

(4.12) w = g(xayb, z) + xcyd

where g is a series and rank(u, xcyd) = 2.

Proof. Choose permissible parameters x, y, z for u, v, w at p. The lemmafollows from an explicit calculation of the Jacobian determinant

J = Det

∂u∂x

∂u∂y

∂u∂z

∂v∂x

∂v∂y

∂v∂z

∂w∂x

∂w∂y

∂w∂z

,

and a change of variables of x, y, z. Observe that J = 0 is supported on DX , sincethe singular locus of f is contained in DX .

We indicate the proof if (4.4) holds at p. There exists a unit series γ in x, y, zand l,m, n ∈ N such that J = γxlymzn. Write w as a series

w =∑

cijkxiyjzk

with cijk ∈ k. We compute

J =∑

cijkDet

a b cd e fi j k

xa+d+i−1yb+e+j−1zc+f+k−1 = γxlymzn

from which we obtain forms (4.4) and (4.10), after making a change of variables inx, y, z, multiplying x, y, z by appropriate roots of γ.

4. TOROIDAL MORPHISMS AND PREPARED MORPHISMS 21

Definition 4.3. ([KKMS], [AK]) A normal variety X with a SNC divisorDX on X is called toroidal if for every point p ∈ X there exists an affine toricvariety Xσ, a point p′ ∈ Xσ and an isomorphism of k-algebras

OX,p ∼= OXσ,p′

such that the ideal of DX corresponds to the ideal of Xσ −T (where T is the torusin Xσ). Such a pair (Xσ, p

′) is called a local model at p ∈ X. DX is called atoroidal structure on X.

A dominant morphism Φ : X → Y of toroidal varieties with SNC divisors DY

on Y and DX = Φ−1(DY ) on X, is called toroidal at p ∈ X, and we will say thatp is a toroidal point of Φ if with q = Φ(p), there exist local models (Xσ, p

′) at p,(Yτ , q′) at q and a toric morphism Ψ : Xσ → Yτ such that the following diagramcommutes:

OX,p ← OXσ,p′

Φ∗ ↑ Ψ∗ ↑OY ,q ← OYτ ,q′ .

Φ : X → Y is called toroidal (with respect to DY and DX) if Φ is toroidal at allp ∈ X.

The following is the list of toroidal forms for a dominant morphism f : X → Yof nonsingular 3-folds with toroidal structures DY and DX = f−1(DX). Supposethat p ∈ DX , q = f(p) ∈ DY , and f is toroidal at p. Then there exist permissibleparameters u, v, w at q and permissible parameters x, y, z for u, v, w at p such thatone of the following forms hold:

1. p is a 3-point and q is a 3-point,

u = xaybzc

v = xdyezf

w = xgyhzi,

where a, b, d, e, f, g, h, i ∈ N and

Det

a b cd e fg h i

6= 0.


u = xayb

v = xdye

w = xgyh(z + α)

with 0 6= α ∈ k and a, b, d, e, g, h ∈ N satisfy ae− bd 6= 0.3. p is a 1-point and q is a 3-point,

u = xa

v = xd(y + α)w = xg(z + β)

with 0 6= α, 0 6= β ∈ k, a, d, g > 0.



u = xayb

v = xdye

w = z

with ae− bd 6= 05. p is a 1-point and q is a 2-point,

u = xa

v = xd(y + α)w = z

with 0 6= α ∈ k, a, d > 0.6. p is a 1-point and q is a 1-point,

u = xa

v = yw = z

with a > 0.We obtain the above list by observing that the germ of f at p is formally

isomorphic to a localization of a monomal mapping k3 → k3 at a point on thecoordinate axis.

Definition 4.4. Let notation be as in Definition 4.1. u, v, w have a (nontoroidal) monomial form at p ∈ f−1(q) if

1. u, v have a form (4.1) at p, q is a 2-point and w = xc(z + β) for somec ∈ N with c > 0, β ∈ k.

2. u, v have a form (4.2) at p, q is a 2-point and w = xeyf (z + β) for somee, f ∈ N, e+ f > 0, β ∈ k.

3. u, v have a form (4.3) at p, q is a 2-point and w = xcyd with ad− bc 6= 0.4. u, v have a form (4.4) at p, q is a 2-point and w = xgyhzi with

det

a b cd e fg h i

6= 0.

5. u, v have a form (4.5) at p and w = xb(z + β) with b ∈ N, b > 0, β ∈ k.6. u, v have a form (4.6) at p and w = xcyd with ad− bc 6= 0.

A prepared morphism ΦX : X → S from a nonsingular 3-fold X to a nonsin-gular surface S (with respect to toroidal structures DS and DX = Φ−1

X (DS)) isdefined in Definition 6.5 [C3].

Remark 4.5. Suppose that f : X → Y is a dominant proper morphism ofnonsingular 3-folds with toroidal structures determined by SNC divisors DY on Yand DX = f−1(DY ) on X, and DX contains the singular locus of the morphismf . With our assumptions on f , f is generically finite. The nonfinite locus NFfis a closed set of codimension ≥ 2 in Y . Let X be the normalization of Y in thefunction field of X, with induced finite morphism λ : X → Y . The branch locus ofλ (which is a divisor by the purity of the branch locus [N]) is contained in the SNCdivisor DY . Let E be an irreducible component of DY . By Abhyankar’s lemma([Ab4], XIII 5.3 [G]), the irreducible components of λ−1(E) are disjoint. Thus theirreducible components of DX which dominate E are disjoint.


Definition 4.6. A dominant morphism f : X → Y of nonsingular 3-folds withtoroidal structures determined by SNC divisors DY on Y , and DX = f−1(DY ) onX such that the singular locus of f is contained in DX is prepared for DY andDX if:

1. If q ∈ Y is a 3-point, u, v, w are permissible parameters at q and p ∈f−1(q), then u, v and w are each a unit (in OX,p) times a monomial inlocal equations of the toroidal structure DX at p . Furthermore, thereexists a permutation of u, v, w such that u, v are toroidal forms at p.

2. If q ∈ Y is a 2-point, u, v, w are permissible parameters at q and p ∈f−1(q), then either(a) u, v are toroidal forms at p or(b) p is a 1-point and there exist regular parameters x, y, z ∈ OX,p such

that there is an expression

u = xa

v = xc(γ(x, y) + xdz)w = y

where γ is a unit series and x = 0 is a (formal) local equation of DX ,or

(c) p is a 2-point and there exist regular parameters x, y, z in OX,p suchthat there is an expression

u = (xayb)k

v = (xayb)l(γ(xayb, z) + xcyd)w = z

where a, b > 0, gcd(a, b) = 1, ad − bc 6= 0, γ is a unit series andxy = 0 is a (formal) local equation of DX .

3. If q ∈ Y is a 1-point, and p ∈ f−1(q), then there exist permissible param-eters u, v, w at q such that u, v are toroidal forms at p.

Remark 4.7. Suppose that f : X → Y is a dominant morphism of nonsingular3-folds, with toroidal structures determined by SNC divisors DY on Y and DX =f−1(DY ) on X such that the singular locus of f is contained in DX . Suppose thatq ∈ Y is a 2-point, and u, v, w are permissible parameters at q. it follows from acomputation of the Jacobian determinant that the following holds.

1. Suppose that p ∈ f−1(q) is a 1-point, and there exist permissible parame-ters x, y, z at p such that

u = xa, v = xcγ,w = y

where γ ∈ OX,p is a unit series. Then f is prepared at p of type 2 (b),2. Suppose that p ∈ f−1(q) is a 2-point, and there exist permissible parame-

ters x, y, z at p such that

u = xeyf , v = xgyhγ,w = z

where γ ∈ OX,p is a unit series.Then f is either torodial at p (so that it is prepared of type 2 (a) at

p) or f is prepared of type 2 (c) at p.

Lemma 4.8. Suppose that X, Y are projective, f : X → Y is birational, pre-pared and q ∈ Y is a 1-point such that f is not an isomorphism over q. Then the


fundamental locus Ff−1 = NFf of f contains a unique curve C passing through qand C is nonsingular at q.

Furthermore, there exist algebraic permissible parameters u, v, w at q such thata form (4.5) or (4.6) of Definition 4.1 (for this fixed choice of u, v, w) holds at pfor all p ∈ f−1(q). u = w = 0 are local equations of C at q.

Proof. As f is birational and Y is nonsingular, the exceptional locus of f isa divisor on X, which is necessarily supported on DX .

Since for all p ∈ f−1(q), there exist permissible parameters at q such that aform (4.5) or (4.6) of Definition 4.1 holds at p, we have that dim f−1(q) = 1. Thusif E′ is an exceptional component of f such that q ∈ f(E′), then f(E′) is a curve.

Let D be the component of DY containing q. Let F be the strict transform ofD on X and let p ∈ f−1(q)∩F . Then p must be a 2-point, and since f is birational,there exist permissible parameters u, v, w at q such that there is an expression inOX,p

u = xayv = zw = φ(xay, z) + xdye

where y = 0 is a (formal) local equation of F , and x = 0 is a (formal) localequation of the other component E of DX containing p. Computing the Jacobiandeterminant of f at p, we see that e = 0. We consider the morphism f∗ : OD,q →OF,p. f∗ : OD,q → OF,p is the k-algebra homomorphism f∗ : k[[v, w]] → k[[x, z]]given by v = z, w = φ(0, z) + xd. Thus OF,p is finite over OD,q. It follows thatf∗ : OD,q → OF,p is quasi-finite, and thus OF,p ∼= OD,q by Zariski’s Main Theorem.In particular, p = f−1(q) ∩ F . Let C = f(E). C is an algebraic curve through qwhich has formal local equations u = w − φ(u, v) = 0 at q. Thus C is nonsingularat q.

Suppose that C ′ ⊂ F is a curve containing q such that q ∈ C ′ and C ′ 6= C. LetC ′′ be the unique curve on D containing p which dominates C ′. A general point ofC ′′ is then not on an exceptional component of f . Thus f is an isomorphism abovea general point of C ′ be Zariski’s main theorem.

If E′ is a component of the exceptional locus of f such that q ∈ f(E′), we musthave that f(E′) = C. Now let u, v, w be permissible parameters at q such thatu = w = 0 are local equations of C at q. We see that u, v, w must have a form (4.5)or (4.6) of Definition 4.1 for all p ∈ f−1(q), since w is divisible by a local equationof a component of DX .

Lemma 4.9. Suppose that X,Y are projective, f : X → Y is a prepared mor-phism of 3-folds, and q ∈ DY is a 1-point. Then there exist algebraic permissibleparameters u, v, w at q such that a form (4.5) or (4.6) of Definition 4.1 (for thisfixed choice of u, v, w) holds at p for all p ∈ f−1(q).

Proof. Let q ∈ Y be a 1-point, u, v, w be permissible parameters at q, X bethe normalization of Y in k(X). We have a natural commutative diagram

Xf→ Y

g ↑ λX

.


Let λ−1(q) = q1, . . . , qn. Abhyankar’s Lemma implies thatX is nonsingular abovea neighborhood of q, and there exist (formal) permissible parameters u1, vi, wi atqi and ri ∈ N such that

u = urii , v = vi, w = wi.

Lemma 4.8 (applied to g : X → X) implies ui, αvi + βwi are toroidal forms at allpoints of g−1(qi) for a dense open set of (α, β) ∈ k2. The condition is that ui =αvi + βwi = 0 intersects the (nonsingular at qi) nonfinite locus of g transversallyat qi. Thus there exist α, β ∈ k such that u, αv + βw are toroidal forms at allp ∈ f−1(q).

Lemma 4.10. Suppose that f : X → Y is a prepared morphism of 3-folds andE is a component of DX such that f(E) = C is a curve which is not a 2-curve.Suppose that q ∈ C is a 1-point. Then all formal branches of C are smooth at q,and there exist permissible parameters u, v, w at q such that u = w = 0 are (formal)local equations of C at q.

Proof. Suppose that p ∈ f−1(q) ∩ E. There exist permissible parametersu, v, w at q such that one of the following forms hold.

p is a 1-point:

(4.13) u = xa, v = y, w = g(x, y) + xnz

where x = 0 is a local equation of E, orp is a 2-point:

(4.14) u = (xayb)i, v = z, w = g(xayb, z) + xcyd

where a, b > 0, ad− bc 6= 0 and x = 0 is a local equation of E.In case (4.13), we have an expression w = φ(y)+xΩ in OX,p (n > 0 since f(E)

is a curve). u = w − φ(y) = 0 is a local equation of a branch of C.In case (4.14), one of the following must hold in OX,p:

(4.15) w = φ(z) + xyΩ,

(4.16) w = φ(z) + xyΩ + xc

where c > 0, or

(4.17) w = φ(z) + xyΩ + yd

where d > 0.In cases (4.15) or (4.16), we have

0 = u = w − φ(z)

is a formal local equation of a branch of C at q.Suppose that case (4.17) holds. f(E) = C a curve implies that there exists an

irreducible series Ψ(x, y) in a power series ring k[[x, y]] such that x divides Ψ(v, w)in OX,p.

Ψ(v, w) = Ψ(z, φ(z) + xyΩ + yd)implies Ψ(0, yd) = 0 (set x = z = 0), which implies that x | Ψ(x, y). SinceΨ(x, y) is irreducible, we have that Ψ = xλ where λ is a unit series. This is acontradiction.

CHAPTER 5

Toroidal ideals

Lemma 5.1. Suppose that X is a nonsingular 3-fold with SNC divisor DX ,defining a toroidal structure on X. Suppose that I is an ideal sheaf on X whichis locally generated by monomials in local equations of components of DX . Thenthere exists a sequence of blow ups of 2-curves Φ1 : X1 → X such that IOX1 isan invertible ideal sheaf. If I is locally generated by two equations, then Φ1 is anisomorphism away from the support of I.

This lemma is an extension of Lemma 18.18 [C3], and is generalized to alldimensions in [Go].

Proof. X has a cover by affine open sets U1, . . . , Un such that there existgi,1, . . . , gi,l ∈ Γ(Ui,OX) such that gi,j = 0 are local equations in Ui of irreduciblecomponents of DX , and there exist fi,1, . . . , fi,m(i) ∈ Γ(Ui,OX) such that the fi,jare monomials in the gi,k and Γ(Ui, I) = (fi,1, . . . , fi,m(i)).

Let Dij be an effective divisor supported on the components of DX such thatthere is equality of divisors Dij ∩ Ui = (fi,j) ∩ Ui. Let Ii ⊂ OX be the ideal sheafwhich is locally generated by local equations of Di,1, . . . , Di,m(i). By construction,Ii | Ui = I | Ui for all i.

We will show that for an ideal sheaf of the form I1, there exists a sequenceof blow ups of 2-curves, π : X1 → X such that I1OX1 is invertible. Since IiOX1

are locally generated by local equations of π∗(Di,1), . . . , π∗(Di,m(i)), there existsπ2 : X2 → X which is a sequence of blow ups of 2-curves such that IiOX2 isinvertible for all i. Since IiOX2 | π−1

2 (Ui) = IOX2 | π−12 (Ui) for all i, IOX2 is

invertible.We may now suppose that there exists n > 0 and effective divisors D1, . . . , Dn

on X whose supports are unions of components of DX , such that I is locallygenerated by local equations of D1, . . . , Dn.

First suppose that n = 2. Suppose that p ∈ X is a general point of a 2-curveC. Let x = 0, y = 0 be local equations of the components of DX containing p.x = y = 0 are local equations of C at p. Then there exist a, b, c, d ∈ N such thatD1 is defined near p by the divisor of xayb, and D2 is defined near p by the divisorof xcyd. Define

ω(C) =

max(|a− c|, |b− d|), (|b− d|, |a− c|) if a− c, b− d are nonzeroand have opposite signs,

−∞ otherwise

Here the maximum is computed in the lexicographic order. We see that the stalkIp is invertible if and only if ω(C) = −∞.

Further, if ω(C) = −∞ for all 2-curves C of X, then I is invertible, as followssince the divisors D1 and D2 are given locally at a 3-point p by the divisors of

27

28 5. TOROIDAL IDEALS

monomials xaybzc and xdyezf where xyz = 0 is a local equation of DX at p. a− dand b − e have the same signs, a − d, c − f have the same signs, and b − e, c − fhave the same signs, so xaybzc|xdyezf or xdyezf | xaybzc.

Now defineω(X) = maxω(C) | C is a 2-curve of X.

We have seen that I is invertible if and only if ω(X) = −∞. Suppose that ω(X) 6=−∞ and C is a 2-curve of X such that ω(C) = ω(X). Let π : X1 → X be the blowup of C. Let DX1 = π−1(DX) = π∗(DX)red, D′

1 = π∗(D1), D′2 = π∗(D2).

We can define the function ω for 2-curves on X1, relative to D′1 and D′

2, anddefine ω(X1).

By a local calculation (as shown in the proof of Lemma 18.18 [C3]) we see thatω(C1) < ω(X) if C1 is a 2-curve which is contained in the exceptional divisor of π.

Suppose that C1, . . . , Cr are the 2-curves C on X such that ω(C) = ω(X). Weobtain a reduction ω(X1) < ω(X) after blowing up (the strict transforms of) theser curves. By induction on ω(X), we must obtain that IOX2 is invertible after anappropriate sequence of blow ups of 2-curves X2 → X.

Now suppose that I is locally generated by local equations of D1, . . . , Dn (withn > 2). Let I1 ⊂ I be the ideal sheaf which is locally generated by local equationsof D1 and D2. We have seen that there exists a sequence of blow ups of 2-curvesπ1 : X1 → X such that I1OX1 is invertible. Thus there exists a divisor D onX1 whose support is a union of components of DX1 such that I1OX1 is locallygenerated by a local equation of D.

Let Di = π∗1(Di) for 3 ≤ i ≤ n. Then IOX1 is locally generated by localequations of the n−1 divisors D,D3, . . . , Dn. By induction, there exists a sequenceof blow ups of 2-curves X2 → X such that IOX2 is invertible.

When I is locally generated by two equations, we restrict to blowing up 2-curves C for which a general point of C is in Ui to obtain the conclusion that Φ1 isan isomorphism away from the support of OX/I.

Lemma 5.2. Suppose that f : X → Y is a prepared morphism of nonsingular3-folds, and C is a 2-curve in Y . Then there exists a sequence of blowups of 2-curves Φ : X1 → X such that ICOX1 is invertible and Φ is an isomorphism overf−1(Y − C).

Proof. The lemma is a consequence of Lemma 5.1.

Lemma 5.3. Suppose that X is a nonsingular 3-fold with SNC divisor DX ,defining a toroidal structure on X. Suppose that I is an ideal sheaf on X whichis locally generated by monomials in local equations of components of DX . Thenthere exists a sequence of blow ups of 2-curves and 3-points Φ1 : X1 → X such thatIOX1 is an invertible ideal sheaf and Φ1 is an isomorphism away from the supportof OX/I.

The proof of this lemma follows from the proof of principalization of ideals, asin [BEV] or [BrM] (cf. the proof of Theorem 6.3 [C6]) in the case when the idealto be principalized is locally generated by monomials in the toroidal structure.

CHAPTER 6

Toroidalization of morphisms from 3-folds tosurfaces

The following theorem is Theorem 19.11 [C3].

Theorem 6.1. Suppose that Φ : X → S is a dominant morphism from anonsingular 3-fold X to a nonsingular surface S and DS is a SNC divisor on Ssuch that DX = Φ−1(DS) is a SNC divisor which contains the singular locus of Φ.

1. There exists a sequence of blow ups of possible centers α1 : X1 → X suchthat(a) The fundamental locus of α−1

1 is contained in the union of irreduciblecomponents E of DX such that E contains a point p such that Φ isnot prepared at p (Definition 6.5 [C3]).

(b) Φ1 = Φ α1 : X1 → S is prepared (Definition 6.5 [C3])2. Further, there exist sequences of blow ups of possible centers, α2 : X2 →X1 and β : S1 → S such that:(a) There is a commutative diagram

X2Φ2→ S1

α2 ↓ ↓ βX1

Φ1→ S

such that Φ2 is toroidal,(b) β is an isomorphism away from β−1(Φ1(Z)) where Z is the locus

where Φ1 is not toroidal.(c) α2 is an isomorphism away from α−1

2 (Φ−11 (Φ1(Z)))

Proof. The proof is immediate from the proof of Theorem 19.11 [C3], whichwe outline below.

We first prove 1 of the theorem. By Lemma 6.2 [C3], there exists a commutativediagram

X0

α0 ↓ Φ0

XΦ→ S

such that Φ0 : X0 → S is a weakly prepared morphism (Definition 6.1 [C3]), and thefundamental locus of α−1

0 is contained in the locus where Φ is not weakly prepared,(which is contained in the locus where Φ is not prepared). It is not necessary toblow up points on S since DS , DX are SNC divisors. Let DX0 = α−1

0 (DX).

29

30 6. TOROIDALIZATION OF MORPHISMS FROM 3-FOLDS TO SURFACES

By Theorem 17.2 [C3] there exists a commutative diagram

X1

α ↓ Φ1

X0Φ0→ S

such that Φ1 is prepared. The algorithm consists of a sequence

X1 = Ynαn→ Yn−1 → · · · → Y1

α1→ Y0 = X0

of blow ups of points and nonsingular curves which are possible centers. They makeSNCs with the preimage DYi of DX0 and are contained in a component E of DYi

such that E contains a point which is not prepared for

Φi = Φ0 α1 · · · αi : Yi → S.

If p ∈ Yi is prepared for Φi then all points of α−1i+1(p) are prepared for Φi+1. Thus

conditions (a) and (b) of 1 hold.We now verify 2 of the theorem. We first observe that Φ1 (from the conclusions

of part 1 of this theorem) is strongly prepared (Definition 18.1 [C3]). We nowexamine the monomialization algorithm of Chapter 18 [C3] and the toroidalizationalgorithm of Chapter 19 [C3], applied to Φ1 : X1 → S.

The monomialization algorithm of Theorem 18.19 and Theorem 18.21 [C3]consists in constructing a commutative diagram

(6.1)X2

π2→ X1

Φ2 ↓ ↓ Φ1

S1Ψ1→ S

such that Φ2 is monomial (all points of X2 are good for Φ2 as defined in Definition18.5 [C3]).

(6.1) has a factorization

(6.2)X2 = Zl

εl→ · · · → Z1ε1→ Z0 = X1

Φ2 ↓ Ωl ↓ Ω1 ↓ Ω0 ↓ Φ1 ↓S1 = Tl

δl→ · · · → T1δ1→ T0 = S

where each Ωi is strongly prepared, each δi+1 is the blow up of a point qi such thatΩ−1i (qi) contains a point pi at which Ωi is not monomial and εi+1 is a sequence of

blow ups of curves which are exceptional to such qi. This step is accomplished byperforming the algorithms of Lemmas 18.16, 18.17 and 18.18 of [C3].

Each map Zi+1 → Zi has a factorization

(6.3) Zi+1 = Zmλm→ Zm−1 → · · ·

λ1→ Z0 = Zi

where each λj+1 is a blow up of a 2-curve or of a curve Cj which contains a 1-point,makes SNCs with the preimage DZj

of DX on Zj , and is contained in a componentof DZj

. Cj is contained in the locus where mqiOZjis not invertible. To construct

(6.3) we successively apply Lemmas 18.16, 18.17, 18.18 of [C3].The algorithms of Lemma 18.16 and Lemmas 18.18 [C3] consist of blow ups of

2-curves.The algorithm of Lemma 18.17 [C3] consists of a sequence of blow ups of curves

λj+1 : Zj+1 → Zj of Cj ⊂ DZjwhich are not 2-curves, and are contained in the

locus where mqiOZjis not invertible.

6. TOROIDALIZATION OF MORPHISMS FROM 3-FOLDS TO SURFACES 31

Theorem 19.9 [C3] and Theorem 19.10 [C3] imply there exists a commutativediagram

(6.4)X3

π3→ X2

Φ3 ↓ ↓ Φ2

S2ψ2→ S1

such that 2 of the conclusions of the theorem holds.

CHAPTER 7

Preparation above 2 and 3-points

Lemma 7.1. Suppose that f : X → Y is a dominant morphism of nonsingu-lar projective 3-folds with toroidal structures determined by SNC divisors DY andDX = f−1(DY ) such that DX contains the singular locus of f . Then there exist acommutative diagram

X1f1→ Y1

Φ ↓ ↓ Ψ

Xf→ Y

such that Φ and Ψ are products of blow ups of 2-curves and f1 is toroidal above all3-points of Y1.

Proof. Suppose that ν is a 0-dimensional valuation of k(X). We will say thatν is resolved for f if the center of ν on Y is not a 3-point or if the center of ν on Yis a 3-point, and f is toroidal at the center of ν on X.

Being resolved is an open condition on the Zariski-Riemann manifold of X, andif ν is resolved for f and

X1f1→ Y1

Φ1 ↓ ↓ Ψ1

Xf→ Y

is a commutative diagram such that Φ1 and Ψ1 are products of blow ups of 2-curves,then ν is resolved for f1.

Suppose that ν is a 0-dimensional valuation of k(X) such that the center q ofν on Y is a 3-point. Let p be the center of ν on X. Let u, v, w be permissibleparameters at q.

Case 1. Suppose that ν(u), ν(v), ν(w) are rationally independent. Since uvw = 0is a local equation of DX at p, there exist regular parameters x, y, z in OX,p suchthat xyz = 0 contains the germ of DX at p, and we have an expression

u = xaybzcγ1

v = xdyezfγ2

w = xgyhziγ3

where the γi are units in OX,p. Since ν(u), ν(v), ν(w) are rationally independent,ν(x), ν(y), ν(z) are also rationally independent and

Det

a b cd e fg h i

6= 0

which implies that p is a 3-point and f is toroidal at p. Thus ν is resolved for f .

33

34 7. PREPARATION ABOVE 2 AND 3-POINTS

Case 2. Suppose that ν(u), ν(v) are rationally dependent. After possibly inter-changing u, v, w we reduce to this case. Let C be the 2-curve of Y with local equa-tion u = v = 0 at q. There exists a sequence of blow ups of 2-curves Ψν : Yν → Ywhich are sections over C such that the center of ν on Yν is not a 3-point.

Yν is the blow up of a toroidal ideal sheaf Jν of OY . Since f−1(DY ) = DX ,JνOX is also a toroidal ideal sheaf. By Lemma 5.1, there exists a sequence ofblow ups of 2-curves Φν : Xν → X such that there is a commutative diagram ofmorphisms

Xνfν→ Yν

Φν ↓ ↓ Ψν

Xf→ Y.

Thus ν is resolved for fν .It follows from compactness of the Zariski Riemann manifold of X [Z], that

there exists a positive integer n and commutative diagrams

Xifi→ Yi

Φi ↓ ↓ Ψi

Xf→ Y

for 1 ≤ i ≤ n such that Φi and Ψi are products of blow ups of 2-curves, and every0-dimensional valuation ν of k(X) is resolved for some fi.

Yi is the blow up of a toroidal ideal sheaf Ji of OY and Xi is the blow up of atoroidal ideal sheaf Ii of OX . Thus there exists a sequence of blow ups of 2-curvesY ∗ → Y such that

∏i JiOY ∗ is invertible, by Lemma 5.1. Y ∗ is thus the blow

up of a toroidal ideal sheaf J ⊂ OY , so that JOX is also a toroidal ideal sheaf.By Lemma 5.1, there exists a sequence of blow ups of 2-curves X∗ → X such thatJ∏IiOX∗ is invertible. Thus for 1 ≤ i ≤ n there exist commutative diagrams of

morphisms

X∗ f∗→ Y ∗

Φ∗i ↓ ↓ Ψ∗

i

Xifi→ Yi

↓ ↓X → Y.

Suppose that ν is a 0-dimensional valuation of k(X). If the center of ν on Y ∗

is a 3-point, then the center of ν on Yi is a 3-point for all i, since Ψ∗i is toroidal.

There exists an i such that ν is resolved for fi. Thus fi is toroidal at the center ofν on Xi. Since Φ∗

i and Ψ∗i are toroidal, f∗ is toroidal at the center of ν. Thus ν

is resolved for f∗. Since all 0-dimensional valuations of k(X) are resolved for f∗,it follows that f∗ is toroidal above all 3-points of Y ∗, and we have achieved theconclusions of the lemma.

Lemma 7.2. Suppose that f : X → Y is a dominant morphism of nonsingularprojective 3-folds, with toroidal structures determined by SNC divisors DY andDX = f−1(DY ) such that DX contains the singular locus of f . Further suppose

7. PREPARATION ABOVE 2 AND 3-POINTS 35

that f is toroidal above all 3-points of Y . Then there exists a commutative diagram

X1f1→ Y1

Φ ↓ ↓ Ψ

Xf→ Y

such that Ψ and Ψ are products of blow ups of 2-curves, f1 is toroidal above all3-points of Y1, and f1 is prepared (and satisfies 2 (a) of Definition 4.6) above all2-points of Y1.

Proof. Suppose that ν is a 0-dimensional valuation of k(X). We will say thatν is resolved for f if the center of ν on Y is a 1-point or if the center of ν on Y is a2-point and f is prepared at the center of ν on X (and satisfies 2 (a) of Definition4.6), or if the center of ν on Y is a 3-point, and f is toroidal at the center of ν onX.

Being resolved is an open condition on the Zariski-Riemann manifold of X.Suppose that

X1f1→ Y1

Φ ↓ ↓ Ψ

Xf→ Y

is a commutative diagram of morphisms such that Φ and Ψ are products of blowups of 2-curves. If ν is a 0-dimensional valuation of k(X) such that ν is resolvedfor f , then ν is resolved for f1.

Suppose that q ∈ Y is a 2-point, and ν is a 0-dimensional valuation of k(X)such that q is the center of ν on Y . Let p be the center of ν on X. Let u, v, w bepermissible parameters at q, so that u = v = 0 are local equations of the 2-curve Cthrough q.

Case 1. Suppose that ν(u), ν(v) are rationally independent. Since uv = 0 is alocal equation of DX at p, there exist regular parameters x, y, z in OX,p such thatxyz = 0 contains the germ of DX in OX,p, and we have an expression

u = xaybzcγ1

v = xdyezfγ2

where the γi are units in OX,p. Since ν(u), ν(v) are rationally independent,

rank(a b cd e f

)= 2

which implies that p is a 2 or 3-point and f is prepared at p (and satisfies 2 (a) ofDefinition 4.6). Thus ν is resolved for f .

Case 2. Suppose that ν(u), ν(v) are rationally dependent. There exists a sequenceof blow ups of 2-curves which are sections over C, Ψν : Yν → Y such that the centerof ν on Yν is a 1-point.

Yν is the blow up of a toroidal ideal sheaf Jν of OY . Since f−1(DY ) = DX , Jνis also a toroidal ideal sheaf. By Lemma 5.1, there exists a sequence of blow ups of


2-curves Φν : Xν → X such that there is a commutative diagram of morphisms

Xνfν→ Yν

Φν ↓ ↓ Ψν

Xf→ Y.

Thus ν is resolved for fν .It follows from compactness of the Zariski Riemann manifold of X [Z] that

there exists a positive integer n and commutative diagrams

Xifi→ Yi

Φi ↓ ↓ Ψi

Xf→ Y

for 1 ≤ i ≤ n such that Φi and Ψi are products of blow ups of 2-curves, and everyvaluation ν of k(X) is resolved for some fi.

Yi is the blow up of a toroidal ideal sheaf Ji of OY and Xi is the blow up of atoroidal ideal sheaf Ii of OX . Thus there exists a sequence of blow ups of 2-curvesY ∗ → Y such that

∏i JiOY ∗ is invertible, by Lemma 5.1. Y ∗ is thus the blow up

of a toroidal ideal sheaf J ⊂ OY . Thus JOX is also a toroidal ideal sheaf. ByLemma 5.1, there exists a sequence of blow ups of 2-curves X∗ → X such thatJ∏IiOX∗ is invertible. Thus for 1 ≤ i ≤ n, there exist commutative diagrams of

morphisms

X∗ f∗→ Y ∗

Φ∗i ↓ ↓ Ψ∗

i

Xifi→ Yi

↓ ↓X → Y.

Since Φ∗i and Ψ∗

i are the blow ups of toroidal ideal sheaves, they are toroidal mor-phisms.

Suppose that ν is a 0-dimensional valuation of k(X). If the center of ν on Y isa 3-point then f∗ is resolved at the center of ν on X∗. In particular, since Ψi Ψ∗

i istoroidal, if the center of ν on Y ∗ is a 3-point, then the center of ν on Y is a 3-pointand ν is resolved for f∗. Suppose that the center of ν on Y ∗ is 2-point, and thecenter of ν on Y is not a 3-point. Then the center of ν on Yi is a 2-point for all i.There exists an i such that ν is resolved for fi. Thus fi is prepared (and satisfies2 (a) of Definition 4.6) at the center of ν on Xi. Since Φ∗

i and Ψ∗i are toroidal,

f∗ is prepared (and satisfies 2 (a) of Definition 4.6) at the center of ν. Thus ν isresolved for f∗. Since all 0-dimensional valuations of k(X) are resolved for f∗, itfollows that f∗ is toroidal above all 3-points of Y ∗, and prepared above all 2-pointof Y ∗, and we have achieved the conclusions of the lemma.

Lemma 7.3. Suppose that f : X → Y satisfies the conclusions of Lemma 7.2.Suppose that H is a general hyperplane section of Y . Then f is prepared above allpoints of H.

Proof. Bertini’s theorem implies that H is nonsingular and makes SNCs withDY . Further, H ′ = f−1(H) is nonsingular and makes SNCs with DX . Thus Hcontains no 3-points of Y and H ′ contains no 3-points of X.


Suppose that q ∈ H ∩DY is a 1-point, and p ∈ f−1(q). Let u, v, w be regularparameters in OY,q such that u = 0 is a local equation of DY at q, and w = 0is a local equation of H. Then we have regular paramaters x, y, z in OX,p suchthat either p is a 1 point with x = 0 a local equation of DX or p is a 2-point withxy = 0 a local equation of DX at p, and we have an expression u = xaγ,w = z oru = xaybγ,w = z where γ is a unit in OX,p. Thus f is prepared at p.

Corollary 7.4. Suppose that f : X → Y satisfies the conclusions of Lemma7.2. Then there exists a finite set of 1-points Ω ⊂ Y such that f is prepared aboveY − Ω.

Proof. The locus of points in X where f is prepared is an open set. Since fis proper, the image Ω of the closed set of points where f is not prepared is closedin Y . Since a general hyperplane section of Y is disjoint from Ω by Lemma 7.3, Ωmust be a finite set of points.

Lemma 7.5. Suppose that f : X → Y is a proper dominant morphism of non-singular 3-folds and π : Y → S is a smooth dominant morphism onto a nonsingularsurface S. Let g = π f . Suppose that C is a nonsingular curve of S, D = π−1(C)and D′ = f−1(D). Suppose that D′ is a SNC divisor on X which contains thesingular locus of g and the singular locus of f . Suppose that q ∈ C is a point,and that g is toroidal and prepared (with respect to C and D′) away from pointsabove finitely many points Ω = q1, . . . , qm ⊂ γ = π−1(q). Further suppose thatf is finite above a general point of γ. Then there exists a commutative diagram ofmorphisms

X1Φ1→ X

g1 ↓ gS

such that Φ1 is a product of possible blow ups for the preimage of D′ supportedabove Ω and g1 is prepared (with respect to C and Φ−1

1 (D′)) in a neighborhoodof all components F of Φ−1

1 (D′) which dominate D and in a neighborhood of allcomponents F of Φ−1

1 (D′) which dominate a curve of Y .

Proof. Let u,w be regular parameters in OS,q such that u = 0 is a localequation of C at q. Let C ′ be the curve on S with local equation w = 0 at q. LetA = π−1(C ′).

Since it suffices to prove the lemma above a neighborhood of q in S, we mayassume that E = C+C ′ is a SNC divisor on S whose only singular point is q. Sinceg is toroidal away from points above Ω, we have that g−1(E) defines a SNC divisoron X away from points above Ω. There exists a morphism Φ1 : X1 → X whichis a sequence of possible blow ups for the preimage of D′ supported above Ω suchthat with g1 = g Φ1 : X1 → S, g−1

1 (E) is a SNC divisor, and (f Φ1)−1(qi) aredivisors for all qi ∈ Ω. We may further assume that the union A of codimension 1subvarieties of X1 which dominate A are disjoint, since the fact that g is toroidalaway from q implies they are disjoint away from the preimage of Ω.

Let D be the union of codimension 1 subvarieties of X1 which dominate D, sothat D is a disjoint union of irreducible components of D′′ = g−1

1 (C) (by Remark4.5).


Suppose that p ∈ D and f Φ1(p) = qi ∈ Ω. Then p must be a 2-point or a3-point. We have regular parameters x, y, z in OX1,p such that one of the followingcases hold:

1. p is a 2-point andu = xayb, w = yc

where x = 0 is a local equation of D, u = 0 is a local equation of D′′ anda, b > 0.

2. p is a 2-point,u = xayb, w = ycz

where x = 0 is a local equation of D, u = 0 is a local equation of D′′,a, b, c > 0 and z = 0 is a local equation of A.

3. p is a 3-point and

u = xaybzc, w = ydze

where x = 0 is a local equation of D, u = 0 is a local equation of D′′ anda, b, c > 0.

Thus g1 is prepared in a neighborhood of D.Now suppose that F is a component of D′′ which dominates a curve of Y and

p ∈ F is such that f Φ1(p) = qi ∈ Ω. Then p must be a 2-point or a 3-point.By our assumption that f is finite above a general point of γ, F dominates thecurve C of S. Thus we have regular parameters x, y, z in OX1,p such that one ofthe following cases hold:

1. p is a 2-point andu = xayb, w = yc

where x = 0 is a local equation of F , u = 0 is a local equation of D′′ anda, b > 0.

2. p is a 2-point,u = xayb, w = ycz

where x = 0 is a local equation of F , u = 0 is a local equation of D′′,a, b, c > 0 and z = 0 is a local equation of A.

3. p is a 3-point and

u = xaybzc, w = ydze

where x = 0 is a local equation of F , u = 0 is a local equation of D′′ anda, b, c > 0.

Thus g1 is prepared in a neighborhood of F .

Lemma 7.6. Suppose that f : X → Y is a dominant morphism of nonsingular3-folds with toroidal structures determined by SNC divisors DY and DX = f−1(DY )such that DX contains the singular locus of f . Further suppose that f : X → Y istoroidal and q ∈ Y is a 2-point. Let Ψ : Y1 → Y be the blow up of q. Then thereexists a commutative diagram of morphisms

X1f1→ Y1

Φ ↓ ↓ Ψ

Xf→ Y

such that Φ is a sequence of possible blow ups for the preimage of DX supportedabove q and f1 is toroidal with respect to DY1 = Ψ−1(DY ) and DX1 = Φ−1(DX).


Proof. There exist permissible parameters u, v, w at q such that if p ∈ f−1(q)then there exist permissible parameters x, y, z for u, v, w such that if p is a 1-point,then we have a form

(7.1) u = xa, v = xb(α+ y), w = z

with 0 6= α ∈ k, and if p is a 2-point,

(7.2) u = xayb, v = xcyd, w = z,

with ad− bc 6= 0. We first show that there exists a sequence of possible blow ups

(7.3) XmΦm→ Xm−1 → · · · → X1

Φ1→ X

obtained by blow ups of possible centers supported above q such that the rationalmap Xm → Y1 is toroidal wherever it is defined, and if IqOXm,p is not invertible,then there exist regular parameters x, y, z in OXm,p such that one of the followingforms hold:

p is a 1-point

(7.4) u = xa, v = xb(α+ y), w = xcz

with 0 6= α ∈ k, and c = 0 or 1, or p is a 2-point

(7.5) u = xayb, v = xcyd, w = xz

with a, c ≥ 1, or p is a 2-point

(7.6) u = xayb, v = xcyd, w = xyz

with a, c ≥ 1 and b, d ≥ 1.After possibly interchanging u and v, the points p ∈ f−1(p) such that u, v, w

do not have a form (7.4), (7.5) or (7.6) at p are 2-points of one of the followingforms:

(7.7) u = xa, v = yb, w = z,

in which case V (x, y, z) is the locus in spec(OX,p) where IqOX,p is not invertible,

(7.8) u = xa, v = xbyc, w = z

with b, c > 0, in which case V (x, z) is the locus in spec(OX,p) where IqOX,p is notinvertible,

(7.9) u = xayb, v = xcyd, w = z

with a, b, c, d > 0 in which case V (x, z) ∪ V (y, z) is the locus in spec(OX,p) whereIqOX,p is not invertible,

Let Z be the closed locus of points r in X such that IqOX,r is not invertible.The isolated points p in Z have a form (7.7). If p is a non isolated point in Z whichis a 2-point, then p has a form (7.8) or (7.9).

Suppose that E is a curve in Z such that E contains a 2-point p satisfying (7.8)or (7.9). Then a generic point of E satisfies (7.1) and all 2-points of E must havea form (7.8) or (7.9).

Let Φ1 : X1 → X be the blow up of the finitely many points p ∈ X of theform (7.7). Suppose that p ∈ X is such a point, and p1 ∈ Φ−1

1 (p). After possiblyinterchanging u and v, we may assume that a ≤ b in (7.7). There are regularparameters x1, y1, z1 in OX1,p1 of one of the following forms:

(7.10) x = x1, y = x1(y1 + α), z = x1(z1 + β)


with α, β ∈ k,

(7.11) x = x1y1, y = y1, z = y1(z1 + α)

with α ∈ k or

(7.12) x = x1z1, y = y1z1, z = z1.

Suppose that (7.10) holds. Then u, v, w have a form

u = xa1 , v = xb1(y1 + α)b, w = x1(z1 + β)

at p1. If a = 1, then f Φ1 factors through Y1 at p1 and we have one of the followingtoroidal forms:

1-point maps to 2-point:

u1 = u = x1, v1 =v

u= xb−1

1 (y1 + α)b, w1 =w

u− β = z1

if b > a = 1 and α 6= 0,1-point maps to 1-point:

u1 = u = x1, v1 =v

u− α = y1, w1 =

w

u− β = z1

if b = a = 1, α 6= 0,2-point maps to 2-point:

u1 = u = x1, v1 =v

u= xb−1

1 yb1, w1 =w

u− β = z1

if a = 1 and α = 0.Suppose that (7.10) holds and a > 1. If β 6= 0, we have that Φ1 f factors

through Y1 at p1 and we have a toroidal form, obtained from a change of variablein

u1 =u

w= xa−1

1 (z1 +β)−1, v1 =v

w= xb−1

1 (z1 +β)−1(y1 +α)b, w1 = w = x1(z1 +β)

where p1 is 1-point mapping to a 3-point if α 6= 0 and p1 is 2-point mapping to a3-point if α = 0.

If β = 0 (and a > 1) then we have

u = xa1 , v = xb1(y1 + α), w = x1z1

of the form (7.4) if α 6= 0 and of the form (7.5) if α = 0.Suppose that (7.11) holds. Then at p1, u, v, w have a form:

u = xa1ya1 , v = yb1, w = y1(z1 + α).

Assume b = 1 (which implies a = 1). then f Φ1 factors through Y1 at p1, andthere is a toroidal form:

u1 =u

v= x1, v1 = v = y1, w1 =

w

v− α = z1

where p1 is 2-point mapping to a 2-point.Assume that b > 1 and α 6= 0. Then f Φ1 factors through Y1 at p1, and there

is a toroidal form, obtained from a change of variable in

u1 =u

w= xa1y

a−11 (z1 + α)−1, v1 =

v

w= yb−1

1 (z1 + α)−1, w1 = w = y1(z1 + α)

where p1 is a 2-point mapping to a 3-point.If b > 1 and α = 0, then we have a form:

u = xa1ya1 , v = yb1, w = y1z1


of the form (7.5).Suppose that (7.12) holds. Then p1 is a 3-point and u, v, w have a form

u = xa1za1 , v = yb1z

b1, w = z1.

Thus Φ1 f factors through Y1 at p1 by

u1 =u

w= xa1z

a−11 , v1 =

v

w= yb1z

b−11 , w1 = w = z1,

where p1 is a 3-point mapping to a 3-point.We have thus completed the analysis of Φ1.We now construct (7.3) by induction. Each Xi will be such that the rational

map Xi → Y1 is toroidal wherever it is defined, and if p ∈ Xi is a 2-point such thatIqOX1,p is not invertible, then there exist regular parameters x, y, z at p such thatu, v, w have one of the forms (7.4), (7.5), (7.6), (7.8) or (7.9) at p. If a form (7.8)or (7.9) holds at p, then Φ1 · · · Φi is an isomorphism near p.

Each Φi+1 : Xi+1 → Xi for i ≥ 1 will be the blow up of a curve Ei which is apossible center and is the strict transform of a component of Z ⊂ X.

Suppose that we have constructed (7.3) out to Xi, and p ∈ Xi is a 2-point suchthat IqOXi,p is not invertible, and u, v, w do not have a form (7.4), (7.5) or (7.6)at p. Then u, v, w have a form (7.8) or (7.9) at p. Let E = Ei be a curve in thelocus where IqOXi

is not invertible which contains p. Let F be the component ofDXi containing Ei We necessarily have ordFw = 0 and ordFu > 0, ordF v > 0.Further, Φ1 · · · Φi is an isomorphism near p. Thus E is the strict transform ofa component of Z.

Suppose that p′ ∈ Ei is another 2-point. Then at p′, since ordFw = 0, u, v, wmust have a form (7.8), (7.9) or (7.5), where in this last case, y = z = 0 is a localequation of E at p′ and b, d ≥ 1 (since ordFw = 0, ordFu > 0 and ordF v > 0). Ifp′ ∈ Ei is a 1-point, then u, v, w have a form (7.1) at p′, since ordFw = 0.

Let Φi+1 : Xi+1 → Xi be the blow up of E and Φi+1 = Φ1 · · · Φi+1.Suppose that p ∈ E is a 1-point and p1 ∈ Φ−1

i+1(p). We have that there is aform (7.4) at p with c = 0. Then f Φi+1 is toroidal whenever it is defined, andpoints above p where f Φi+1 does not factor through Yi have a form (7.4) (withc = 1). A detailed analysis of a case including this one is given later in the proof,after (7.19).

Suppose that p ∈ E is a 2-point of the form (7.9) and p1 ∈ Φ−1i+1(p).

There are regular parameters x1, y1, z1 in OXi,p1 of one of the following forms:

(7.13) x = x1, z = x1(z1 + α)

with α ∈ k or

(7.14) x = x1z1, z = z1.

Suppose that (7.13) holds. We have that p1 is a 2-point, and

u = xa1yb, v = xc1y

d, w = x1(z1 + α).

If α 6= 0, we have that f Φi+1 factors through Y1 at p1. We have a form:

u1 =u

w= xa−1

1 yb(z1 + α)−1, v1 =v

w= xc−1

1 yd(z1 + α)−1, w1 = x1(z1 + α)

at the 2-point p1, which maps to a 3-point, and thus is toroidal, after a change ofvariables.


If α = 0 in (7.13), we have

u = xa1yb, v = xc1y

d, w = x1z1

of the form (7.5).If (7.14) holds, then p1 is a 3-point and

u = xa1ybza1 , v = xc1y

dzc1, w = z1.

Thus f Φi+1 factors through Y1 at p1, and we have a toroidal form:

u1 =u

w= xa1y

bza−11 , v1 =

v

w= xc1y

dzc−11 , w1 = w = z1

at the 3-point p1, which maps to a 3-point.The analysis of Φi+1 above points (7.8) and above points satisfying (7.5) where

y = z = 0 are local equations of E (and b, d ≥ 1) is similar. This last case willlead to a form (7.6). Since Z has only finitely many components, we inductivelyconstruct (7.3).

There now exists a sequence of blow ups of 2-curves Xr → Xm which aresupported above q such that the rational map Xr → Y1 is toroidal where everit is defined, and if IqOXr,p is not invertible, then there there exist permissibleparameters x, y, z at p for u, v, w such that one of the following forms hold:

p a 1-point

(7.15) u = xa, v = xb(α+ y), w = xdz

with 0 6= α ∈ k and d < mina, b orp a 2-point

(7.16) u = xayb, v = xcyd, w = xeyfz

with (e, f) < (a, b) < (c, d) or (e, f) < (c, d) < (a, b).We accomplish this as follows. We first consider u and v. Suppose that p ∈ Xm

is a 2-point such that IqOXm,p is not invertible. We have forms

(7.17) u = xayb, v = xcyd, w = xeyfz

with e+f > 0 at 2-points pi above p in the construction of the sequence Xr → Xm.At pi we have an invariant (a− c)(b− d). This is a nonnegative integer if and onlyif (a, b) ≤ (c, d) or (c, d) ≤ (a, b). Further, if (a− c)(b− d) < 0, then after blowingup the 2-curve E which has local equations x = y = 0 at pi, we obtain that all2-points above pi have a form (7.17), but (a − c)(b − d) has increased. Further Econtracts to q on Y since e+ f > 0.

After a finite number of blow ups of 2-curves above Xm (which must contractto q) we achieve that all 2-points pi above a 2-point p ∈ Xm such that IqOXm,p isnot invertible have a form (7.17) with (a, b) ≤ (c, d) or (c, d) ≤ (a, b).

We now apply this algorithm to the pairs u, xeyf and v, xeyf in (7.17) toconstruct Xr → Xm.

We will now inductively construct Xn → Xr so that IqOXn is invertible ev-erywhere and the morphism Xn → Y1 is toroidal. We will construct a sequence ofblow ups

(7.18) Xn → Xn−1 → · · · → Xr+1 → Xr

so that each Φi+1 : Xi+1 → Xi is the blow up of a nonsingular curve λi which is apossible center and is contained in the locus where IqOXi is not invertible. We will


have that the rational map fi : Xi → Y1 is toroidal where ever it is defined, and allpoints p ∈ Xi where IqOXi,p is not invertible have a form (7.15) or (7.16).

Suppose that we have inductively constructed (7.18) up to Xi and IqOXi isnot invertible. We will construct Φi+1 : Xi+1 → Xi.

Inspection of the forms (7.15) and (7.16) shows that the locus in Xi whereIqOXi

is not invertible is a union of nonsingular curves which are possible centers.For such a curve λ, let η be a general point of λ, so that a form (7.15) holds at η.Let A(λ) = mina, b − d > 0.

Choose a curve λi which maximizes A(λ) on Xi. Let Φi+1 : Xi+1 → Xi be theblow up of λi. Suppose that pi ∈ λi and pi+1 ∈ Φ−1

i+1(λi).First suppose that pi has the form (7.15). After possibly interchanging u and

v, we may assume that a ≤ b. There are regular parameters x1, y, z1 in OXi+1,pi+1

satisfying

(7.19) x = x1, z = x1(z1 + β)

for some β ∈ k, or

(7.20) x = x1z1, z = z1.

Suppose that (7.19) holds. pi+1 is then a 1-point, and

(7.21) u = xa1 , v = xb1(α+ y), w = xd+11 (z1 + β).

If d + 1 = a = b in (7.21), then Xi+1 → Y1 is a morphism near pi+1, whichmaps pi+1 to a 1-point, and has a toroidal form

u1 = u = xa1 , v1 =v

u− α = y1, w1 =

w

u− β = z1.

If d + 1 = a < b in (7.21), then Xi+1 → Y1 is a morphism near pi+1, whichmaps pi+1 to a 2-point, and has a toroidal form

u1 = u = xa1 , v1 =v

u= xb−a1 (α+ y1), w1 =

w

u− β = z1.

If d+ 1 < a ≤ b and β 6= 0 in (7.21) then Xi+1 → Y1 is a morphism near pi+1,which maps pi+1 to a 3-point, and has a toroidal form obtained from a change ofvariable in

u1 =u

w= xa−d−1

1 (z1 + β)−1, v1 =v

w= xb−d−1

1 (α+ y1)(z1 + β)−1,

w1 = w = xd+11 (z1 + β).

If d + 1 < a ≤ b and β = 0 then (7.21) has a form (7.15) with d < d + 1 <mina, b.


u = xa1za1 , v = xb1z

b1(α+ y), w = xd1z

d+11 .

Xi+1 → Y1 is thus a morphism near pi+1, which maps pi+1 to a 3-point, andhas a toroidal form

u1 =u

w= xa−d1 za−d−1

1 , v1 =v

w= xb−d1 zb−d−1

1 (α+ y1), w1 = w = xd1zd+11 .

Now suppose that pi has the form (7.16). After possibly interchanging u and v,we may assume that (a, b) < (c, d). After possibly interchanging x and y, we mayassume that there are regular parameters x1, y, z1 in OXi+1,pi+1 satisfying (7.19) or(7.20) (so that e < a).


Suppose that (7.19) holds. Then pi+1 is a 2-point. We have

(7.22) u = xa1yb, v = xc1y

d, w = xe+11 yf (z1 + β).

If (e+ 1, f) = (a, b) in (7.22), then Xi+1 → Y1 is a morphism near pi+1, whichmaps pi+1 to a 2-point, and has a toroidal form

u1 = u = xa1yb, v1 =

v

u= xc−a1 yd−b1 , w1 =

w

u− β = z1.

If (e + 1, f) < (a, b) and β 6= 0 in (7.22), then Xi+1 → Y1 is a morphism nearpi+1, which maps pi+1 to a 3-point, and has a toroidal form obtained from a changeof variable in

u1 =u

w= xa−e−1

1 yb−f (z1 + β)−1, v1 =v

w= xc−e−1

1 yd−f1 (z1 + β)−1,

w1 = w = xe+11 yf (z1 + β).

If (e + 1, f) < (a, b) and β = 0 then (7.22) has a form (7.16) with (e, f) <(e+ 1, f) < (a, b).


u = xa1ybza1 , v = xc1y

dzc1, w = xe1yfze+1

1 .

Xi+1 → Y1 is thus a morphism near pi+1, which maps pi+1 to a 3-point, andhas a toroidal form

u1 =u

w= xa−e1 yb−fza−e−1

1 , v1 =v

w= xc−e1 yd−fzc−e−1

1 , w1 = w = xe1yfze+1

1 .

In summary, we have that all points where IqOXi+1 is not invertible have aform (7.15) or (7.16) and if λi+1 ⊂ Φ−1

i (λi) is a curve such that IqOXi+1 is notinvertible along λi, we have 0 < A(λi+1) < A(λi). Thus after a finite number ofblow ups, we construct the desired sequence (7.18), completing the proof of thelemma.

Lemma 7.7. Suppose that f : X → Y is a dominant morphism of nonsingular3-folds with toroidal structures determined by SNC divisors DY and DX = f−1(DY )such that DX contains the singular locus of f . Further suppose that f : X → Yis toroidal and C ⊂ Y is a possible center for DY which is a curve and containsa 1-point. Let Ψ : Y1 → Y be the blow up of C. Then there exists a commutativediagram of morphisms

X1f1→ Y1

Φ ↓ ↓ Ψ

Xf→ Y

such that Φ is a sequence of possible blow ups for the preimage of DX supportedabove C and f1 is toroidal with respect to DY1 = Ψ−1(DY ) and DX1

= Φ−1(DX).Further, Φ has a factorization

X1 = Xn → Xn−1 → · · · → X1 → X

where each Φi+1 : Xi+1 → Xi is either the blow up of a section Ei over C such thatICOXi is not invertible, or Φi+1 : Xi+1 → Xi is the blow up of a curve Ei whichmaps to a 2-point of Y and such that Ei is contained in the locus where ICOXi isinvertible.


Proof. We follow the algorithm of Lemma 18.17 [C3] to construct Φ.Suppose that q ∈ C and p ∈ f−1(q). Then there are permissible parameters

u, v, w for DY in OY,q and regular parameters x, y, z in OX,p such that one of thefollowing cases holds:

q is a 2-point and p is a 2-point,

(7.23) u = xayb, v = xdye, w = z

where uv = 0 is a local equation of DY and u = w = 0 is a local equation of C.q is a 2-point and p is a 1-point,

(7.24) u = xa, v = xb(y + α), w = z

where 0 6= α ∈ k, uv = 0 is a local equation of DY and u = w = 0 is a localequation of C.

q is a 1-point and p is a 1-point,

(7.25) u = xa, v = y, w = z

where u = 0 is a local equation of DY and u = w = 0 is a local equation of C.We will construct a sequence of morphisms

(7.26) · · · → XnΦn→ Xn−1

Φn−1→ · · · → X1Φ1→ X

where each Φi+1 is the blow up of a nonsingular curve Ei contained in the locuswhere ICOXi is not invertible, and for each q ∈ C and p ∈ (f Φ1 · · · Φi)−1(q)such that ICOXi,p is not invertible, there are permissible parameters u, v, w for DY

in OY,q and permissible parameters x, y, z in OX,p such that one of the forms (7.27)- (7.29) below hold.


(7.27) u = xayb, v = xdye, w = xgyhz

with ae− ba 6= 0, (g, h) < (a, b).q a 2-point, p a 1-point

(7.28) u = xa, v = xb(y + α), w = xdz

with 0 6= α ∈ k, d < a,q a 1-point, p a 1-point

(7.29) u = xa, v = y, w = xdz

with d < a. Further in the locus where the rational map Xi → Y1 is a morphism,Xi → Y1 is toroidal.

Observe that the forms (7.23), (7.24) and (7.25) are special cases of (7.27),(7.28) and (7.29) respectively.

The locus of points where ICOXi is not invertible is a union of nonsingularcurves which intersect transversally. If E is a curve in this locus, and p′ ∈ E isa general point, then u, v, w have a form (7.28) or (7.29) at p′. In either case, wedefine an invariant

Ω(E) = a− d > 0.

Let Φi+1 : Xi+1 → Xi be the blow up of a curve Ei such that Ω(Ei) is maximal.Suppose that p1 ∈ Ei, p2 ∈ Φ−1

i+1(p1) and q = (f Φ1 · · · Φi)(p1).


Suppose that p1 has a form (7.28). Then OXi+1,p2 has regular parametersx1, y, z1 such that

(7.30) x = x1, z = x1(z1 + β)

with β ∈ k or

(7.31) x = x1z1, z = z1.

Suppose that (7.30) holds. Then p2 is a 1-point,

(7.32) u = xa1 , v = xb1(y + α), w = xd+11 (z1 + β).

If d + 1 = a in (7.32), then Xi+1 → Y1 is a morphism near p2, mapping p2 to a2-point, and at p2, we have a toroidal form

u1 = u = xa1 , v = xb1(y + α), w1 =w

u− β = z1.

If d + 1 < a and β 6= 0 in (7.32) then Xi+1 → Y1 is a morphism near p2,mapping p2 to a 3-point, and at p2, we have a toroidal form obtained from achange of variable in

u1 =u

w= xa−d−1

1 (z1 + β)−1, v = xb1(y1 + α), w1 = w = xd+11 (z1 + β).

If d + 1 < a and β = 0 in (7.32), then we have a form (7.28) with d increasedto d+ 1. The curve E′ containing p2 in the locus where ICOXi+1 is not invertiblesatisfies

0 < Ω(E′) = a− (d+ 1) < Ω(E).

Suppose that (7.31) holds. Then p2 is a 2-point.

(7.33) u = xa1za1 , v = xb1z

b1(y + α), w = xd1z

d+11 .

Further, Xi+1 → Y1 is a morphism near p2, mapping p2 to a 3-point, and atp2, we have a toroidal form

u1 =u

w= xa−d1 za−d−1

1 , v = xb1zb1(y + α), w1 = w = xd1z

d+11 .

There is a similar argument if p1 satisfies (7.29).Suppose that p1 has a form (7.27) and x = z = 0 are local equations of Ei (so

that g < a). OXi+1,p2 has regular parameters x1, y1, z1 satisfying (7.30) or (7.31).Suppose that (7.30) holds. Then p2 is a 2-point,

(7.34) u = xa1yb, v = xd1y

e, w = xg+11 yh(z1 + β).

If (g+1, h) = (a, b) in (7.34), then Xi+1 → Y1 is a morphism near p2, mappingp2 to a 2-point, and at p2, we have a toroidal form

u1 = u = xa1yb, v1 = v = xd1y

e, w1 =w

u− β = z1.

If (g + 1, h) < (a, b) and β 6= 0 in (7.34), then Xi+1 → Y1 is a morphism nearp2, mapping p2 to a 3-point, and we have a toroidal form obtained from a changeof variable in

u1 =u

w= xa−g−1

1 yb−h(z1 + β)−1, v = xd1ye, w1 = w = xg+1

1 yh(z1 + β).

If (g + 1, h) < (a, b) and β = 0 in (7.34), then (7.34) has the form (7.27) withg increased to g + 1.


Suppose that (7.31) holds. Then p2 is a 3-point,

u = xa1ybza1 , v = xd1y

ezd1 , w = xg1yh1 z

g+11 .

Xi+1 → Y1 is a morphism near p2, mapping p2 to a 3-point, and at p2, we have atoroidal form

u1 =u

w= xa−g1 yb−h1 za−g−1

1 , v = xd1yezd1 , w1 = w = xg1y

h1 z

g+11 .

By descending induction on max(Ω(E)), we see that the sequence (7.26) mustterminate after a finite number of blow ups, and we complete the proof of thelemma.

CHAPTER 8

Preparation

In this chapter we prove Theorem 1.3 stated in the introduction, giving theconstruction of a prepared morphism of 3-folds.

To prove this theorem, we may assume by Lemmas 7.1 and 7.2 that f is pre-pared (of type 2 (a) of Definition 4.6) above 2 points and toroidal above 3-pointsof Y . By Corollary 7.4, f only fails to be prepared above a finite set of 1-pointsΣ ⊂ Y .

Suppose that q ∈ Σ. Let D be the component of DY containing q. There existsa very ample effective divisor L on Y such that q 6∈ L and D + L ∼ H where H isa very ample effective divisor such that q 6∈ H. Let α : Z → Y be the blow up ofq, with exceptional divisor E. We may replace L with a high multiple of L so thatα∗H−E is very ample on Z. Let N be a general member of α∗H−E. By Bertini’stheorem, N is nonsingular, makes SNCs with DZ = α−1(DY ), intersects 2-curvesof DZ transversally at general points, does not contain a component of the stricttransform on Z of the nonfinite locus of f , and is disjoint from α−1(Σ− q). LetM = α(N). Then M ∼ H, M is nonsingular and intersects D transversally in anonsingular curve γ which contains q, M contains no other points of Σ, containsno 3-points of DY , intersects 2-curves of DY transversally at general points, doesnot contain a component of the nonfinite locus of f and by Bertini’s theorem, atpoints which are not above q, f∗(M) is nonsingular and f∗(M) + DX is a SNCdvisor. After possibly replacing L and H with effective divisors linearly equivalentto L and H respectively, we may assume that γ ∩ (L + H) consists of 1-points ofDY and is disjoint from the nonfinite locus of f .

U = Y − (L +H) is an affine neighborhood of q. Let γ = γ ∩ U . There existf, g ∈ Γ(Y,OY (H)) such that (f) = D + L −H and (g) = M −H. We can thusdefine a morphism π : U → S = A2 by π(a) = (f(a), g(a)) for a ∈ U . Let q = π(q).π−1(q) = γ (scheme theoretically) so π is smooth in a neighborhood of γ. We maythus replace U with an open neighborhood of γ so that π is smooth.

LetX = f−1(U), and f = f | X. LetDU = DY ∩U , D∗U = D∩U , D∗

S = π(D∗U ),

g = π f : X → S, D∗X

= g−1(D∗S), DX = DX ∩X. The map π is toroidal with

respect to D∗S and D∗

U .LetD1, . . . , Dm be the components ofDY other thanD which intersect γ. Since

γ intersects these components transversally, we may assume then that π | Di ∩ Uis etale onto its image for 1 ≤ i ≤ m. We further may assume that Σ ∩ U = q,and (by Bertini’s theorem) for q′ ∈ D∗

S − q, there exist regular parameters u,wat q′ such that u = 0 is a local equation of D∗

S , and if E is the curve w = 0 on S,then E is nonsingular, D∗

S + E is a SNC divisor, D∗U + π−1(E) is a SNC divisor

on U , g−1(E) is nonsingular, and g−1(E) + DX is a SNC divisor on X. Thus ifq′ ∈ π−1(q′), there exist permissible parameters u, v, w in OU,q′ (for DU ) such that

49

50 8. PREPARATION

if p ∈ f−1(q′) then there exist regular parameters x, y, z in OX,p such that

(8.1) u = xayb, w = z

where u = xayb = 0 is a local equation ofDX at p (with a > 0, b ≥ 0) if q′ ∈ D−∪Di

and

(8.2) u = xayb, v = xcydγ1, w = z

where γ1 ∈ OX,p is a unit and uv = xa+cyb+d = 0 is a local equation of DX at p ifq′ ∈ D ∩Di for some i.

Since γ intersects the 2-curves Di∩D∩U of U at general points of the 2-curves,after possibly replacing U with a smaller open neighborhood of γ, we have that theintersection of the nonfinite locus of f with U is contained in D ∩ U .

We will now establish that g is toroidal and prepared with respect to D∗S and

D∗X

away from the preimages of finitely many 1-points Ω ⊂ γ of DU .

Suppose that q′ ∈ (Di − D) ∩ U , p ∈ f−1(q′), and q′ = π(q′), which implies

that there exist regular parameters u,w at q′, u, v, w at q′ such that v = 0 is a localequation of Di. q′ is not in the nonfinite locus of f , and q′ is a 1-point of DU , so byAbhyankar’s lemma ([Ab4] and XIII 5.3 [G]) there exist regular parameters x, y, zin OX,p such that

(8.3) u = x, v = yb, w = z.

g is defined by u = x,w = z near p, which implies that g is smooth, and thusprepared and toroidal for D∗

S and D∗X

at p.

Suppose that q′ ∈ (D − γ) ∩ U , p ∈ f−1(q′), q′ = π(q′). Then we have a form

(8.1) or (8.2) at p, so that g is prepared and toroidal for D∗S and D∗

Xat p.

Let δ = D ∩ Di ∩ U for some 1 ≤ i ≤ m. Suppose that q′ ∈ δ ∩ γ andp ∈ f−1

(q′). Then π(q′) = q. There exist regular parameters u,w in OS,q such thatu = 0 is a local equation of D on U , w = 0 is a local equation of M on U , thereexists v ∈ OU,q′ such that v = 0 is a local equation of Di and u, v, w are regularparameters in OU,q′ . By our choice of M , f

∗(M) is nonsingular and makes SNCs

with DX at p. Since uv = 0 is a local equation of DX at p, and w = 0 is a localequation of f

∗(M) at p, there exist regular parameters x, y, z in OX,p such that

u = xaybγ1, v = xcydγ2, w = z

with γ1, γ2 units in OX,p. Thus g is prepared and toroidal for D∗S and D∗

Xat p.

Suppose that q′ ∈ γ is a general point. Then q′ is a 1-point of DU and f isfinite above q′. There exist regular parameters u, v, w in OU,q′ such that u,w arepermissible parameters for D∗

S at q = π(q′), and if p ∈ f−1(q′), then there exist

permissible parameters x, y, z in OX,p such that

(8.4) u = xa, v = y, w = z

by Abhyankar’s lemma, which implies that g is prepared and toroidal at p for D∗S

and D∗X

.We conclude that g is toroidal and prepared with respect to D∗

S and D∗X

awayfrom points above finitely many 1-points Ω ⊂ γ of DU .

By Lemma 7.5 (applied to a neighborhood of Ω), there exists a morphism Φ1 :X1 → X such that Φ1 is a sequence of possible blow ups for the preimage of D∗

Xof

8. PREPARATION 51

points and nonsingular curves supported above Ω such that if g1 = g Φ1 : X1 → Sand f1 = f Φ1 : X1 → U , then in a neighborhood of all components of D∗

X1which

do not map to a point of Ω, g1 is prepared for D∗S and D∗

X1= Φ−1

1 (D∗X

)

By 1 of Theorem 6.1, there exists a morphism Φ2 : X2 → X1 which is asequence of possible blow ups for the preimage of D∗

X1of points and nonsingular

curves supported above Ω, such that g2 = π f1 Φ2 : X2 → S is prepared for D∗S

and D∗X2

= Φ−12 (D∗

X1). Let f2 = f1 Φ2 : X2 → U .

Now by 2 of Theorem 6.1, there exists a commutative diagram

X3g3→ S1

Φ3 ↓ ↓ λ1

X2g2→ S

such that λ1 is a sequence of possible blow ups for the preimage of D∗S of points

supported above q, Φ3 is a sequence of possible blow ups for the preimage of D∗X2

of points and nonsingular curves supported above γ, and g3 is toroidal with respectto D∗

S1= λ−1

1 (D∗S) and D∗

X3= Φ

−1

3 (D∗X2

). Let f3 = f2 Φ3 : X3 → U .Consider the commutative diagram

X3f3→ Y 1

π1→ S1

Φ ↓ Ψ ↓ ↓ λ1

Xf→ U

π→ S

where Φ = Φ1 Φ2 Φ3, Y 1 = U ×S S1 and Ψ : Y 1 → U , π1 : Y 1 → S1 are thenatural projections, and f3 = f3 × g3. D∗

Y 1= Ψ

−1(D∗

U ) and DY 1= Ψ

−1(DU ) are

SNC divisors. Y 1 is nonsingular, and is obtained from U by possible blow ups forthe preimage of DU of sections over γ. Since g3 is toroidal with respect to D∗

S1and

D∗X3

, f3 is prepared with respect to D∗Y 1

and D∗X3

.

Suppose that q′ ∈ γ is a general point, and p ∈ (f Φ)−1(q′). Then f has atoroidal and prepared form (8.4) at Φ(p). By Theorem 6.1, and the form of thefactorization (6.3), we see that f3 is toroidal and prepared at p with respect to D∗

Y 3

and D∗X3

. Also, over a general point of γ, Φ is a sequence of possible blow ups forthe preimages of D∗

Xof sections over γ.

Recall that DY = D+D1 + · · ·+Dm+G, where G consists of the componentsof DY disjoint from U , and that the Di ∩ U are etale over their images in S. LetDi be the strict transform of Di on Y 1 for 1 ≤ i ≤ m. Let

DY 1= Ψ

−1(DU ) = D∗

Y 1+D1 + · · ·+Dm,

DX3= Φ

−1(DX).

We will now verify that DX3is a SNC divisor on X3 and that f3 is prepared

for DY 1and DX3

. Since f3 is prepared for D∗Y 1

and D∗X3

, we need only verify that

f3 is prepared for DY 1and DX3

at points p′ ∈ X3 such that q′ = f Φ(p′) ∈ Di

for some i.First suppose that q′ ∈ Di−γ for some i. Then Φ and Ψ are isomorphisms near

p′ and q′ respectively. Suppose that q′ 6∈ D. Then we have permissible parametersv, u, w for DU at q′ which have an expression (8.3) at p′. Thus f3 has an expression

52 8. PREPARATION

3 of Definition 4.6 at p′. Suppose that q′ ∈ D∩Di− γ. Then q′ is a 2-point of DU ,so that f is prepared above q′ for DU and DX . Thus f3 is prepared above q′ (forDY 1

and DX3).

Suppose that q′ = f Φ(p′) ∈ γ ∩ Di for some i. Without loss of generality,we may assume that Di = D1. Recall that q′ ∈ γ ∩ D1 is a general point of the2-curve D ∩D1, f is prepared above q′ and f

∗(M) is nonsingular and makes SNCs

with DX above q′. Since q′ is a general point of D ∩D1, there are no 3-points inf−1

(q′). Let D′1 be the reduced divisor on X whose components dominate D1. The

irreducible components of D′1 are disjoint by Remark 4.5.

There exist permissible parameters u, v, w in OU,q′ for the two point q′ of DU

such that u = 0 is a local equation of D, v = 0 is a local equation of D1, w = 0 isa local equation of M on U , and u,w are regular parameters in OS,q such that ifp = Φ(p′) ∈ f−1

(q′), then there exist regular parameters x, y, z in OX,p such thatone of the following prepared forms for f hold at p. u, v are toroidal forms for DU

and DX in all cases.

1. p is a 1-point of DX

(8.5)u = xa

v = xbγw = z

where γ ∈ OX,p is a unit and x = 0 is a local equation of DX .2. p is a 2-point of DX which is not on D′

1

(8.6)u = xayb

v = xcydγw = z

with a, b > 0, γ ∈ OX,p is a unit and xy = 0 is a local equation of DX .3. p is a 2-point which is on D′

1

(8.7)u = xa

v = xbyc

w = z

where xy = 0 is a local equation of DX and y = 0 is a local equation ofD′

1.

Ψ is the sequence of monodial transforms induced by a sequence of quadratictransforms,

S1 = Sn → · · · → S0 = S.

Each map Sj+1 → Sj is the blow up of the ideal sheaf mj of a point qj above q.Let

(8.8) Y 1 = Yn → · · · → Y0 = U

be the induced factorization of Ψ, where Ψj+1 : Yj+1 = U×SSj+1 → Yj = U×SSj ,is the blow up of a curve Cj . Let πj : Yj → Sj be the natural projection.

Φ is a sequence of morphisms

X3 = Xn → · · · → X0 = X2 → X.

8. PREPARATION 53

where Φj+1 : Xj+1 → Xj is a principalization of mjOXj, with natural morphism

fj : Xj → Yj . Let DXj, DYj

be the respective preimages of DU , and let D∗Xj

, D∗Yj

be the respective preimages of D∗U . Let D∗

Sjbe the preimage of D∗

S in Sj . The

principalizations Φj are explicitly described in the proof of Theorem 6.1. We havea factorization

(8.9) Xj+1 = Xnj ,j → · · · → X0,j = Xj .

where each Φi+1,j : Xi+1,j → Xi,j is the blow up of a single curve Eij which isa possible center for the preimage D∗

Xijof D∗

U on Xi,j . Eij is in the locus where

mjOXi,jis not locally principal. Let DXij

be the preimage of DU on Xij .Recall that X2 → X is an isomorphism above q′.We will prove that f3 : X3 → Y 1 is prepared for DY 1

and DX3above q′ by

induction on j in the morphisms fj : Xj → Yj .Recall that we have a fixed choice of regular parameters u = u0, v, w = w0 in

OU,q′ , which are permissible parameters for DY at the 2-point q′, and one of theforms (8.5) - (8.7) holds at all points of X above q′.

Suppose by induction that DXjis a SNC divisor, fj : Xj → Yj is prepared for

DYjand DXj

, and if qj ∈ Yj and Ψ1 · · · Ψj(qj) = q′, then

1. qj is a 2-point or a 3-point of DYjand there exist regular parameters

uj , wj in OSj ,q′j, where πj(qj) = q′j , such that uj , v, wj are permissible

parameters for DYjin OYj ,qj

.2. If qj ∈ Cj , then uj = wj = 0 are local equations of Cj at qj .3. If pj ∈ f−1

j (qj), then there exist regular parameters xj , yj , zj in OXj ,pj

such that one of the following forms hold:

Case 1. qj is a 2-point of DYj, and ujv = 0 is a local equation of DYj

(sothat πj(qj) = q′j is a 1-point), and pj is a 1-point of DXj

with

(8.10) uj = xaj , v = xbjγj , wj = zj

where xj = 0 is a local equation of DXj, γj is a unit in OXj ,pj

or pj is a2-point of DXj

with

(8.11) uj = xaj ybj , v = xcjy

dj γj , wj = zj

where xjyj = 0 is a local equation of DXj, γj is a unit in OXj ,pj

.

Case 2. qj is a 3-point of DYj, and ujvwj = 0 is a local equation of DYj

(so that πj(qj) = q′j is a 2-point), and pj is a 1-point of DXjwith

(8.12) uj = xaj , v = xbjγj , wj = xcj(zj + β)

where xj = 0 is a local equation of DXj, γj is a unit in OXj ,pj

and0 6= β ∈ k or pj is a 2-point of DXj

with

(8.13) uj = xaj zbj , v = xcjz

dj γj , wj = xejz

fj

54 8. PREPARATION

where af − be 6= 0, xjzj = 0 is a local equation of DXj, γj is a unit in

OXj ,pjor pj is a 2-point of DXj

with

(8.14) uj = (xaj ybj)k, v = xdjy

ejγj , wj = (xaj y

bj)t(zj + β)

where xjyj = 0 is a local equation of DXj, γj is a unit in OXj ,pj

,gcd(a, b) = 1 and 0 6= β ∈ k or pj is a 3-point of DXj

with

(8.15) uj = xaj ybjzcj , v = xdjy

ejzfj γj , wj = xgjy

hj z

ij

where xjyjzj = 0 is a local equation of DXj, γj is a unit in OXj ,pj

and

rank(a b cg h i

)= 2.

All of the forms of Case 1 and Case 2 are prepared for DYjand DXj

. In case1, this follows from Remark 4.7. In case 2, uj and wj are toroidal forms.

We will prove that the above statements hold for fj+1 : Xj+1 → Yj+1. Theargument is essentially the same as in the proof of Lemma 7.7.

Suppose that qj ∈ Cj is a 2-point (and Ψ1 · · · Ψj(qj) = q′), so that Case 1holds, and pj ∈ f−1

j (qj).ICjOXj ,pj

is not invertible, and uj , wj satisfy (185) [C3] at pj if (8.10) holds,uj , wj satisfy (190) [C3] at pj if (8.11) holds and a, b > 0 (so that pj is a 2-pointof D∗

Xj), uj , wj satisfy (185) [C3] at pj if (8.11) holds and b = 0 (so that pj is a

1-point of D∗Xj

).The algorithm of Lemma 18.17 [C3] (as used in the proof of Theorem 6.1) is

applied to construct Φj+1 : Xj+1 → Xj and fj+1 : Xj+1 → Yj+1 above qj . Supposethat qj+1 ∈ Ψ−1

j+1(qj), and πj+1(qj+1) = q′j+1 ∈ Sj+1. Then there exist regularparameters uj+1, wj+1 in OSj+1,q′j+1

such that uj+1, v, wj+1 are regular parametersin OYj+1,qj+1

and one of the following forms hold:

q′j+1 is a 1-point of D∗Sj+1

(8.16) uj = uj+1, wj = uj+1(wj+1 + α)

with α ∈ k, or q′j+1 is a 2-point for D∗Sj+1

(8.17) uj = uj+1wj+1, wj = wj+1.

If (8.16) holds at q′j+1 and pj+1 ∈ f−1j+1(qj+1), then an analysis of the algorithm

of Lemma 18.17 [C3] and Theorem 6.1 shows that uj+1, v, wj+1 satisfy one of theforms (8.10) or (8.11) at pj+1.

If (8.17) holds at q′j+1, and pj+1 ∈ f−1j+1(qj+1), then uj+1, v, wj+1 satisfy one of

the forms (8.12) - (8.15) at pj+1.If qj+1 ∈ Cj+1, then uj+1 = wj+1 = 0 are local equations of Cj+1.Now suppose that qj ∈ Cj is a 3-point (and Ψ1 · · · Ψj(qj) = q′), so that Case

2 holds, and pj ∈ f−1j (qj).

If ICjOXj ,pjis not invertible, then after possibly interchanging uj and wj , then

we have one of the following forms. uj , wj satisfy (187) [C3] at pj if (8.13) holdsand a, b > 0, uj , wj satisfy (191) [C3] at pj if (8.13) holds and b = e = 0 (in both

8. PREPARATION 55

cases, pj is a 2-point of D∗Xj

). uj , wj satisfy (187) or (191) [C3] if (8.15) holds andpj is a 2 point of D∗

Xj. uj , wj satisfy (193), (194) or (195) [C3] at pj if (8.15) holds

and pj is a 3-point of D∗Xj

.

The algorithm of Lemma 18.18 [C3] is then applied to construct Φj+1 : Xj+1 →Xj and fj+1 : Xj+1 → Yj+1 above qj . Suppose that qj+1 ∈ Ψ−1

j+1(qj), and π(qj+1) =q′j+1 ∈ Sj+1. Then there exist regular parameters uj+1, wj+1 in OSj+1,q′j+1

suchthat uj+1, v, wj+1 are regular parameters in OYj+1,qj+1

and one of the followingforms hold:

q′j+1 is a 1-point of D∗Sj+1

(8.18) uj = uj+1, wj = uj+1(wj+1 + α)

with 0 6= α ∈ k, or q′j+1 is a 2-point for D∗Sj+1

(8.19) uj = uj+1, wj = uj+1wj+1,

or q′j+1 is a 2-point for D∗Sj+1

(8.20) uj = uj+1wj+1, wj = wj+1.

If (8.18) holds at q′j+1 and pj+1 ∈ f−1j+1(qj+1), then an analysis of the algorithm

of Lemma 18.18 [C3] shows that uj+1, v, wj+1 satisfy one of the forms (8.10) or(8.11) at pj+1.

If (8.19) or (8.20) holds at q′j+1, and pj+1 ∈ f−1j+1(qj+1), then uj+1, v, wj+1

satisfy one of the forms (8.12) - (8.15) at pj+1.We have shown that DXj+1

= f−1j+1(DYj+1

) is a SNC divisor above qj and that

fj+1 is prepared for DYj+1and DXj+1

.We have thus established that f3 is prepared for DY 1

and DX3.

Recall that Φi+1 : Xi+1 → Xi is a principalization of miOXiwhich in a neigh-

borhood of a general point of γ is a sequence of blow ups of sections over γ wheremiOXi

is not invertible. Each Ψi+1 : Yi+1 → Yi is the blow up of a curve Ci whichis a section over γ and is a possible center for DYi

.We will construct Ψ1 : Y1 → Y such that Ψ−1

1 (U) ∼= Y 1, Ψ1 | Ψ−11 (U) = Ψ and

Ψ−11 (DY ) is a SNC divisor by constructing a sequence of morphisms

(8.21) Y1 = YnΨn→ Yn−1 → · · ·

Ψ1→ Y

where each Ψi+1 is a product of blow ups of possible centers for the preimage ofDY , and Ψ−1

i+1(Yi) ∼= Yi+1, Ψi+1 | Yi+1 = Ψi+1 for all i.We will inductively construct (8.21). Suppose that we have constructed Ψi :

Yi → Yi−1.Let γi be the Zariski closure of Ci in Yi. Then γi is a section over γ, and is

thus a nonsingular curve. We construct Ψi+1 by first blowing up points on (thestrict transform of) γi above γ−γ where (the strict transform of) γi does not makeSNCs with (the preimage of) DYi

, and then blowing up the strict transform of γi.X2 → X is an isomorphism away from the preimage of Ω. Thus the sequence

of blow ups X2 → X extends trivially to a morphism X2 → X, so that X2 → X isan isomorphism away from the preimage of Ω.

56 8. PREPARATION

Now we construct Φ3 : X3 → X2 such that Φ−13 (X2) = X3, Φ3 | X3 = Φ3 and

Φ−13 (DX3) is a SNC divisor by constructing a sequence of morphisms

X3 = XnΦn→ Xn−1 → · · ·

Φ1→ X0 = X2

where Φ−1i (Xi−1) ∼= Xi and Φi | Xi = Φi for all i, and so that there are morphisms

Xi → Yi which are toroidal (with respect to the preimages of DY and DX) overpoints of γ − γ. This follows from application of Lemmas 7.6 and 7.7, and the factthat the case when γi is a 2-curve (or a 3-point is blown up) extends directly to atoroidal morphism.

The resulting morphism Φ3 is an isomorphism away from the preimage of γ.We have constructed a diagram

X3f3→ Y1

Φ ↓ ↓ Ψ1

Xf→ Y

such that Φ and Ψ1 are isomorphisms away from the preimage of γ and f3 isprepared with respect to DY1 = Ψ−1

1 (DY ) and DX3 = Φ−1(DX) away from thepoints in Σ− q.

By induction on |Σ|, we repeat this construction to prove Theorem 1.3.

CHAPTER 9

The τ invariant

Throughout this chapter, we assume that f : X → Y is a dominant morphismof nonsingular 3-folds, with toroidal structures DY , DX = f−1(DY ) such that DX

contains the singular locus of f .

Definition 9.1. Suppose that f : X → Y is prepared.Suppose that p ∈ DX . Define τf (p) = τ(p) = −∞ if there exist permissible

parameters u, v, w at q = f(p) such that u, v, w are toroidal forms at p. (The exis-tence of such formal parameters u, v, w implies the existence of algebraic permissibleparameters which are toroidal forms at p).

Suppose that p ∈ X is a 3-point, and τ(p) 6= −∞. Suppose that u, v, w arepermissible parameters at q = f(p). Then there is an expression (after possiblypermuting u, v, w if q is a 3-point)

(9.1)u = xaybzc

v = xdyezf

w =∑i≥0 αiMi +N

where x, y, z are permissible parameters at p for u, v, w, rank(u, v) = 2, the sum inw is over (possibly infinitely many) monomials Mi = xaiybizci in x, y, z such thatrank(u, v,Mi) = 2 for all i, deg(Mi) ≤ deg(Mj) if i < j, N is a monomial in x, y, zsuch that rank(u, v,N) = 3, 0 6= αi ∈ k for all i and N 6 | Mi for any Mi in theseries

∑αiMi.

If q is a 3-point and u, v, w is not a toroidal form at p, we necessarily have(since f is prepared) that

(9.2)∑

αiMi = M0γ

where γ is a unit series in the monomials Mi

M0in x, y, z such that

rank(u, v,M0) = rank(u, v,Mi

M0) = 2

for all i, and M0 | N .Define a group Hp = Hf,p as follows. The Laurent monomials in x, y, z form a

group under multiplication. We define Hp = Hf,p to be the subgroup of Z3 definedby

Hp = Hf,p = (a, b, c)Z + (d, e, f)Z +∑i≥0

(ai, bi, ci)Z.

Define a subgroup Ap of Hp by:

Ap = Af,p =

(a, b, c)Z + (d, e, f)Z + (a0, b0, c0)Z if q is 3-point(a, b, c)Z + (d, e, f)Z if q is a 2-point.

57

58 9. THE τ INVARIANT

DefineLp = Lf,p = Hp/Ap,

τ(p) = τf (p) = |Lp|.Suppose that p ∈ X is a 2-point, and τ(p) 6= −∞. Suppose that u, v, w are

permissible parameters at q = f(p) (which satisfy the conclusions of Lemma 4.9 iff(p) = q is a 1-point). Then there is an expression (after possibly permuting u, v, wif q = f(p) is a 3-point) of one the following forms:

q is 2-point or a 3-point:

(9.3)u = xayb

v = xcyd

w =∑i≥0 αix

aiybi + xeyf (z + β)

where x, y, z are permissible parameters at p for u, v, w, 0 6= αi ∈ k for all i andxeyf 6 | xaiybi for all i, or

q is a 2-point or a 3-point:

(9.4)u = (xayb)k

v = (xayb)t(z + β)w =

∑i≥0 αi(z)(x

ayb)i + xcyd

where gcd(a, b) = 1, 0 6= β ∈ k, x, y, z are permissible parameters at p for u, v, w,αi(z) ∈ k[[z]] for all i and xcyd 6 | (xayb)i for all i, or

q is a 2-point:

(9.5)u = (xayb)k

v =∑i≥0 αi(z)(z

ayb)i + xcyd

w = z

where gcd(a, b) = 1, x, y, z are permissible parameters at p for u, v, w, αi(z) ∈ k[[z]]for all i and xcyd 6 | (xayb)i for all i, or

q is a 1-point:

(9.6)u = (xayb)k

v = zw =

∑i≥0 αi(z)(x

ayb)i + xcyd

where gcd(a, b) = 1, x, y, z are permissible parameters at p for u, v, w, αi(z) ∈ k[[z]]for all i and xcyd 6 | (xayb)i for all i.

If q is a 3-point, we necessarily have (since f is prepared) that

(9.7) w = M0γ

where γ is a unit series and M0 is a monomial in x and y.In equation (9.5), we have

(9.8) v = (xayb)i0γ

where γ is a unit series.Suppose that (9.3) holds. Then define

τ(p) = |((a, b)Z + (c, d)Z +∑i≥0

(ai, bi)Z)/((a, b)Z + (c, d)Z)|

9. THE τ INVARIANT 59

if q is a 2-point,

τ(p) = |((a, b)Z + (c, d)Z +∑i≥0

(ai, bi)Z)/((a, b)Z + (c, d)Z + (a0, b0)Z)|

if q is a 3-point, and M0 = xa0yb0 in (9.7).Suppose that (9.4) holds. Then define

τ(p) = |(kZ + tZ +∑αi 6=0

iZ)/(kZ + tZ)|

if q is a 2-point,

τ(p) = |(kZ + tZ +∑αi 6=0

iZ)/(kZ + tZ + i0Z)|

if q is a 3-point, and M0 = xi0γ in (9.7).Suppose that (9.5) (and (9.8)) hold. Then define

τ(p) = |(kZ +∑αi 6=0

iZ)/(kZ + i0Z)|.

Suppose that (9.6) holds. Then define

τ(p) = |(kZ +∑αi 6=0

iZ)/kZ|.

Suppose that p ∈ X is a 1-point, and τ(p) 6= −∞. Suppose that u, v, w arepermissible parameters at q = f(p) which satisfy the conclusions of Lemma 4.9 iff(p) = q is a 1-point. Then there is an expression (after possibly permuting u, v, wif q = f(p) is a 3-point) of one the following forms:

q is 2-point or a 3-point

(9.9)u = xa

v = xb(α+ y)w =

∑i<c αi(y)x

i + xc(z + β)

where x, y, z are permissible parameters at p for u, v, w, 0 6= α ∈ k, β ∈ k andαi(y) ∈ k[[y]] or

q is 2-point

(9.10)u = xa

v =∑i<c αi(y)x

i + xc(z + β)w = y

where x, y, z are permissible parameters at p for u, v, w, β ∈ k and αi(y) ∈ k[[y]],or

q is a 1-point

(9.11)u = xa

v = yw =

∑i<c αi(y)x

i + xc(z + β)

where x, y, z are permissible parameters at p for u, v, w, β ∈ k, αi(y) ∈ k[[y]].


In case (9.9) we have

(9.12) w = xi0γ

where γ is a unit series if q is a 3-point. In case (9.10) we have

(9.13) v = xi0γ

where γ is a unit series.Suppose that (9.9) holds. Then define

τ(p) = |(aZ + bZ +∑αi 6=0

iZ)/(aZ + bZ)|

if q is a 2-point,

τ(p) = |(aZ + bZ +∑αi 6=0

iZ)/(aZ + bZ + i0Z)|

if q is a 3-point.Suppose that (9.10) holds. Then define

τ(p) = |(aZ +∑αi 6=0

iZ)/(aZ + i0Z)|.

Suppose that (9.11) holds. Then define

τ(p) = |(aZ +∑αi 6=0

iZ)/aZ|.

Observe that τ(p) < ∞ in Definition 9.1, since τ(p) is the order of a finitelygenerated group.

We defineτ(X) = τf (X) = supτf (p) | p ∈ X.

It will follow from Lemma 9.3 that τ(X) <∞.

Lemma 9.2. Suppose that f : X → Y is prepared and p ∈ DX . Then τf (p) isindependent of choice of permissible parameters u, v, w at q = f(p) and permissibleparameters x, y, z at p for u, v, w.

Proof. First suppose that q ∈ Y is a 3-point, and p ∈ X is a 3-point.The condition τ(p) = −∞ is independent of permuting u, v, w, multiplying

u, v, w by units in OY,q and multiplying x, y, z by units in OX,p so that the con-ditions of (9.1) hold. Thus τ(p) = −∞ is independent of choice of permissibleparameters at (the 3-points) q and p.

Suppose that u, v, w are permissible parameters at q and x, y, z are permissibleparameters at p for u, v, w satisfying (9.1). Let τ be the computation of τ(p) forthese variables.

Suppose that u, v, w is another set of permissible parameters at q, and x, y, z arepermissible parameters for u, v, w at p, satisfying (9.1). Let τ1 be the computationfor τ(p) with respect to these variables. We must show that τ = τ1. We mayassume that τ ≥ 1 and τ1 ≥ 1.

Since p and q are 3-points, u, v, w can be obtained from u, v, w by permutingthe variables u, v, w and then multiplying u, v, w by unit series (in u, v, w). x, y, zcan be obtained from x, y, z by permuting x, y, z and then multiplying x, y, z byunit series (in x, y, z).

We then reduce to proving the following:


1. Suppose that u, v, w is a permutation of u, v, w such that u, v are toroidalforms at p. Then there exist permissible parameters x, y, z at p for u, v, wsuch that a form (9.1) holds, and τ1 = τ .

2. Suppose that u, v, w are obtained from u, v, w by multiplying u, v, w byunit series. Then there exist permissible parameters x, y, z at p for u, v, wsuch that a form (9.1) holds, and τ1 = τ .

3. Suppose that u = u, v = v and w = w and x, y, z, x, y, z are two sets ofpermissible parameters for u, v, w. Then τ1 = τ .

We now verify 1. The case when

u = v, v = u, w = w

is immediate. We will verify the case when

u = w, v = v, w = u.

Since the symmetric group S3 is generated by the permutations (1, 2) and (1, 3),the remaining cases of 1 will follow.

Since τ ≥ 1, and u, v, w have a form (9.1) at p, we have rank(v,M0) = 2. Sincew, v are (by assumption) toroidal forms at p, there exist permissible parametersx, y, z at p such that w, v, u (in this order) have an expression of the form of (9.1)in terms of x, y, z. We will show that there is an isomorphism of the correspondinggroup H (computed for these variables) and the group H computed for u, v, w andx, y, z which takes the corresponding group A to A.

With the notation of (9.2), we have

w = M0(γ +N0)

where N0 = NM0

is a monomial in x, y, z.Set M0 = M0, M i = Mi

M0for all Mi appearing in the series

∑i≥1 αiMi. Thus

γ = α0 +∑i≥1 αiM i. There exist ai, bi, ci ∈ N such that

M i = xaiybizci

for 0 ≤ i. Let mi = (ai, bi, ci) for i ≥ 0,

H = (a, b, c)Z + (d, e, f)Z +∑i≥0

miZ.

Define a finite sequence

1 = µ(1) < µ(2) < · · · < µ(r)

(for appropriate r) so thatµ(r)∑i=1

miZ =∞∑i=1

miZ,

µ(j)∑i=1

miZ =n∑i=1

miZ

if µ(j) < n < µ(j + 1) and

µ(j)∑i=1

miZ 6=µ(j+1)∑i=1

miZ.


Set

Gj =µ(j)∑i=1

miZ.

There exist ki, li ∈ Z and ei ∈ N with gcd(ei, ki, li) = 1 such that Mei

i = ukivli

for i ∈ 0, µ(1), . . . , µ(r), and there exist g, h, i ∈ N such that N0 = xgyhzi. ThusH is the subgroup of Z3 generated by

δ1 = (a, b, c), δ2 = (d, e, f), ε0 = (a0, b0, c0),εµ(1) = (aµ(1), bµ(1), cµ(1)), . . . , εµ(r) = (aµ(r), bµ(r), cµ(r)).

A is the subgroup with generators δ1, δ2 and ε0.Since rank(v,M0) = 2, we can make a change of variables

x = xλ1, y = yλ2, z = zλ3

where λi = (γ +N0)βi for some βi ∈ Q, so that

λa01 λ

b02 λ

c03 = (γ +N0)−1

λd1λe2λf3 = 1

λa1λb2λc3 = (γ +N0)t

for some t ∈ Q. Since u, v, w are algebraically independent in OX,p (by Zariski’ssubspace theorem, Theorem 10.14 [Ab3]), and

rank(u, v,M0) = 2,

we have that t 6= 0,w = xa0yb0zc0

v = xdyezf

u = xaybzc(γt + xgyhziγ2)where γ2 is a unit series in x, y, z.

Set Mi = xaiybizci for i ≥ 1. In the Taylor’s series expansion of h(ζ) = (α0+ζ)t

we substitute ζ =∑∞i=1 αiM i to see that

γ(x, y, z)t ≡∞∑j=0

qj mod (x, y, z)N0

where each qj is a series in monomials of degree j in Mi | i ≥ 1 with coefficientsin k, and

q0 = αt0, q1 =∞∑i=1

σiMi

withσi = tαt−1+aiβ1+biβ2+ciβ3

0 αi

for i ≥ 1. Let

ω =∞∑j=0

qj .

There exists a unit series γ3 such that

u = xaybzc(ω + xgyhziγ3).

We now see that the coefficient of Mµ(j) for 1 ≤ j ≤ r in the expansion of ω as aseries in x, y, z is σµ(j). If not, there would exist a relation Mµ(j) = Mi1 · · · Min forsome n > 1. Thus i1, . . . , in < µ(j) and Gj = Gj−1, a contradiction.


Since

Det

a b cd e fg h i

6= 0,

there exists a change of variables

x = xφ1, y = yφ2, z = zφ3

where φ1, φ2, φ3 are unit series in x, y, z (fractional powers of γ3) such that

φa

e01 φ

be02 φ

ce03 = 1

φd

e01 φ

ee02 φ

fe03 = 1

φg1φh2φ

i3 = γ−1

3 .

Since ei(ai, bi, ci) = ki(a, b, c) + li(d, e, f) for i ≥ 0, we have an expression

w = xa0 yb0 zc0

v = xdyezf

u = xaybzc(ω + xg yhzi)

and Mi = xai ybi zciηi for some ei-th root of unity ηi in k for all 1 ≤ i.H is thus the subgroup of Z3 with generators

δ1, δ2, ε0, εµ(1), . . . , εµ(r),

defined by

δ1 = (a0, b0, c0), δ2 = (d, e, f), ε0 = (a, b, c),εµ(1) = (aµ(1), bµ(1), cµ(1)), . . . , εµ(r) = (aµ(r), bµ(r), cµ(r)).

A is the subgroup of H generated by δ1, δ2 and ε0. Thus H = H and A = A.We have thus completed the verification of 1. The verification of 2 and 3 follow

from simpler calculations. We thus obtain the conclusions of the lemma when p isa 3-point and q is a 3-point.

Now assume that q ∈ Y is a 2-point, and p is a 3-point. τ(p) is independent ofinterchanging u and v, multiplying u and v by unit series in OY,q, and permutingx, y, z and multiplying x, y, z by unit series in OX,p, so that the conditions of (9.1)hold. If we replace w by w′ ∈ OY,q so that u, v, w′ are permissible parameters at q,then by the Weierstrass preparation theorem, there exists a unit series α(u, v, w) ∈OY,q and a series β(u, v) ∈ k[[u, v]] such that w = α−1(w′ − β(u, v)).

There exists a series φ in x, y, z such that

α(u, v, w) = α(u, v,∑

αiMi +N) = α(u, v,∑

αiMi) +Nφ.

Then

w′ = β(u, v)+α(u, v,∑

αiMi)(∑

αiMi)+N [α(u, v,∑

αiMi)+(∑

αiMi)φ+Nφ].

Now as in the calculation we make in the verification that τ1 = τ in the case whenp is a 3-point and q is a 3-point, we see that there exist permissible parametersx, y, z for u, v, w′ at p such that τ1 = τ (where τ is computed for u, v, w and x, y, zand τ ′ is computed for u, v, w′ and x, y, z). Thus τ(p) is independent of choice ofpermissible parameters at q and p when p is a 3-point and q is a 2-point.

Now assume that p is a 1-point and q = f(p) is a 2-point or a 3-point.


Invariance of τf (p) is not difficult to prove when there exists a computation ofτf (p) which gives −∞, so we will assume that τf (p) ≥ 1 for all computations ofτf (p).

Assume that p is a 1-point, and that q = f(p) is a 2-point or a 3-point. Supposethat u, v, w are permissible parameters at q, such that u, v, w have an expression ofthe form (4.1) and (9.9), or of the form 2 (b) of Definition 4.6 and (9.10).

We will first show that for fixed permissible parameters u, v, w at q, τf (p) isindependent of choice of permissible parameters x, y, z for u, v, w at p which havean expression of the form (4.1) and (9.9).

We have that u, v, w have an expression

(9.14)u = xa

v = xb(α+ y)w =

∑nj=0 αij (y)x

ij + xc(z + β)

where ij < c for all j, and αij (y) 6= 0 for all j.If x, y, z are other permissible parameters for u, v, w at q, which also give an

expansion of the form (4.1) and (9.9), then we have x = ωx where ωa = 1, andthus y = ω−by, z = ω−cz. Thus the computation of τf (p) for x, y, z is the same asfor x, y, z.

By a similar calculation, for fixed permissible parameters u, v, w at q, τ isindependent of choice of permissible parameters x, y, z for u, v, w at p which havean expression of type 2 (b) of Definition 4.6 and (9.10) at p.

Now suppose that q is a 2-point, and u, v, w admit expressions of both theform 2 (b) of Definition 4.6 and of the form (4.1) and (9.9) at p. Let τ1 be thecomputation of τf (p) from a representation of u, v, w in the form 2 (b) of Definition4.6, and let τ2 be the computation of τf (p) from a representation of u, v, w in theform of (4.1) and (9.9).

Since u, v, w have an expression of type 2 (b) at p, there are permissible pa-rameters x, y, z at p such that there is an expression

(9.15)u = xa

v =∑nj=0 αij (y)x

ij + xc(z + β)w = y

of the form (9.10), where αij are all nonzero and ij < c for all j. τ1 is the com-putation of τf (p) for this expression. Since u, v, w also have an expression of type(4.1) at p, u, v are toroidal forms at p. Since τ1 ≥ 1, we have

v = (α+ yλ(y))xi0 + αi1(y)xi1 + · · ·+ αin(y)xin + xc(z + β)

where 0 6= α = αi0(0) ∈ k and λ(y) is a unit series. Set

y = yλ(y) + αi1(y)xi1−i0 + · · ·+ αin(y)xin−i0 + xc−i0(z + β).

Then(9.16)y = λ(y)−1y−λ(y)−1αi1(y)x

i1−i0 −· · ·−λ(y)−1αin(y)xin−i0 −xc−i0λ(y)−1(z+β).

Iterate, substituting (9.16) into successive iterations of (9.16), to get a series P suchthat

y = P (y, xi1−i0 , . . . , xin−i0) + xc−i0Ω

for some Ω ∈ OX,p.


We compute the Jacobian determinant

(9.17) Det(∂(u, v, w)∂(x, y, z)

)= −axa+c−1.

We haveu = xa

v = xi0(y + α)w = P (y, xi1−i0 , . . . , xin−i0) + xc−i0Ω,

and

(9.18) Det(∂(u, v, w)∂(x, y, z)

)= axa+c−1 ∂Ω

∂z.

Since (9.17) and (9.18) differ by multiplication of a unit series, ∂Ω∂z (0, 0, 0) 6= 0. Set

β = Ω(0, 0, 0), z = Ω− β. x, y, z are regular parameters in OX,p. We have

u = xa

v = xi0(y + α)w = P (y, xi1−i0 , . . . , xin−i0) + xc−i0(β + z).

τ2 is the value of τf (p) for this expression, computed from (9.9). We have

(9.19)τ2 ≤ |(aZ + i0Z +

∑nj=1(ij − i0)Z)/(aZ + i0Z)|

= |(aZ +∑nj=0 ijZ)/(aZ + i0Z)|

= τ1.

We now show that τ1 ≤ τ2. Since u, v, w have an expression of type (4.1) at p,we have an expression

(9.20)u = xa

v = xb(α+ y)w = P (x, y) + xc(z + β)

of the form (9.9), where

P = αi0(y) + αi1(y)xi1 + · · ·+ αin(y)xin ,

αij (y) 6= 0 for all j, and in < c.τ2 is the computation of τf (p) for this expression. Since τ2 ≥ 1, and u, v, w also

have an expression of type 2 (b) of Definition 4.6, we have ∂P∂y (0, 0) 6= 0. We then

haveαi0(y) = yλ(y)

where λ is a unit series. Set z = P + xc(z + β). z, x, z are regular parameters inOX,p. We have(9.21)y = λ(y)−1[z − αi1(y)xi1 − · · · − αin(y)xin − xc(z + β)]

= λ(y)−1z − λ(y)−1αi1(y)xi1 − · · · − λ(y)−1αin(y)xin − λ(y)−1xc(z + β).

Iterate, substituting (9.21) into successive iterations of (9.21), to get series P andΩ such that

y = P (z, xi1 , . . . , xin) + xcΩ(z, x, z).Thus we have an expression

u = xa

v = xb(α+ P (z, xi1 , . . . , xin)) + xb+cΩw = z.


We compute the Jacobian determinant from (9.20),

Det(∂(u, v, w)∂(x, y, z)

)= axa+b+c−1.

We compare with

Det(∂(u, v, w)∂(x, z, z)

)= axa+b+c−1 ∂Ω

∂z,

to see that ∂Ω∂z (0, 0, 0) 6= 0. Set β = Ω(0, 0, 0), y = Ω − β. x, y, z are regular

parameters in OX,p. We have

u = xa

v = xb(α+ P (z, xi1 , . . . , xin)) + xb+c(β + y)w = z

in the form of 2 (b) of Definition 4.6. τ1 is the value of τf (p) for this expression.We compute from (9.10),

(9.22)τ1 ≤ |(aZ + bZ +

∑nj=0 ijZ)/(aZ + bZ)|

= τ2.

By (9.19) and (9.22) we see that τ1 = τ2.From our calculations so far, we conclude that for fixed u, v, w, assuming that

p is a 1-point and q is a 2-point or a 3-point, τf (p) is independent of choice ofpermissible parameters x, y, z for u, v, w at p.

Now we will show that if u1, v1, w1 are a permutation of u, v, w such thatu1, v1, w1 have one of the forms (4.1) and (9.9) or 2 (b) of Definition 4.6 and (9.10)at p, then we obtain the same value of τf (p).

First suppose that u1 = v, v1 = u, w1 = w, and u, v are toroidal forms at p (sothat (4.1) holds). Then u, v, w have an expression (9.14) at p. We have

u1 = v = xb

v1 = u = xa(α+ y)

where x = x(α+y)−1b , α = α−

ab , and y = λ(y)y for an appropriate unit series λ(y).

Setβ = βα−

cb , z = (z + β)(α+ λ(y)y)−

cb − β.

We have

w =∑nj=0 αij (λ(y)y)(α+ λ(y)y)−

ijb xij + xc(z + β)(α+ λ(y)y)−

cb

=∑nj=0 αij(y)x

ij + xc(β + z)

where αij(y) 6= 0 for all j.Let τ1 be the calculation of τf (p) for the variables u, v, w, and let τ2 be the

calculation of τ for the variables u1, v1, w1. We see that τ1 = τ2.Now suppose that u1 = v, v1 = u,w1 = w, and u, v, w have an expression of

the form 2 (b) of Definition 4.6 and (9.10). u, v, w have an expression (9.15) at p.We have

u = xa

v = xi0(∑nj=0 αij (y)x

ij−i0 + xc−i0(z + β))w = y


where αi0(y) is a unit series. Set

γ =n∑j=0

αij (y)xij−i0 + xc−i0(z + β).

Define x by

(9.23) x = xγ−1i0 .

We have a series P and Ω ∈ OX,p such that

u1 = v = xi0

v1 = u = xaγ−ai0 = xa(P (y, xi1−i0 , . . . , xin−i0) + xc−i0Ω)

w = y

By iterating substitution of (9.23) in P (y, xi1−i0 , . . . , xin−i0), we see that there is aseries P (y, xi1−i0 , . . . , xin−i0) and Ω ∈ OX,p such that

v1 = xa(P (y, xi1−i0 , . . . , xin−i0) + xc−i0Ω).

Since the Jacobians

Det(∂(u, v, w)∂(x, y, z)

)= −axa+c−1

and

Det(∂(u1, v1, w1)∂(x, y, z)

)= −i0xa+c−1 ∂Ω

∂z

differ by multiplication by a unit, ∂Ω∂z (0, 0, 0) 6= 0.

Setβ = Ω(0, 0, 0), z = Ω− β.

Then x, y, z are regular parameters in OX,p, and

u1 = xi0

v1 = xaP (y, xi1−i0 , . . . , xin−i0) + xa+c−i0(β + z)w = y.

Let τ1 be the calculation of τf (p) for the variables u, v, w, and let τ2 be thecalculation of τf (p) for the variables u1, v1, w1 (from (9.10). We see that τ2 ≤ τ1.From this argument for the change of variables u1, v1, w1 to u, v, w, we see thatτ1 ≤ τ2, so that τ1 = τ2.

Now suppose that q is a 3-point, u, v, w have an expression (4.1) and (9.9), andu1 = w, v1 = v, w1 = u also have an expression (4.1) and (9.9).

We have

(9.24)u = xa

v = xb(α+ y)w = xi0(

∑nj=0 αij (y)x

ij−i0 + xc−i0(z + β))

where in < c, αij (y) 6= 0 for all j, and αi0(y) is a unit series. Set

γ =n∑j=0

αij (y)xij−i0 + xc−i0(z + β).


Set x = xγ−1i0 . We have

u1 = w = xi0

v1 = v = xbγ−b

i0 (α+ y)w1 = u = xaγ−

ai0 .

As we are assuming that all calculations of τf (p) are ≥ 1, we have that c > 0.Since w, v are assumed to be toroidal forms at p, we then have

∂

∂y(γ−

bi0 (α+ y))(0, 0, 0) 6= 0.

Setα = γ−

bi0 (0, 0, 0)α, y = γ−

bi0 (α+ y)− α.

We have

(9.25) x = x[(α+ y)−1(y + α)]1b .

There exists a unit series λ(y), series αr1,...rn(y), and Ω ∈ OX,p, such that

y = yλ(y) +∑

r1,...,rn>0

αr1,...,rn(y)x(i1−i0)r1 · · ·x(in−i0)rn + xc−i0Ω.

Thus

(9.26)y = yλ(y)−1 −

∑r1,...,rn>0

λ(y)−1αr1,...,rn(y)x(i1−i0)r1 · · ·x(in−i0)rn − xc−i0λ(y)−1Ω.

There exists a series

(9.27) P (y, xi1−i0 , . . . , xin−i0)

such that u = xa(P + xc−i0Ω) for some Ω ∈ OX,p.Substituting (9.25) and (9.26) into successive iterations of (9.27), we see that

there is a series P (y, xi1−i0 , . . . , xin−i0) and Ω′ ∈ OX,p such that

w1 = u = xa(P (y, xi1−i0 , . . . , xin−i0) + xc−i0Ω′).

Thus there is an expression

(9.28)u1 = w = xi0

v1 = v = xb(α+ y)w1 = u = xa(P (y, xi1−i0 , . . . , xin−i0) + xc−i0Ω′).

We compute the Jacobian determinants

Det(∂(u, v, w)∂(x, y, z)

)= axa+b+c−1

and

Det(∂(u1, v1, w1)∂(x, y, z)

)= i0x

a+b+c−1 ∂Ω′

∂z.

Thus ∂Ω′

∂z (0, 0, 0) 6= 0. Set

β = Ω′(0, 0, 0), z = Ω′ − β.


x, y, z are regular parameters in OX,p. We have an expression

(9.29)u1 = w = xi0

v1 = v = xb(α+ y)w1 = u = xaP (y, xi1−i0 , . . . , xin−i0) + xa+c−i0(β + z)

of the form (9.9).Let τ1 be the computation of τf (p) from (9.24), τ2 be the computation of τf (p)

from (9.29). We see that τ2 ≤ τ1.By this argument applied to the change of variables u1 = w, v1 = v, w1 = u to

u, v, w, we see that τ1 ≤ τ2, so that τ1 = τ2.There is a similar calculation if u1 = u, v1 = w,w1 = v.Now all other permutations u1, v1, w1 of u, v, w such that u1, v1, w1 have a form

(4.1) and (9.9) at p (so that u1, v1 are toroidal forms at p) are a composition ofchange of variables of the three types analyzed above. We have shown that τf (p)is invariant under such change of variables. Hence we get the same evaluation ofτf (p) for all possible permuations of the variables u, v, w and u1, v1, w1.

Now assume that u1 = u, v1 = v and w1 are permissible parameters at q.Suppose that u, v, w have an expression of the form (4.1) at p. We then have an

expression (9.14) of u, v, w. Let τ1 be the computation of τf (p) for this expression.Set

P =n∑j=0

αij(y)xij .

We have an expansion

(9.30) w1 =∞∑i=0

gi(u, v)wi

and w1 = wγ where γ(u, v, w) is a unit series if q = f(p) is a 3-point.Substitute w = P + xc(z + β) into (9.30) to get

w1 =∞∑i=0

i∑j=0

gi(u, v)(i

j

)P i−jxjc(z + β)j .

We see that there exists a series P (y, xa, xb, xi0 , . . . , xin) such that

w1 = P (y, xa, xb, xi0 , . . . , xin) + xcΩ

for some series Ω. Comparing the Jacobian determinants

Det(∂(u, v, w)∂(x, y, z)

)= axa+b+c−1

and

Det(∂(u, v, w1)∂(x, y, z)

)= axa+b+c−1 ∂Ω

∂z,

we see that ∂Ω∂z (0, 0, 0) 6= 0. Set

β = Ω(0, 0, 0), z = Ω− β.We have

u = xa

v = xb(α+ y)w1 = P (y, xa, xb, xi0 , . . . , xin) + xc(z + β)


of the form of (9.9). Further, we have w1 = xi0γ where γ(x, y, z) is a unit seriesif q = f(p) is a 3-point. Let τ2 be the computation of τf (p) for this expression.We see that τ2 ≤ τ1. Now applying this argument to the change of variables fromu, v, w1 to u, v, w, we see that τ1 = τ2.

Now suppose that u, v, w have an expression of the form 2 (b) of Definition 4.6and (9.10) (so that q = f(p) is a 2-point). We have

u = xa

v =∑nj=0 αij (y)x

ij + xc(z + β)w = y

where αij (y) are all nonzero and ij < c for all j.Let τ1 be the computation of τf (p) for u, v, w.By the Weierstrass preparation theorem, there exists a unit series γ(u, v, w)

and a series φ(u, v) with φ(0, 0) = 0 such that w1 = γ(w − φ). We have

w1 = γ(0, 0, y)y + xΩ′

for some series Ω′(x, y, z). We may thus set y = w1, and have that x, y, z are regularparameters in OX,p.

There exist a unit series α0(y) and series αr0,...,rn(y) such that there is anexpression

y = α0(y)y +∑

s+r0+···+rn>0

αr0,...,rn(y)xas+i0r0+···+inrn + xcΩ

with α0(0) 6= 0.We thus have an expression

(9.31)y = α0(y)−1y −

∑s+r0+···+rn>0

α0(y)−1αr0,...,rn(y)xas+i0r0+···+inrn + xcα0(y)−1Ω.

We substitute (9.31) into successive iterations of

v =n∑j=0

αij (y)xij + xc(z + β)

to obtain an expression

v = P (y, xa, xi0 , . . . , xin) + xcΩ.

Comparing the Jacobian determinants

Det(∂(u, v, w)∂(x, y, z)

)= −axa+c−1

and

Det(∂(u, v, w1)∂(x, y, z)

)= −axa+c−1 ∂Ω

∂z,

we see that ∂Ω∂z (0, 0, 0) 6= 0. Set

β = Ω(0, 0, 0), z = Ω− β.

x, y, z are regular parameters in OX,p, and we have an expression

u = xa

v = P (y, xa, xi0 , . . . , xin) + xc(z + β)w1 = y


of the form of (9.10). Let τ2 be the computation of τf (p) for u, v, w1. We haveτ2 ≤ τ1. Now applying this argument to the change of variables from u, v, w1 tou, v, w, we see that τ1 = τ2.

Using the above techniques, we can show that we have the same computationof τf (p) for a change of variables

u1 = λ1u, v1 = λ2v, w1 = w

where λ1, λ2 are unit series.Finally, suppose that u1, v1, w1 are arbitrary permissible parameters at q. Then

u1, v1, w1 may be obtained from u, v, w be a series of changes of variables of thetypes considered above. It follows that the computation of τf (p) for the variablesu, v, w and u1, v1, w1 are the same.

We leave to the reader the verification of the remaining cases (p a 1-point andq a 1-point, p a 2-point).

Lemma 9.3. Suppose that f : X → Y is prepared. Then τf is upper semicontinuous on DX .

Proof. This follows from a local calculation, computing a deformation in etalelocal coordinates of the possible prepared forms.

Suppose that p∗ ∈ X. We must find a Zariski open neighborhood U of p∗ in Xsuch that p ∈ U ∩DX implies τf (p) ≤ τf (p∗).

We work out the case when p∗ is a 3-point. The other cases are similar. Sup-pose that p∗ ∈ X is a 3-point, τf (p∗) ≥ 1 and there exist (algebraic) permissibleparameters u, v, w at q∗ = f(p∗) such that u, v are toroidal forms at p∗. Then thereexist regular parameters x, y, z ∈ OX,p∗, and units γ1, γ2 ∈ OX,p∗ such that

u = xaybzcγ1

v = xdyezfγ2

with

rank(a b cd e f

)= 2.

There exist rational numbers aij such that if we set

x = xγa111 γa12

2 , y = xγa211 γa22

2 , z = xγa311 γa32

2 ,

thenu = xaybzc, v = xdyezf .

From the Jacobian determinant

Det(∂(u, v, w)∂(x, y, z)

)we see that we have an expansion

(9.32)u = xaybzc

v = xdyezf

w =∑i≥0 αix

aiybizci + xgyhziγ

where γ is a unit series,

Det

a b cd e fg h i

6= 0,


and for all i, 0 6= αi ∈ k,

Det

a b cd e fai bi ci

= 0,

and (ai, bi, ci) 6≥ (g, h, i).We can compute τf (p∗) by changing variables, multiplying x, y, z by appropriate

rational powers of γ. We obtain that

u = xaybzc

v = xdyezf

w =∑i≥0 αix

ai ybi zci + xg yhzi.

Thus

τf (p∗) = |((a, b, c)Z + (d, e, f)Z +∑i≥0

(ai, bi, ci)Z)/((a, b, c)Z + (d, e, f)Z)|

if q∗ is a 2-point,

τf (p∗) = |((a, b, c)Z+(d, e, f)Z+∑i≥0

(ai, bi, ci)Z)/((a, b, c)Z+(d, e, f)Z+(a0, b0, c0)Z)|

if q∗ is a 3-point.We compute the Jacobian

(9.33) Det(∂(u, v, w)∂(x, y, z)

)= Det

a b cd e fg h i

xa+d+g−1yb+e+h−1zc+f+i−1.

Let e be a common denominator of the aij . There exists an affine neighborhoodU = spec(A) of p∗ such that x, y, z are uniformizing parameters on U , and γ1, γ2

are units on U . LetB = A[γ

1e1 , γ

1e2 ],

V = spec(B). After possibly shrinking U to a smaller neighorhood of p∗, we mayassume that V is an etale cover of U and x, y, z are uniformizing parameters on V .Let Λ : V → U be the natural morphism.

Suppose that p ∈ DX ∩U is a 2-point. We will show that τf (p) ≤ τf (p∗). Afterpossibly interchanging x, y, z, we may assume that OU,p has regular parametersx, y, z = z − α for some 0 6= α ∈ k. After interchanging u, v if necessary, we havethree possible cases:

1. ae− bd 6= 0,2. ae− bd = 0, (a, b) 6= (0, 0) and (d, e) 6= (0, 0),3. (d, e) = 0.

We will analyze these three cases in turn.

Suppose that ae − bd 6= 0, Define (formal) regular parameters x, y, z at p bychoosing λ1, λ2 ∈ Q so that

x = xzλ1 , y = yzλ2 , z = z + α

satisfyu = xayb, v = xdye.


We thus have an expression

(9.34) u = xayb, v = xdye, w = P (x, y) + xmynΩ

for some series P (x, y) and Ω ∈ OX,p, where ∂Ω∂z (p) 6= 0.

We compute the Jacobian determinant

Det(∂(u, v, w)∂(x, y, z)

)= (ae− bd)∂Ω

∂zxa+d+m−1yb+e+n−1

to see that ∂Ω∂z (p) 6= 0. Comparing with (9.33), we see that m = g and n = h.

We compute that

∂

∂x= zλ1

∂

∂x,∂

∂y= zλ2

∂

∂y,∂

∂z= λ1

x

z

∂

∂x+ λ2

y

z

∂

∂y+

∂

∂z.

Thus for m,n ∈ N,∂m+nw

∂xm∂yn= zmλ1+nλ2

∂m+nw

∂xm∂yn.

From (9.32) we see that ∂m+nw∂xm∂yn ∈ (x, y) if (m,n) 6= (ai, bi) for any i, and

(m,n) 6≥ (g, h). We have an expansion

P (x, y) =∑

(m,n) 6≥(g,h)

1m!n!

∂m+nw

∂xm∂yn(p)xmyn.

Thus there is an expansionP =

∑i≥0

βixaiybi

with

βi = αmλ1+nλ2∂m+nw

∂xm∂yn(p)

if (m,n) = (ai, bi).We conclude that we may make a formal change of variables in (9.34), setting

β = Ω(p) and z∗ = Ω− β, to get

(9.35)u = xayb

v = xdye

w =∑βix

aiybi + xgyh(z∗ + β).

We now compare τf (p) to τf (p∗). Let q = f(p).Suppose that q∗ is a 3-point. If q is a 2-point, then either (a0, b0) = (0, 0), or

g = h = 0 in (9.35). (a0, b0) = (0, 0) implies ae−bd = 0 which is not possible. Thusg = h = 0, and we see that τf (p) = −∞ ≤ τf (p∗). Suppose that q is a 3-point.Then we compute τf (p) from (9.35) and (9.3), to get

τf (p) = |((a, b)Z + (d, e)Z +∑βi 6=0,(ai,bi) 6≥(g,h)(ai, bi)Z)

/((a, b)Z + (d, e)Z + (a0, b0)Z)|≤ τf (p∗).

Suppose that q∗ is a 2-point. Then q is a 2-point, and we have from (9.35) and(9.3) that

τf (p) = |((a, b)Z + (d, e)Z +∑βi 6=0,(ai,bi) 6≥(g,h)(ai, bi)Z)/((a, b)Z + (d, e)Z)|

≤ τf (p∗).


Suppose that ae− bd = 0, (a, b) 6= (0, 0) and (d, e) 6= (0, 0).There exist a, b, t, k such that (a, b) = k(a, b), (d, e) = t(a, b) with gcd(a, b) = 1,

k, t 6= 0.Define (formal) regular parameters x, y, z at p by choosing λ1, λ2, λ3 ∈ Q so

thatx = xzλ1 , y = yzλ2 , zλ3 = z + α

satisfy

u = (xayb)k, v = (xayb)t(z + α).


(9.36) u = (xayb)k, v = (xayb)t(z + α), w = P (xayb, z) + xmynΩ

where Ω is a unit series and an− bm 6= 0.We compute the Jacobian determinant

Det(∂(u, v, w)∂(x, y, z)

)= (xayb)t+k−1xm+a−1yn+b−1γ,

where

γ = k(bm− an)Ω + kbx∂Ω∂x− kay ∂Ω

∂y

is a unit series. Comparing with (9.33), we see that m = g and n = h.We compute that

∂

∂x= zλ1

∂

∂x,∂

∂y= zλ2

∂

∂y,∂

∂z=λ1

λ3xz−λ3

∂

∂x+λ2

λ3yz−λ3

∂

∂y+

1λ3z1−λ3

∂

∂z.



∂m+nw

∂xm∂yn.

From (9.32) we see that ∂m+nw∂xm∂yn ∈ (x, y) if (m,n) 6= (ai, bi) for any i, and

(m,n) 6≥ (g, h). There is thus an expansion

P (xayb, z) =∑j,k

βj,k(xayb)jzk

with

βj,k =1

k!(ja)!(jb)!∂k

∂zk

(∂j(a+b)w

∂xja∂yjb

)(p).

Thus βj,k = 0 if (ja, jb) 6= (ai, bi) for any i, and (ja, jb) 6≥ (g, h).We conclude that we may make a formal change of variables in (9.36), setting

x = x∗γ1 and y = y∗γ2,

with appropriate unit series γ1 and γ2, to get

(9.37)u = ((x∗)a(y∗)b)k

v = (x∗)a(y∗)b)t(z + α)w = P ((x∗)a(y∗)b, z) + (x∗)g(y∗)h.

We now compare τf (p) to τf (p∗). Let q = f(p).


Suppose that q∗ is a 3-point and q is a 3-point. We compute τf (p) from (9.37)and (9.4), to get

τf (p) = |(kZ + tZ +∑

βj,k 6=0,(ja,jb) 6≥(g,h)

jZ)/(kZ + tZ + j0Z)|,

where j0(a, b) = (a0, b0). We have a surjection

((a, b, c)Z + (d, e, f)Z +∑

(ai, bi, ci)Z)/((a, b, c)Z + (d, e, f)Z + (a0, b0, c0)Z)→ (kZ + tZ +

∑j(a,b)=(ai,bi) for some i jZ)/(kZ + tZ + j0Z)

defined by j(a, b, 0) + l(0, 0, 1) 7→ j. Thus τf (p) ≤ τf (p∗).If q∗ is a 3-point and q is a 2-point, then we must have that (a0, b0) = (0, 0) or

g = h = 0. But we cannot have g = h = 0 since

Det

a b cd e fg h i

6= 0.

If (a0, b0) = (0, 0), then we have (a0, b0, c0) = (0, 0, c0), from which we concludeby comparison with (9.37) and (9.4) that τf (p) ≤ τf (p∗).

Suppose that q∗ is a 2-point. Then q is a 2-point.We compute τf (p) from (9.37) and (9.4), to get

τf (p) = |(kZ + tZ +∑

βj,k 6=0,(ja,jb) 6≥(g,h)

jZ)/(kZ + tZ)|.

As in the case when q∗ is a 3-point, we conclude that τf (p) ≤ τf (p∗).

Suppose that d = e = 0. We have (a, b) = k(a, b) with a, b > 0 and gcd(a, b) = 1.Define regular parameters x, y, z in OX,p by choosing λ1, λ2 ∈ Q and α = αf ,

so thatx = xzλ1 , y = yzλ2 , zf = z + α

satisfyu = (xayb)k, v = z + α.


(9.38) u = (xayb)k, v = v − α = z, w = P (xayb, z) + xmynΩ

where u, v are toroidal forms at q = f(p), P is a series, Ω is a unit series andan− bm 6= 0.

We compute the Jacobian

Det(∂(u, v, w)∂(x, y, z)

)= xa+m−1yb+n−1γ.

where

γ = (mb− an)Ω + bx∂Ω∂x− ay ∂Ω

∂y

is a unit series. Comparing with (9.33), we see that m = g and n = h.We compute that

∂

∂x= zλ1

∂

∂x,∂

∂y= zλ2

∂

∂y,∂

∂z=λ1

f

x

zf∂

∂x+λ2

f

y

zf∂

∂y+

1fz1−f ∂

∂z.




∂m+nw

∂xm∂yn.

From (9.32) we see that ∂m+nw∂xm∂yn ∈ (x, y) if (m,n) 6= (ai, bi) and (m,n) 6≥ (g, h).

There is an expansionP =

∑j,k

βj,k(xayb)jzk

with

βj,k =1

k!(ja)!(jb)!∂k

∂zk

(∂j(a+b)w

∂xja∂yjb

)(p).

Thus βj,k = 0 if (ja, jb) 6= (ai, bi) for any i, and (ja, jb) 6≥ (g, h).We conclude that we may make a formal change of variables in (9.38), setting

x = x∗γ1 and y = y∗γ2,

with appropriate unit series γ1 and γ2, to get

(9.39)u = ((x∗)a(y∗)b)k

v = v − α = z

w = P ((x∗)a(y∗)b, z) + (x∗)g(y∗)h.

We now compare τf (p) to τf (p∗). Let q = f(p).Suppose that q∗ is a 3-point. Then q is a 1-point or a 2-point. If q is a 2-point,

then u,w, v are permissible parameters at q such that there is an expression

u = ((x∗)a(y∗)b)k

w = P ((x∗)a(y∗)b, z) + (x∗)g(y∗)h

v = z

of the form of 2 (c) of Definition 4.6, and (9.5).We have

τf (p) = |(kZ +∑

βj,k 6=0,(ja,jb) 6≥(g,h)

jZ)/(kZ + j0Z)|,

where j0(a, b) = (ai0 , bi0).We thus have a surjection

((a, b, c)Z + (d, e, f)Z +∑

(ai, bi, ci)Z)/((a, b, c)Z + (d, e, f)Z + (a0, b0, c0)Z)→ (kZ +

∑j(a,b)=(ai,bi) for some i jZ)/(kZ + j0Z)

defined by j(a, b, 0) + l(0, 0, 1) 7→ j. Thus τf (p) ≤ τf (p∗).If q∗ is a 3-point and q is a 1-point, then we have permissible parameters u, v, w

at q, defined by

u = ((x∗)a(y∗)b)k

v = z

w = w − β = P ((x∗)a(y∗)b, z)− β + (x∗)g(y∗)h

of the form (4.6) and (9.6), where 0 6= β = P (0, 0) so that (aj0 , bj0) = (0, 0).We have

τf (p) = |(kZ +∑

βj,k 6=0,(ja,jb) 6≥(g,h)

jZ)/kZ|.


We have a surjection

((a, b, c)Z + (d, e, f)Z +∑

(ai, bi, ci)Z)/((a, b, c)Z + (d, e, f)Z + (a0, b0, c0)Z)→ (kZ +

∑j(a,b)=(ai,bi) for some i jZ)/(kZ + j0Z)

defined by j(a, b, 0) + l(0, 0, 1) 7→ j. Thus τf (p) ≤ τf (p∗).There is a similar analysis if q∗ is a 2-point.We have completed the analysis for a 2-point p ∈ DX ∩ U .Now suppose that p ∈ DX ∩ U is a 1-point. Recall that we are assuming that

p∗ is a 3-point. We will show that τf (p) ≤ τf (p∗). After possibly interchangingx, y, z, we may suppose that OU,p has regular parameters x, y − α, z − β for some0 6= α ∈ k, 0 6= β ∈ k. After interchanging u, v and y, z if necessary, we may assumethat a 6= 0 and ea− bd 6= 0.

Define regular parameters x, y, z in OX,p by choosing λij ∈ Q, α = αλ21βλ22 ,so that

x = xyλ11zλ12 , y + α = yλ21zλ22 , z + β = z

satisfyu = xa, v = xd(y + α).

We may do this by setting

λ11 = − ba, λ12 = − c

a,

λ21 = dλ11 + e =ea− bd

a6= 0, λ22 = dλ12 = f.


(9.40)u = xa

v = xd(y + α)w = P (x, y) + xmΩ

where P (x, y) is a series and ∂Ω∂z (p) 6= 0.

We compute the Jacobian

Det(∂(u, v, w)∂(x, y, z)

)= axa+d+m−1 ∂Ω

∂z.

Comparing with (9.33), we see that m = g.We have

x = x(y + α)λ11λ21 (z + β)λ12−λ11λ22

λ21

y = (y + α)1

λ21 (z + β)−λ22λ21

z = z + β,

from which we see that

∂∂x = yλ11zλ12 ∂

∂x∂∂y = λ11

λ21xy−λ21z−λ22 ∂

∂x + 1λ21

y1−λ21z−λ22 ∂∂y

∂∂z =

(λ12λ21−λ11λ22

λ21

)xz−1 ∂

∂x −λ22λ21

yz−1 ∂∂y + ∂

∂z .

Thus

∂mw

∂xm= ymλ11zmλ12

∂mw

∂xm.


From (9.32) we see that ∂mw∂xm ∈ (x) if m 6= ai for some i and m 6≥ g. There is

an expansionP =

∑j,k

βj,kxjyk

with

βj,k =1k!j!

∂k

∂yk

(∂jw

∂xj

)(p).

Thus βj,k = 0 if j 6= ai and j 6≥ g.We may make a formal change of variables in (9.40), setting β = Ω(p) and

z∗ = Ω− β, to get

(9.41)u = xa

v = xd(y + α)w =

∑βj,kx

jyk + xg(z∗ + β).

We now compare τf (p) to τf (p∗). Let q = f(p).Suppose that q∗ is a 3-point. If q is a 2-point then we have that a0 = 0, and

by (9.9), we have

τf (p) = |(aZ + dZ +∑

βjk 6=0,j<g

jZ)/(aZ + dZ)| ≤ τf (p∗).

If q∗ is a 3-point and q is a 3-point, then by (9.9), we have

τf (p) = |(aZ + dZ +∑

βjk 6=0,j<g

jZ)/(aZ + dZ + a0Z)| ≤ τf (p∗).

Suppose that q∗ is a 2-point. Then q is a 2-point, and by a similar calculation,τf (p) ≤ τf (p∗).

This completes the analysis that τf (p) ≤ τf (p∗) if p∗ is a 3-point and p ∈ U .The analysis when p∗ is a 2-point or a 1-point is simpler, and we leave it to the

reader.We conclude that τf is upper semi-continuous.

Definition 9.4. Suppose that f : X → Y is a prepared, proper morphism,and τ ∈ N is such that τf (X) ≤ τ . Let GX(f, τ) = p ∈ DX | τf (X) = τ,GY (f, τ) = f(GX(f, τ)). We will say that f is τ -prepared if GY (f, τ) contains no2-curves and no 3-points.

By Lemma 9.3, GX(f, τ) is a closed subset of X, and since f is proper, GY (f, τ)is a closed subset of Y .

CHAPTER 10

Super parameters

Throughout this chapter, we assume that f : X → Y is a dominant, propermorphism of nonsingular 3-folds.

Definition 10.1. Assume that f : X → Y is prepared.Suppose that q ∈ Y is a 2-point. Permissible parameters u, v, w at q are super

parameters for f at q if at all p ∈ f−1(q), there exist permissible parametersx, y, z for u, v, w at p such that we have one of the forms:

1. p is a 1-point

(10.1)u = xa

v = xb(α+ y)w = xcγ(x, y) + xd(z + β)

where γ is a unit series (or zero), 0 6= α ∈ k and β ∈ k,2. p is a 2-point of the form of (4.2) of Definition 4.1

(10.2)u = xayb

v = xcyd

w = xeyfγ(x, y) + xgyh(z + β)

where ad− bc 6= 0, γ is a unit series (or zero), and β ∈ k.3. p is a 2-point of the form of (4.3) of Definition 4.1

(10.3)u = (xayb)k

v = (xayb)t(α+ z)w = (xayb)lγ(xayb, z) + xcyd

where 0 6= α ∈ k, gcd(a, b) = 1, ad − bc 6= 0 and γ is a unit series (orzero).

4. p is a 3-point

(10.4)u = xaybzc

v = xdyezf

w = xgyhziγ + xjykzl

where rank(u, v, xjykzl) = 3, rank(u, v, xgyhzi) = 2 and γ is a unit seriesin monomials M such that rank(u, v,M) = 2 (or γ is zero).

Suppose that q ∈ Y is a 1-point. Permissible parameters u, v, w at q are superparameters for f at q if at all p ∈ f−1(q), there exist permissible parametersx, y, z for u, v, w at p such that we have one of the forms:

5. p is a 1-point

(10.5)u = xa

v = yw = xcγ(x, y) + xd(z + β)

79

80 10. SUPER PARAMETERS

where γ is a unit series (or zero) and β ∈ k,6. p is a 2-point

(10.6)u = (xayb)k

v = zw = (xayb)lγ(xayb, z) + xcyd

where gcd(a, b) = 1, ad− bc 6= 0 and γ is a unit series (or zero).

Lemma 10.2. Suppose that f : X → Y is prepared, and Φ : X1 → X is theblow up of a 2-curve or a 3-point. Let f1 = f Φ : X1 → Y . Then f1 is prepared,and τf1(p) ≤ τf (Φ(p)) for all p ∈ DX1 .

If q ∈ Y and u, v, w are super parameters for f at q, then u, v, w are superparameters for f1 at q.

Proof. We prove this in the case when Φ : X1 → X is the blow up of a 2-curveC. The case when Φ is the blow up of a 3-point is similar.

Suppose that p ∈ C. Let q = f(p). Then there are permissible parametersu, v, w at q and x, y, z for u, v, w at p such that either u, v are toroidal forms at p,or a form 2 (c) of Definition 4.6 holds at q. Further, x = y = 0 are local equationsof C at p.

The most difficult case is when p is a 3-point, q = f(p) is a 3-point and τf (p) ≥1. The other cases are similar. Assume that this case holds.

We have an expansion of the form of (9.1)

(10.7)u = xaybzc

v = xdyezf

w =∑

(ai,bi,ci) 6≥(g,h,i) αixaiybizci + xgyhzi

with αi 6= 0 for all i. There are φi, ψi ∈ Q such that

(ai, bi, ci) = φi(a, b, c) + ψi(d, e, f)

for all i.Suppose that p1 ∈ Φ−1(p). Then (after possibly interchanging x and y) OX1,p1

has regular parameters x1, y1, z where

x = x1, y = x1(y1 + α)

for some α ∈ k. We have

u = xa+b1 (y1 + α)bzc

v = xd+e1 (y1 + α)ezf

w =∑

(ai,bi,ci) 6≥(g,h,i) αixai+bi1 (y1 + α)bizci + xg+h1 (y1 + α)hzi.

We may assume that τf1(p1) ≥ 1.

Case 1. Assume that 0 6= α and (a + b)f − c(d + e) 6= 0. There exist regularparameters x1, y1, z1 in OX1,p1 and β ∈ k, 0 6= αi ∈ k, such that

u = xa+b1 zc1v = xd+e1 zf1w =

∑(ai+bi,ci) 6≥(g+h,i) αix

ai+bi1 zci

1 + xg+h1 zi1(y1 + β)

10. SUPER PARAMETERS 81

of the form of (9.3). The homomorphism Λ : Z3 → Z2 defined by (x, y, z) 7→(x+ y, z) induces a surjection

((a, b, c)Z + (d, e, f)Z +∑i≥0(ai, bi, ci)Z)/((a, b, c)Z + (d, e, f)Z + (a0, b0, c0)Z)

→ ((a+ b, c)Z + (d+ e, f)Z +∑i≥0(ai + bi, ci)Z)/

((a+ b, c)Z + (d+ e, f)Z + (a0 + b0, c0)Z).

Thusτf (p) ≥ |((a+ b, c)Z + (d+ e, f)Z +

∑i≥0(ai + bi, ci)Z)

/((a+ b, c)Z + (d+ e, f)Z + (a0 + b0, c0)Z)|≥ |((a+ b, c)Z + (d+ e, f)Z +

∑(ai+bi,ci) 6≥(g+h,i)(ai + bi, ci)Z)

/((a+ b, c)Z + (d+ e, f)Z + (a0 + b0, c0)Z))|= τf1(p1).

Case 2. Assume 0 6= α, and (a+ b)f − c(d+ e) = 0. Then there exist a, b ∈ Nsuch that (a+ b, c) = k(a, b), (d+ e, f) = t(a, b) with k, t 6= 0, gcd(a, b) = 1.

There exist regular parameters x1, y1, z1 in OX1,p1 and 0 6= α ∈ k such that

u = xa+b1 zc1 = (xa1zb1)k

v = xd+e1 zf1 (y1 + α) = (xa1zb1)t(y1 + α)

w =∑

(ai+bi,ci) 6≥(g+h,i) αi(xa1zb1)φik+ψit(y1 + α)ψi + xg+h1 zi1

of the form of (9.4). We have

τf1(p1) ≤ |kZ + tZ +∑

(ai+bi,ci) 6≥(g+h,i)

(φik + ψit)Z)/(kZ + tZ + (φ0k + ψ0t)Z)|.

As in the argument of Case 1, we see that

τf (p) ≥ |((a+ b, c)Z + (d+ e, f)Z +∑i≥0(ai + bi, ci)Z)

/((a+ b, c)Z + (d+ e, f)Z + (a0 + b0, c0)Z)|= |(k(a, b)Z + t(a, b)Z +

∑i≥0(φik(a, b) + ψit(a, b)Z)

/(k(a, b)Z + t(a, b)Z + (φ0k(a, b) + ψ0t(a, b))Z))|= |(kZ + tZ +

∑i≥0(φik + ψit)Z)

/(kZ + tZ + (φ0k + ψ0t)Z)|≥ τf1(p1).

Case 3. 0 = α. There exist regular parameters x1, y1, z1 in OX1,p1 such thatWe have

u = xa+b1 yb1zc

v = xd+e1 ye1zf

w =∑

(ai+bi,bi,ci) 6≥(g+h,h,i) αixai+bi1 ybi

1 zci + xg+h1 yh1z

i1

of the form of (9.1). Thus

τf1(p1) = |((a+ b, b, c)Z + (d+ e, e, f)Z +∑

(ai+bi,bi,ci) 6≥(g+h,h,i)(ai + bi, bi, ci)Z)/((a+ b, b, c)Z + (d+ e, e, f)Z + (a0 + b0, b0, c0)Z)|≤ τf (p).

In all of the above cases, we see that if u, v, w are super parameters at q, thenu, v, w satisfy one of the cases (10.2) - (10.4) of Definition 10.1.


Lemma 10.3. Suppose that f : X → Y is prepared, q ∈ Y and u, v, w are superparameters at q. Then there exists a sequence of blow ups of 2-curves Φ : X1 → Xsuch that f1 = f Φ : X1 → Y is prepared, τf1(p) ≤ τf (Φ(p)) for p ∈ DX1 , u, v, ware super parameters at q for f1 and if p ∈ f−1

1 (q) with τf1(p) > 1, then w = 0 isa divisor supported on DX at p.

Proof. The fact that for a sequence of blow ups of 2-curves Φ : X1 → Y ,f1 = f Φ : X1 → Y is prepared and τf1(p) ≤ τf (Φ(p)) for p ∈ DX1 followsfrom Lemma 10.2. Further, the condition that u, v, w are super parameters at q ispreserved by blowup of 2-curves above X.

If p ∈ f−11 (q) is a 1-point, then 1 or 5 of Definition 10.1 hold, and if τf1(p) > 1,

then w = 0 is a divisor supported on DX at p.By Lemma 5.1, we may construct Φ so that if p ∈ f−1

1 (q), then (xeyf , xgyh)OX1,p

is principal if 2 of Definition 10.1 holds, ((xayb)l, xcyd)OX,p is principal if 3 holds,

(xgyhzi, xjykzl)OX1,p

is principal if 4 holds, ((xayb)l, xcyd)OX1,p is principal if 6 holds.We see that in all these cases that w = 0 is a divisor supported on DX1 at p if

τf1(p) > 1, so that the conclusions of the lemma hold.

Lemma 10.4. Suppose that f : X → Y is prepared and C ⊂ Y is a 2-curve.Then there exists a commutative diagram

X1f1→ Y1

Φ1 ↓ ↓ Ψ1

Xf→ Y

where Ψ1 : Y1 → Y is the blow up of C, Φ1 : X1 → X is a product of blow ups of2-curves, Φ1 is an isomorphism above f−1(Y − C) and f1 is prepared. If p1 ∈ X1

and p = Φ1(p1), thenτf1(p1) ≤ τf (p).

If f is τ -prepared then f1 is τ -prepared.

Proof. Let Ψ1 : Y1 → Y be the blow up of C. By Lemma 5.2, there exists acommutative diagram

X1f1→ Y1

Φ1 ↓ ↓ Ψ1

Xf→ Y

such that Φ1 : X1 → X is a sequence of blow ups of 2-curves, Φ1 is an isomorphismabove f−1(Y − C) and f1 is a morphism. Further, by Lemma 10.2, f Φ1 isprepared, and τfΦ1(p) ≤ τf (Φ1(p)) for p ∈ DX1 .

We will show that f1 : X1 → Y1 is prepared, and τf1(p) ≤ τf (Φ1(p)) forp ∈ DX1 .

Suppose that q ∈ C ⊂ Y and p ∈ (f Φ1)−1(q). Let q1 = f1(p), p′ = Φ1(p).

Suppose that q is a 3-point. Let u, v, w be permissible parameters at q. Thenafter possibly permuting u, v and w, we have that u, v, w have one of the forms(4.1) - (4.4) of Definition 4.1 at p′, and u, v, w also have one of the forms (4.1) -(4.4) of Definition 4.1 at p.


First assume q1 is a 3-point. Further assume that p is a 3-point. Then u, v, whave the form of (9.1) at p, and have a form (4.4) of Definition 4.1 at p. Furthermore,p′ is a 3-point. If τf (p′) = −∞, then τfΦ1(p) = −∞, and if τf (p) ≥ 1, thenτfΦ1(p) ≥ 1 and Hf,p′ = HfΦ1,p, Af,p′ = AfΦ1,p and τf (p′) = τfΦ1(p). Afterpossibly interchanging u and v, q1 has permissible parameters u1, v1, w1 such thatone of the following equations (10.8), (10.9) or (10.10) hold:

(10.8) u = u1, v = u1v1, w = w1

or

(10.9) u = u1, v = v1, w = u1w1

or

(10.10) u = u1w1, v = v1, w = w1.

Assume that (10.8) or (10.9) holds. Then u1, v1, w1 have a form (4.4) of Definition4.1 and (9.1) at p. If τfΦ1(p) = −∞, then τf1(p) = −∞, and if τfΦ1(p) ≥ 1, then

HfΦ1,p = Hf1,p,

AfΦ1,p = Af1,p and τf1(p) = τfΦ1(p) = τf (p′).Assume that (10.10) holds. If τfΦ1(p) = −∞, then we can interchange u, v

and w to obtain the case (10.8), which we have already analyzed, so we may assumethat τfΦ1(p) ≥ 1. There exist permissible parameters x, y, z for u, v, w at p suchthat u, v, w has an expression (9.1).

If rank(v,M0) = 2, then w, v is a toroidal form at p, so we have, after a changeof variables in x, y, z at p, an expression of w, v, u of the form of (9.1), and (byLemma 9.2) we obtain the same calculation of HfΦ1,p and AfΦ1,p for these newparameters. Thus (10.10) has been transformed into (10.9), from which it followsthat w1, v1, u1 have a form (4.4) of Definition 4.1 at p and τf1(p) = τfΦ1(p) =τf (p′).

If rank(v,M0) = 1, then rank(w, u) = 2 and w, u is a toroidal form at p. Asin the above paragraph, we change variables to obtain a form (10.8), from whichit follows that w1, u1, v1 have a form (4.4) of Definition 4.1 at p and τf1(p) =τfΦ1(p) = τf (p′).

Suppose that q1 is a 3-point and p is a 2-point. Then u, v, w are such that u, vare toroidal forms of type (4.2) or type (4.3) of Definition 4.1 at p, and w = Mγ

where M is a monomial in x, y, and γ is a unit series in OX1,p.First assume that u, v are of type (4.2) of Definition 4.1 at p. After possibly

interchanging u and v, q1 has permissible parameters u1, v1, w1 of one of the forms(10.8) (10.9) or (10.10). In any of these cases, we have an expression

u1 = M1γ1, v1 = M2γ2, w1 = M3γ3

where γ1, γ2, γ3 are unit series and M1,M2,M3 are monomials in x, y with

rank(M1,M2,M3) = 2.

Thus (after possibly interchanging u1, v1, w1) u1, v1, w1 have a form (4.2) of Defi-nition 4.1 at p.

Now assume that u, v is of type (4.3) of Definition 4.1 at p (and there does notexist a permutation of u, v, w such that u, v are of type (4.2) at p. We continue to


assume that q1 is a 3-point. Then there are permissible parameters x, y, z at p suchthat

u = (xayb)k

v = (xayb)t(α+ z)w = (xayb)l[γ(xayb, z) + xcyd]

where γ is a unit series and ad− bc 6= 0.If u = v = 0 are local equations for C at q then after possibly permuting u and

v, we may assume that q1 has permissible parameters u1, v1, w1 defined by

u = u1, v = u1v1, w = w1.

Thus u1, v1, w1 have a form (4.3) of Definition 4.1 at p. Otherwise, we can assume,after possibly interchanging u and v that u = w = 0 are local equations for C. Ifpermissible parameters are defined at q1 by

u = u1, v = v1, w = u1w1,

then u1, v1, w1 have a form (4.3) of Definition 4.1 at p.The remaining possibility is that u1, v1, w1 are permissible parameters at q1,

whereu = u1w1, v = v1, w = w1.

Thenu1 = (xayb)k−l[γ + xcyd]−1

v1 = (xayb)t(α+ z)w1 = (xayb)l[γ + xcyd].

Define new regular parameters x, y, z at p by x = xλx, y = yλy, where λx, λy areunit series in OX1,p such that

xayb = (γ + xcyd)−1l xayb.

Thenu1 = (xayb)k−l(γ + xcyd)−

kl

v1 = (xayb)t[γ + xcyd]−tl (α+ z)

w1 = (xayb)l.

If∂γ

∂z(0, 0, 0) 6= 0,

then w1, u1 have a form (4.3) of Definition 4.1 at p. If

∂γ

∂z(0, 0, 0) = 0,

then w1, v1 have a form (4.3) of Definition 4.1 at p.In all of these cases (q1 a 3-point and p a 2-point) we calculate that τf1(p) ≤

τfΦ1(p).There is a similar analysis (to the case when (4.3) holds at p) if p is a 1-

point (and q1 is a 3-point). After possibly permuting u, v, w, we find permissibleparameters u1, v1, w1 at q1 such that one of the forms (10.8) – (10.10) hold. As inthe above paragraph, we see that (after possibly interchanging u1, v1, w1) u1, v1, w1

have a form (4.1) of Definition 4.1 at p. We have that τf1(p) ≤ τfΦ1(p).


Still assuming that q is a 3-point, assume that q1 is a 2-point. If p is a 3-pointthen there are permissible parameters x, y, z for u, v, w at p such that

u = xaybzcγ1

v = xdyezfγ2

w = xlymznγ3

where γ1, γ2, γ3 are unit series, and

rank

a b cd e fl m n

≥ 2.

After possibly permuting u, v and w, we may assume that q1 has permissible pa-rameters u1, v1, w1 defined by

(10.11) u = u1, v = v1, w = u1(w1 + α)

with 0 6= α ∈ k. Thus (a, b, c) = (l,m, n), so that

rank(a b cd e f

)= 2

and τf1(p) = τfΦ1(p) = τf (p′) ≥ 1. We thus have that u1, v1 are toroidal forms atp, of type (4.4) of Definition 4.1 and τf1(p) = τf (p′).

We have a similar analysis if q1 is a 2-point, p is a 2-point (q is a 3-point),and u, v satisfy (4.2) of Definition 4.1 at p. Then q1 has permissible parametersu1, v1, w1 satisfying (10.11), and u1, v1 are toroidal forms of type (4.2) of Definition4.1 at p.

Now assume that p is a 2-point, q1 is a 2-point, (q is a 3-point) and u, v satisfy(4.3) of Definition 4.1 at p and (4.2) of Definition 4.1 does not hold at p for anypermutation of u, v, w. Then we have permissible parameters x, y, z at p such that

u = (xayb)l

v = (xayb)t(β + z)w = (xayb)m(γ(xayb, z) + xcyd)

where γ is a unit series and β 6= 0. If u = w = 0 are local equations of C at q, thenq1 has regular parameters u1, v1, w1 defined by

(10.12) u = u1, v = v1, w = u1(w1 + α)

with α 6= 0. Thus u1, v1 are toroidal forms of type (4.3) of Definition 4.1 at p.We have a similar analysis if v = w = 0 are local equations of C at q.If u = v = 0 are local equations of C at q, then q1 has permissible parameters

u1, w1, v1 defined by

(10.13) u = u1, v = u1(v1 + β), w = w1

with 0 6= β ∈ k and we thus have t = l. We have

u1 = (xayb)l

w1 = (xayb)m(γ + xcyd)v1 = z

and u1, w1, v1 have the form 2 (c) of Definition 4.6 at p.In all these cases, we have τf1(p) ≤ τfΦ1(p).There is a similar analysis in the case when (q is a 3-point) q1 is a 2-point and p

is a 1-point. Then (after possibly permuting u, v, w) u, v satisfy (4.1) of Definition


4.1 at p. If u = w = 0 are local equations of C at q, then q1 has permissibleparameters u1, v1, w1 defined by (10.12), and u1, v1 are toroidal forms of type (4.1)of Definition 4.1 at p. If u = v = 0 are local equations of C at q, then q1 haspermissible parameters u1, w1, v1 defined by (10.13), and u1, w1, v1 have the form2 (b) of Definition 4.6 at p. We further check that τf1(p) ≤ τfΦ1(p) in all thesecases (when q is a 3-point, q1 is a 2-point, p is a 1-point).

Suppose that q is a 2-pointLet u, v, w be permissible parameters at q. Suppose that p is a 3-point (satis-

fying 1 of Definition 4.6) or p is a 3-point which satisfies 2 (a) of Definition 4.6. q1is a 2-point, which has permissible parameters u1, v1, w1 defined by

u = u1, v = u1v1, w = w1,

or

u = u1v1, v = v1, w = w1,

or q1 is a 1-point, which has permissible parameters u1, v1, w1 defined by

u = u1, v = u1(v1 + α), w = w1,

with 0 6= α. By a similar calculation to the case when q is a 3-point, we see thatu1, v1 are toroidal forms at p, f1 is prepared at p and τf1(p) ≤ τfΦ1(p).

We further check that τf1(p) ≤ τf (p′) if p is a 1-point or a 2-point (and p′

satisfies 2 (a) of Definition 4.6).Now suppose that u, v, w satisfy 2 (b) of Definition 4.6 at p. If permissible

parameters at q1 are u1, v1, w with u = u1, v = u1v1, w = w1, then u1, v1, w1 havethe form 2 (b) at p also. If u = u1v1, v = v1 then we can make an appropriatechange of permissible parameters at p to get a form 2 (b) for v, u, w at p. Thus weobtain a form 2 (b) for u1, v1, w1 at p. If u = u1, v = u1(v1 + α), with 0 6= α ∈ k,then q1 is a 1-point, and u1, w1 are toroidal forms at p. We further check thatτf1(p) ≤ τfΦ1(p).

Suppose that u, v, w satisfy 2 (c) of Definition 4.6 at p. If q1 is a 2-point, whichhas regular parameters u1, v1, w1 defined by

(10.14) u = u1v1, v = v1, w = w1

or

(10.15) u = u1, v = u1v1, w = w1

then at p there is an expression of u1, v1, w1 of the form 2 (c) of Definition 4.6. Ifq1 is a 1-point, which has regular parameters u1, v1, w1 defined by

(10.16) u = u1, v = u1(v1 + α), w = w1,

with 0 6= α ∈ k, then u1, w1 are toroidal forms at p. We further check thatτf1(p) ≤ τfΦ1(p).

Comparing the above expressions, we see that f1 is prepared, and that f1satisfies the conclusions of the lemma.


Lemma 10.5. Suppose that f : X → Y is prepared and q ∈ Y is a 2-point.Then there exists a commutative diagram

(10.17)X1

f1→ Y1

Φ ↓ ↓ Ψ

Xf→ Y

where Φ and Ψ are products of blow ups of 2-curves, such that f1 is prepared,τf1(p) ≤ τf (Φ(p)) for p ∈ DX1 , and there exist no points of form 2 (b) or 2 (c) ofDefinition 4.6 for f1 above points of Ψ−1(q).

Proof. There exist sequences of blow ups of 2-curves Y1 → Y such that therational map X → Y1 is defined at all points p ∈ f−1(q) such that f has at pan expression of form 2 (b) or 2 (c) of Definition 4.6, and p maps to 1-point. ByLemma 10.4, by blowing up 2-curves above X, we can construct f1 which has thedesired property. No new points of the form 2 (b) or 2 (c) of Definition 4.6 arecreated by this construction.

Lemma 10.6. Suppose that f : X → Y is prepared, Ω is a set of 2-points of Y ,and we have assigned to each q0 ∈ Ω permissible parameters u = uq0 , v = vq0 , w =wq0 at q0. Then there exists a commutative diagram

(10.18)X1

f1→ Y1

Φ ↓ ↓ Ψ

Xf→ Y

such that1. f1 is prepared.2. Φ,Ψ are products of blow ups of 2-curves.3. Let

Ω1 =q1 ∈ Y1 such that q1 is a 2-point and q1 = f1(p1)for some 3-point p1 ∈ (f Φ)−1(Ω).

Suppose that q1 ∈ Ω1 with q0 = Ψ(q1) ∈ Ω. Then there exist permissibleparameters u1, v1, w1 at q1 such that

uq0 = ua1vb1

vq0 = uc1vd1

wq0 = w1

for some a, b, c, d ∈ N with ad−bc = ±1, and if p1 ∈ f−11 (q1) is a 3-point,

then there exist permissible parameters x, y, z at p1 for u1, v1, w1 such that

(10.19)u1 = xa1

1 yb11 z

c11

v1 = xd11 ye11 z

f11

w1 = γ1 +N1

where N1 = xg11 yh11 zi11 , with rank(u1, v1, N1) = 3, γ1 =

∑i αiMi where

αi ∈ k and each Mi is a monomial in x1, y1, z1 such that there are expres-sions

(10.20) Meii = uai

1 vbi1

with ai, bi, ei ∈ N and gcd(ai, bi, ei) = 1 for all i. Further, there is a boundr ∈ N such that ei ≤ r for all Mi in expressions (10.20).


4. Φ is an isomorphism above f−1(Y − Σ(Y )).

Proof. Suppose that q0 ∈ Ω. Let the 3-points in f−1(q0) be p1, . . . , pt.Each pi has permissible parameters x, y, z such that there is an expression of theform (9.1),

(10.21)uq0 = xaybzc

vq0 = xdyezf

wq0 = γ +N

where N = xgyhzi, γ =∑αiMi with relations

(10.22) Meii = uaivbi

with ai, bi, ei ∈ Z, ei > 0.We construct an infinite commutative diagram of morphisms

(10.23)

......

↓ ↓Xn

fn→ YnΦn ↓ ↓ Ψn

......

Φ2 ↓ ↓ Ψ2

X1f1→ Y1

Φ1 ↓ ↓ Ψ1

Xf→ Y

as follows. Order the 2-curves of Y , and let Ψ1 : Y1 → Y be the blow up ofthe 2-curve C of smallest order. Then construct (by Lemma 10.4) a commutativediagram

(10.24)X1

f1→ Y1

Φ1 ↓ ↓ Ψ1

Xf→ Y

where f1 is prepared, Φ1 is a product of blow up of 2-curves and Φ1 is an isomor-phism above f−1(Y − C). Order the 2-curves of Y1 so that the 2-curves containedin the exceptional divisor of Ψ1 have larger order than the order of the (stricttransforms of the) 2-curves of Y .

Let Ψ2 : Y2 → Y1 be the blow up of the 2-curve C1 on Y1 of smallest order, andconstruct a commutative diagram

X2f2→ Y2

Φ2 ↓ ↓ Ψ2

X1f1→ Y1

as in (10.24). We now iterate to construct (10.23). Let Ψn = Ψ1 · Ψn : Yn → Y ,Φn = Φ1 · · · Φn : Xn → X.

Let ν be a 0-dimensional valuation of k(X). Let pn be the center of ν on Xn,qn = fn(pn). We will say that ν is resolved on Xn if one of the following holds:

1. Ψn(qn) 6∈ Ω or2. Ψn(qn) = q0 ∈ Ω and


(a) pn is not a 3-point or(b) pn is a 3-point such that a form (10.19) holds for pn and qn = fn(pn)

so that (10.20) holds.Observe that if ν is resolved on Xn, then there exists a neighborhood U of thecenter of ν in Xn such that if ω is a 0-dimensional valuation of k(X) whose centeris in U , then ω is resolved on Xn, and if n′ > n, then ν is resolved on Xn′ .

We will now show that for every 0-dimensional valuation ν of k(X), there existsn ∈ N such that ν is resolved on Xn.

If the center of ν on Y is not in Ω or if the center of ν on X is not a 3-point,then ν is resolved on X, so we may assume that the center of ν on Y is q0 ∈ Ω andthe center of ν on X is a 3-point p.

Suppose that ν(uq0) and ν(vq0) are rationally dependent. Then there exists nsuch that the center of ν on Yn is a 1-point. Thus ν is resolved on Xn.

Suppose that ν(uq0) and ν(vq0) are rationally independent. At the center p ofν on X,

u = uq0 , v = vq0 , w = wq0

have an expression (9.1). We may identify ν with an extension of ν to the quotientfield of OX,p which dominates OX,p. We have

u = xaybzc, v = xdyezf ,

andMeii = ukivli = xaiybizci

with ki, li ∈ Z, ei > 0, ai, bi, ci ∈ N. Thus, for all i, (ki, li) ∈ σ, where

σ = (k, l) ∈ Q2 | ka+ ld ≥ 0, kb+ le ≥ 0, kc+ lf ≥ 0.

Since

rank(a b cd e f

)= 2,

σ is a rational polyhedral cone which contains no nonzero linear subspaces, and iscontained in the (irrational) half space

(k, l) | kν(u) + lν(v) ≥ 0.

Let λ1 = (m1,m2), λ2 = (n1, n2) be integral vectors such that σ = Q+λ1 + Q+λ2.Since λ1, λ2 are rational points in σ, we have ν(um1vm2) > 0 and ν(un1vn2) > 0.

Since ν(u) and ν(v) are rationally independent, there exists (by Theorem 2.7[C2]) a sequence of quadratic transforms k[u, v]→ k[u1, v1] such that the center ofν on k[u1, v1] is (u1, v1), there is an expression

u = ua1vb1, v = uc1v

d1

for some a, b, c, d ∈ N with ad − bc = ±1, and um1vm2 , un1vn2 ∈ k[u1, v1]. Thusthere exists a rational polyhedral cone σ1 ⊂ Q2 containing λ1 and λ2 such thatk[u1, v1] = k[σ1 ∩Z2]. We thus have Mei

i ∈ k[u1, v1] for all i, so that for all i, Meii

is a monomial in u1 and v1.There exists an n such that the center of ν on Yn has permissible parameters

u1, v1, w1 where

(10.25) u = ua1vb1, v = uc1v

d1 , w = w1.

We thus have ai, bi ∈ N such that


(10.26) Meii = uai

1 vbi1

for all i. Let pn be the center of ν on Xn. If pn is not a 3-point, then ν is resolvedon Xn.

If pn is a 3-point, then u1, v1, w1 (defined by (10.25)) are permissible parame-ters at the 2-point fn(pn). There exist permissible parameters x1, y1, z1 at pn foru1, v1, w1 defined by

x = xa111 ya12

1 za131

y = xa211 ya22

1 za231

z = xa311 ya132

1 za331

where aij ∈ N, and Det(aij) = ±1. Substituting into the expression (9.1) of u, v, wat p, we have expressions

u1 = xa1yb1zc1, v1 = xd1y

e1zf1 , N = xg1y

h1 z

i1,

where

Det

a b c

d e f

g h i

6= 0.

Let ~v1 = (a, b, c), ~v2 = (d, e, f),

σ2 = Q+~v1 + Q+~v2 ⊂ Q2.

By Gordan’s Lemma, (Proposition 1 [F]) σ2∩Z3 is a finitely generated semi group.Let ~w1, . . . , ~wn ∈ σ2 ∩ Z3 be generators. There exists 0 6= r ∈ N and δj , εj ∈ Nsuch that

~wj =δjr~v1 +

εjr~v2

for 1 ≤ j ≤ n. Since the exponents of

Mi = uaiei1 v

biei1 = (xa1y

b1zc1)

aiei (xd1y

e1zf1 )

biei

are in σ2 ∩ Z3 for all i, we have an expression

Mi = uair

1 vbir

1

with ai, bi ∈ N for all Mi appearing in the expansion (9.1) of w. Thus ν is resolvedon Xn.

By compactness of the Zariski-Riemann manifold of X ([Z]), there exist finitelymany Xi, 1 ≤ i ≤ t, such that the center of any 0-dimensional valuation ν of k(X)is resolved on some Xi. Thus Xt → Yt satisfies the conclusions of the Lemma.

Lemma 10.7. Suppose that f : X → Y is prepared, q ∈ Y is a 2-point containedin a 2-curve E of Y and u, v, w are permissible parameters at q. Then there existsa commutative diagram

(10.27)X1

f1→ Y1

Φ1 ↓ ↓ Ψ1

Xf→ Y

such that


1. f1 is prepared.2. Φ1 is a product of blow ups of 2-curves and 3-points such that Φ1 is an

isomorphism above f−1(Y −E). Ψ1 is a product of blow ups of 2-curves.3. Suppose that q1 ∈ Ψ−1

1 (q) is a 2-point, so that q1 has permissible parame-ters u1, v1, w1 defined by

(10.28)u = ua1v

b1

v = uc1vd1

w = w1

for some a, b, c, d ∈ N with ad− bc 6= 0. Then u1, v1, w1 are super param-eters at q1.

Proof. By Lemma 10.4 and Lemma 10.2, any diagram (10.27) satisfying 2satisfies 1. Further, if p ∈ f−1(q), u, v, w are super parameters at p and p1 ∈ Φ−1

1 (p)is such that f1(p1) = q1 is a 2-point, then the permissible parameters u1, v1, w1 of(10.28) at q1 are super parameters at p1.

Step 1. We will show that there exists a sequence of blow ups of 2-curves and3-points Φ1 : X1 → X such that Φ1 is an isomorphism over f−1(Y −E) and u, v, ware super parameters at all 3-points p ∈ (f Φ1)−1(q).

Suppose that p ∈ f−1(q) is a 3-point, so that there exist permissible parametersx, y, z ∈ OX,p at p for u, v, w such that

(10.29)u = xaybzc

v = xdyezf

w = g(x, y, z) +N

of the form of (4.4) of Definition 4.1 and (4.10) of Lemma 4.2. There exist regularparameters x, y, z in OX,p, and unit series λ1, λ2, λ3 ∈ OX,p such that

x = xλ1, y = yλ2, z = zλ3.

xyz = 0 is a local equation of DX at p.There is an expression g =

∑αiMi where 0 6= αi ∈ k and Mi = xaiybizci

are monomials in x, y, z such that rank(u, v,Mi) = 2. Let Ip be the ideal in OX,pgenerated by the xaiybizci for ai, bi, ci appearing in some Mi. There exists an rsuch that

Ip = (xa0yb0zc0 , xa1yb1zc1 , . . . , xarybrzcr ).We have relations Mei

i = uαivβi with ei, αi, βi ∈ Z and ei > 0 for all i. Thus fora ∈ spec(OX,p), (Ip)a is principal if u or v is not in a.

By Lemma 5.3, there exists a sequence of blow ups of 2-curves and 3-pointsΦ1 : X1 → X such that Φ1 is an isomorphism over f−1(X − E) and IpOX1,p1 isinvertible for all 3-points p ∈ X such that f(p) = q and p1 ∈ Φ−1

1 (p). f Φ1 : X1 →Y is prepared by Lemma 10.2.

Suppose that p1 ∈ (f Φ1)−1(q) is a 3-point. Then p = Φ1(p1) is also a 3-point,and τfΦ1(p1) = τf (p) (by Lemma 10.2). Let notation be as in (10.29). There existpermissible parameters x1, y1, z1 for the permissible parameters u, v, w at p1 suchthat x1, y1, z1 are defined by

(10.30)x = xa11

1 ya121 za13

1

y = xa211 ya22

1 za231

z = xa311 ya32

1 za331 ,


where aij ∈ N, Det(aij) = ±1. Thus all of the Mi and N are distinct monomialswhen expanded in the variables x1, y1, z1. So that since IpOX1,p1 is principal,g(x, y, z) is a monomial in x1, y1, z1 times a unit series (in x1, y1, z1). Thus u, v, ware super parameters at p1.

Step 2. We will show that there exists a sequence of blow ups of 2-curves Φ2 :X2 → X1, where Φ1 : X1 → X is the map constructed in Step 1, such that Φ2

is an isomorphism over (f Φ1)−1(Y − E) and u, v, w are super parameters at allp ∈ (f Φ1 Φ2)−1(q) for which p is a 3-point or p is a 2-point of type (4.2) ofDefinition 4.1 for u, v, w.

Let f1 = f Φ1. Let ν be a 0-dimensional valuation of k(X). Let p be thecenter of ν on X1. Say that ν is resolved on X1 if

1. f1(p) 6= q, or2. p is not a 2-point of type (4.2) of Definition 4.1, or3. p is a 2-point of type (4.2) and u, v, w are super parameters at p.

The condition of being resolved is an open condition on the Zariski Riemannmanifold of X.

We construct an infinite sequence of morphisms

(10.31) · · · → XnΦn→ · · · Φ3→ X2

Φ2→ X1

as follows. Order the 2-curves C of X1 such that q ∈ (f Φ1)(C) ⊂ E. LetΦ2 : X2 → X1 be the blow up of the 2-curve C1 on X1 of smallest order. Order the2-curves C ′ of X2 such that q ∈ (f Φ1Φ2)(C ′) ⊂ E so that the 2-curves containedin the exceptional divisor of Φ2 have order larger than the order of the (stricttransform of the) 2-curves C of X1 such that q ∈ f1(C) ⊂ E. Let Φ3 : Y3 → Y2 bethe blow ups of the 2-curve C2 on Y3 of smallest order, and repeat to inductivelyconstruct the morphisms Φn : Xn → Xn−1. Let Φn = Φ2 · · ·Φn : Xn → X1. Themorphisms f Φn are prepared by Lemma 10.2.

Let ν be a 0-dimensional valuation of k(X) that is not resolved on X. Let pbe the center of ν on X. Then p ∈ f−1

1 (q) is a 2-point satisfying (4.2) of Definition4.1, so there exist permissible parameters x, y, z at p for u, v, w such that

u = xayb

v = xcyd

w = g(x, y) + xeyfz

with ad − bc 6= 0. There exist regular parameters x, y, z in OX1,p and unit seriesλ1, λ2 ∈ OX1,p such that

x = xλ1, y = yλ2.

xy = 0 is a local equation of DX1 at p.If ν(x), ν(y) are rationally independent, then the center of ν on Xn is a 2-point

for all n, there exists an n such that the center of ν on Xn is a 2-point such thatu, v, w satisfy (4.2) of Definition 4.1 and u, v, w are super parameters at p1, byembedded resolution of plane curve singularities (cf. Section 3.4 and Exercise 3.3[C6]) applied to g(x, y) = 0. If ν(x), ν(y) are rationally dependent, then thereexists an n such that the center p1 of ν on Xn is a 1-point.

By compactness of the Zariski-Riemann manifold of k(X) [Z], there exists ann such that all points of Xn are resolved. Thus there exists a sequence of blow upsof 2-curves Φ2 : X2 → X1 such that the conclusions of Step 2 hold.


Step 3. We will show that there exists a diagram (10.27) satisfying the conclusionsof the lemma.

After replacing f with f Φ1Φ2, we can assume that f satisfies the conclusionsof Step 2. We construct a sequence of diagrams (10.27) satisfying 1, 2 and 3 of theconclusions of the lemma as follows. Let Ψ1 : Y1 → Y be the blow up of the2-curve C containing q. We order the two 2-curves in Y1 which dominate C. LetΨ2 : Y2 → Y1 be the blow up of the 2-curve of smallest order. Now extend theordering to the 2-curves of Y2 which dominate C, by requiring that the two 2-curves on the exceptional divisor of Ψ2 which dominate C have larger order thanthe order of the (strict transform of the) 2-curve on Y1 dominating C (which wasnot blown up by Ψ2). Now let Ψ3 : Y3 → Y2 be the blow up of the 2-curve ofsmallest order. We continue this process to construct a sequence of blow ups of2-curves

· · · → YnΨn→ Yn−1 → · · · → Y1

Ψ1→ Y.

Let Ψn = Ψ1 · · ·Ψn−1 Ψn. Let

U =p ∈ f−1(q) such that u, v, w have a form (4.1) or (4.3) ofDefinition 4.1 or of 2 (b) or 2 (c) of Definition 4.6 at p

.

U is an open subset of f−1(q). For each p ∈ U , there exists n(p) such that n ≥ n(p)implies the rational map Ψ

−1

n f is defined at p, and (Ψ−1

n f)(p1) is a 1-point, forp1 in some neighborhood Up of p in U . Up is an open cover of U , so there existsa finite subcover Up1 , . . . , Upm of U . Let n = maxn(p1), . . . , n(pm). We havethat Ψ

−1

n f(p) is a 1-point if p ∈ U .By Lemma 10.4, we can now construct a commutative diagram

Xnfn→ Yn

Φn ↓ ↓ Ψn

Xf→ Y

such that 1 and 2 of the conclusions of the lemma hold.If p1 ∈ Xn is such that fn(p1) = q1 ∈ Ψ

−1

n (q) is a 2-point, then p = Φn(p1) ∈X − U , and thus u, v, w have a form 2 or 4 of Definition 10.1 at p. As observed atthe beginning of the proof, p1 must then have one of the forms 1 – 4 of Definition10.1 with respect to the permissible parameters u1, v1, w1 at q1 defined by 3 of thestatement of Lemma 10.7. Thus fn satisfies 3 of the statement of Lemma 10.7.

CHAPTER 11

Good and perfect points

Throughout this section, we suppose that f : X → Y is a dominant, propermorphism of nonsingular 3-folds.

Definition 11.1. Suppose that f : X → Y is prepared, p ∈ DX and q = f(p).Let u, v, w be permissible parameters at q. We say that w is good at p for f if oneof the following expressions holds:

p a 1-point, q a 1-point

(11.1) u = xa, v = y, w =∑a6 |i

aijxiyj + xn(z + α)


(11.2) u = (xayb)k, v = z, w =∑k 6 |i

aij(xayb)izj + xcyd


(11.3) u = xa, v = xb(α+ y), w =∑d6 |i

aijxiyj + xn(z + β)

where d = gcd(a, b).p a 2-point, q a 2-point

(11.4) u = xayb, v = xcyd, w =∑

(i,j) 6∈Z(a,b)+Z(c,d)

aijxiyj + xeyfz


(11.5) u = (xayb)k, v = (xayb)t(α+ z), w =∑d6 |i

aij(xayb)izj + xcyd

where d = gcd(k, t).p a 3-point, q a 2-point

(11.6) u = xaybzc, v = xdyezf , w =∑

aijkxiyjzk + xgyhzi

where the sum is over i, j, k such that

Det

a b cd e fi j k

= 0, (i, j, k) 6∈ Z(a, b, c) + Z(d, e, f).

Definition 11.2. Suppose that f : X → Y is prepared, p ∈ DX and q = f(p)is a 1-point. Let u, v, w be permissible parameters at q. We say that w is weaklygood at p for f if one of the following forms hold:

95

96 11. GOOD AND PERFECT POINTS

1. p is a 1-point,

u = xa, v = y, w =m∑j=0

aσj (y)xσj + xn(z + α)

where α ∈ k, σ0 < σ1 < · · · < σm < n, σi are all nonzero, and a 6 | σ0.2. p is a 2-point,

u = (xayb)k, v = z, w =m∑j=0

aσj (z)(xayb)σj + xcyd

where gcd(a, b) = 1, ad − bc 6= 0, σm(a, b) 6> (c, d), σ0 < σ1 < · · · < σm,σi are all nonzero, and k 6 | σ0.

Remark 11.3. Suppose that f : X → Y is prepared and τf (p) = 1, u, v, w arepermissible parameters at q = f(p), and w is good (weakly good) at p for f . Thenu, v, w are monomial forms (Definition 4.4) at p.

Remark 11.4. Suppose that f ;X → Y is prepared. If q is a 1-point, and u, v, ware permissible parameters at q satisfying the conclusions of Lemma 4.9, then forall p ∈ f−1(q), there exists a series φp(u, v) such that w − φp(u, v) is good (weaklygood) at p for f .

Lemma 11.5. Suppose that f : X → Y is prepared, q ∈ Y and u, v, w ∈ OY,qare permissible parameters at q (such that the conclusions of Lemma 4.9 hold if qis a 1-point). Suppose that p ∈ f−1(q) and there exists φ(u, v) ∈ OY,q such thatw − φ(u, v) is good (weakly good) at p for f .

Suppose that p is an n-point. Then there exists an affine neighborhood V =Spec(S) of p such that w− φ(u, v) is good (weakly good) at p′ for f for all n-pointsp′ ∈ f−1(q) ∩ V .

Proof. We will prove this in the case that p and q are 1-points, and a form(11.1) of Definition 11.1 holds for u, v, w− φ in OX,p. The proof in the other casesis similar.

There exists an affine neighborhood V = spec(S) of p, regular parametersx, y, z ∈ OX,p and a finite etale morphism π : V1 = spec(S1) → S such that x, y, zare uniformizing parameters on V1, and regular parameters in OV1,p′ for p′ ∈ π−1(p)such that

u = xa

v = yw =

∑i<n aijx

iyj + xn(γ(x, y, z)z + Ω(x, y))

in OV1,p′ = OX,p where γ is a unit series, Ω(x, y) is a series.Let U = Spec(R) be an affine neighborhood of q such that f(V ) ⊂ U .In OX,p,

w − φ(u, v) =∑

a6 |i,i<n

aijxiyj + xn(γz + Ω)

where Ω(x, y) is a series. We see that

xn divides∂(w − φ)

∂zin OX,p,

so that

11. GOOD AND PERFECT POINTS 97

xn divides∂(w − φ)

∂zin OS1,p′ ⊗Rq

Rq

at all points p′ ∈ π−1(p).We also have that

∂n+1(w − φ)∂z∂xn

(p) 6= 0

which implies∂n+1(w − φ)

∂z∂xn(p′) 6= 0

at all p′ ∈ π−1(p).Finally, we see that

x divides∂i(w − φ)

∂xi

in OS1,p′ ⊗RqRq at all points p′ of π−1(p).

Thus there exists a Zariski closed subset C of V1 which is disjoint from π−1(p)such that if p ∈ (f π)−1(q) ∩ (V1 − C), then

(11.7) xn divides∂(w − φ)

∂zin OS1,p ⊗Rq

Rq

(11.8)∂n+1(w − φ)

∂z∂xnis a unit in OS1,p ⊗Rq

Rq

and

(11.9) x divides∂i(w − φ)

∂xiin OS1,p ⊗Rq Rq

if i < n and a divides i.Let C = π(C). π : Spec(S1)−π−1(C)→ V −C is finite etale. Let V = spec(S)

be an affine neighborhood of p in V −C, and let π−1(V ) = Spec(S1). After replacingV with V , S with S, V1 with π−1(V ) and S1 with S1, we have that (11.7), (11.8)and (11.9) hold at all p ∈ (f π)−1(q).

Now consider the expression of u, v, w − φ(u, v) at p ∈ (f π)−1)(q). Thereexists α ∈ k such that x, y, z − α are regular parameters at p. We have

u = xa

v = y

w − φ =∑

1i!j!k!

∂i+j+k(w−φ)∂xi∂yj∂zk (0, 0, α)xiyj(z − α)k.

(11.7) implies∂i+j+k(w − φ)∂xi∂yj∂zk

(0, 0, α) = 0

if i < n and k ≥ 1, (11.9) implies

∂i+j(w − φ)∂xi∂yj

(0, 0, α) = 0

if a divides i and i < n, and (11.8) implies

∂n+1(w − φ)∂xn∂z

(0, 0, α) 6= 0.


Thus

w − φ =∑

i<n,a6 |i

1i!j!

∂i+j(w − φ)∂xi∂yj

(0, 0, α)xiyj + xn(γ′(z − α) + Ω′(x, y))

where γ′ is a unit series, so that w − φ is good at p.

Lemma 11.6. Suppose that f : X → Y is prepared, and C ⊂ Y is an irreduciblecurve in the nonfinite locus of f such that C contains a 1-point.

Suppose that U ⊂ Y is an affine open subset, with uniformizing parametersu, v, w such that u, v, w are permissible parameters in OY,q for a 1-point q ∈ C ∩Usuch that u = w = 0 are local equations of C. Then for a general point q of C ∩U ,and appropriate (general) α ∈ k, u, v = v − α,w are permissible parameters at qand for p ∈ f−1(q), either p is a 1-point and we have a form at p

(11.10)u = xa

v = yw =

∑i<n φi(y)x

i + xn(z + δ)

with δ ∈ k and φi(0) 6= 0 whenever φi 6= 0, or p is a 2-point with a form at p

(11.11)u = (xayb)t

v = zw =

∑φi(z)(xayb)i + xcyd

with gcd(a, b) = 1, ad− bc 6= 0 and φi(0) 6= 0 whenever φi 6= 0.

Proof. To prove this theorem, we may replace q with a point of C∩U such thatC is the only component of the nonfinite locus of f through q, and q is a nonsingularpoint of C. Suppose p ∈ f−1(q). Then there exists a Zariski open neighborhoodV = V p = Spec(S) of p in X, and an etale neighborhood W = W p = Spec(S1) ofV p with uniformizing parameters x, y, z in S1, with induced morphism

π : Spec(S1)→ Spec(S),

such that x, y, z are regular parameters in OWp,p1 for p1 ∈ π−1(p), and by Lemma4.9 and its proof, we have one of the following cases:

Case 1. Suppose that p is a 1-point. Then we have in OX,p = Sp:

(11.12) u = xa, v = y, w =∑i<n

φi(y)xi + xn(γz + ψ(x, y))

where γ is a unit series. In (11.12), for i < n we have

(11.13)1i!∂iw

∂xi= φi(y) + xΩ ∈ Sp

for some Ω ∈ Sp and∂n+1w

∂xn∂z(0, 0, 0) 6= 0.

We can choose V p so that for p1 ∈W p ∩DX , with regular parameters

x, y = y − α, z = z − βin OWp,p1 , for i < n we have

(11.14)∂i+1w

∂z∂xi(0, α, β) = 0


and

(11.15)∂n+1w

∂z∂xn(0, α, β) 6= 0.

We can choose V p so that for i < n, all irreducible components of x = ∂iw∂xi = 0

in W p contain (a preimage of) p.

Case 2. Suppose that p is a 2-point. Then we have in OX,p = Sp:

(11.16) u = (xayb)t, v = z, w =∑

φi(z)(xayb)i + xcydγ

with gcd(a, b) = 1, ad− bc 6= 0 and c > ai or d > bi for all i in the series. Further,γ is a unit series.

We have Ω1,Ω2 ∈ Sp = OX,p such that(11.17)

1j!k!

∂j+kw

∂xj∂yk=

xΩ1 + yΩ2 if jb− ka 6= 0 and j < c or k < dφi(z) + xΩ1 + yΩ2 if there exists i such that (j, k) = i(a, b)

and j < c or k < d

There exists Ω1 ∈ OX,p such that(11.18)

1j!∂jw

∂xj=

xΩ1 if j < c and there do not exist k, i such that(j, k) = i(a, b)

yibφi(z) + xΩ1 if j < c and there exist k, i such that (j, k) = i(a, b).

There exists Ω1 ∈ OX,p such that

1k!∂kw

∂yk=

yΩ1 if k < d and there do not exist j, i such that(j, k) = i(a, b)

xiaφi(z) + yΩ1 if k < d and there exist j, i such that (j, k) = i(a, b).

Furthermore,∂c+dw

∂xc∂yd(0, 0, 0) 6= 0.

We can choose V p so that

1. for j < c or k < d, all irreducible components of

x = y =∂j+kw

∂xj∂yk= 0

in W p contain (a preimage of) p,2. for j < c, all irreducible components of

x =∂jw

∂xj= 0

in W p contain (a preimage of) p, and3. for k < d, all irreducible components of

y =∂kw

∂yk= 0

in W p contain (a preimage of) p.


There exist V1 = V p1 , . . . , Vn = V pn such that V1, . . . , Vn is an affine coverof f−1(q). We may assume that V1, . . . , Vn is an affine cover of f−1(U).

Suppose that q ∈ C ∩ U is a general point. Then OY,q has regular parametersu, v = v−α,w for a general α ∈ k. u, v, w are permissible parameters at q. Supposethat p ∈ f−1(q). p ∈ V p = Vi for some i. We identify p and p with points in π−1(p),π−1(p).

Suppose that p is a 1-point. There exists β ∈ k such that x, y = y − α, z − βare permissible parameters in OX,p. We have an expression

(11.19)u = xa

v = y = y − αw =

∑1

i!j!k!∂i+j+kw∂xi∂yj∂zk (0, α, β)xi(y − α)j(z − β)k

in OX,p.By (11.14) and (11.15), in (11.19), we have

w =∑i<n

φi(y)xi + xn(zγ + Ψ(x, y))

where γ is a unit series, and

φi(y) =∞∑j=0

1i!j!

∂i+jw

∂xi∂yj(0, α, β)yj .

If φi(y) = 0 we have φi(y) = 0.Suppose that φi(y) 6= 0. We will show that φi(0) 6= 0. If φi(0) 6= 0, then ∂iw

∂xi

does not vanish on W p ∩DX , so that

φi(0) =∂iw

∂xi(p) 6= 0.

Suppose that φi(0) = 0. Further suppose that

φi(0) =∂iw

∂xi(p) = 0.

Then there exists an irreducible curve Λ which is a component of x = ∂iw∂xi = 0 in

W p containing p.By our construction of W p, we may assume that our choice of preimage of p

in W p (which we have identified with p) satisfies p ∈ Λ. Let IΛ be the prime idealof Λ in S1. Let S1 be the completion of S1 at the maximal ideal mp of p in S1.∂iw∂xi , x ∈ IΛS1 implies φi(y) ∈ IΛS1. Since IΛS1 is reduced, we have y ∈ IΛS1. As(S1)mp

→ S1 = OX,p is faithfully flat, we have y ∈ IΛ(S1)mp.

Since IΛ is a prime ideal and (IΛ)mp 6= (S1)mp , we have that y ∈ IΛ.Let mp be the ideal of p in S1. The maximal ideal of (S1)mp is (x, y−α, z−β).

IΛ(S1)mp ⊂ (x, y − α, z − β)

implies y ∈ (x, y−α, z−β) which is impossible since 0 6= α. Thus we have φi(0) 6= 0.Suppose that p is a 2-point and p is a 2-point. Then there is α ∈ k such that

x, y, z − α are regular parameters in OX,p, and we have an expression

(11.20)u = (xayb)t

v = z = z − αw =

∑1

i!j!k!∂i+j+kw∂xi∂yj∂zk (0, 0, α)xiyj(z − α)k


in OX,p.We have

∂c+dw

∂xc∂yd(0, 0, α) 6= 0.

Further if j < c or k < d, jb− ka 6= 0 and l ≥ 0, we have

∂j+k+lw

∂xj∂yk∂zl(0, 0, α) = 0.

Thus in (11.20) we have

w =∑

φi(z)(xayb)i + xcydγ, and

where γ is a unit series, the sum is over i such that ai < c or bi < d, and

φi(z) =∞∑k=0

1(ia)!(jb)!k!

∂ia+jb+kw

∂xia∂yib∂zk(0, 0, α)zk.

By a similar analysis as for Case 1, we see that if φi(z) = 0 then φi(z) = 0 and ifφi(z) 6= 0, we have φi(0) 6= 0.

Suppose that p is a 2-point and p is a 1-point. Then after possibly interchangingx and y, there exist α, β ∈ k and regular parameters x, y − β, z − α in OX,p with0 6= α, β.

Setx = xy

ba , y = y − β, z = z − α

Then x, y, z are regular parameters in OX,p. From the Jacobian of u, v, w we seethat we have an expression

u = xat

v = zw = P (x, z) + xcΩ

where P and Ω are series, ∂Ω∂y (p) 6= 0, and P has degree < c in x.

We havex = x(y + β)−

ba

y = y + βz = z + α.

Further,∂∂x = y−

ba∂∂x

∂∂y = − b

axy−1 ∂

∂x + ∂∂y

∂∂z = ∂

∂z .

We have an expansion

P =∑j<c

φj(z)xj =

∑j<c

( ∞∑k=0

∂j+kw

∂zk∂xj(p)zk

)xj

where∂j+kw

∂zkxj(p) = β−j

ba∂j+kw

∂zk∂xj(p).

By (11.18) we have that φj(z) = 0 if there do not exist k, i such that (j, k) = i(a, b).


Suppose that j < c and there exists k, i such that (j, k) = i(a, b) and φj(z) 6= 0.Suppose that φj(0) = 0. Then

0 = φj(0) =∂jw

∂xj(p) = β−j

ba∂jw

∂xj(p)

implies there exists an irreducible curve Λ which is a component of

x =∂jw

∂xj= 0

in W p which contains p and p. Let IΛ be the prime ideal of Λ in S1. Let S1 be thecompletion of S1 at the maximal ideal mp of p in S1. ∂jw

∂xj , x ∈ IΛS1 implies

yibφi(z) ∈ IΛS1.

Since y 6∈ IΛS1, as Λ is not a 2-curve, we have z ∈ IΛS1. Thus z ∈ IΛ(S1)mp . As(IΛ)mp 6= (S1)mp , we have that z ∈ IΛ, which is a contradiction, since α 6= 0. Thusφi(0) 6= 0.

Lemma 11.7. Suppose that f : X → Y is prepared. Then there exists an opensubset V of Y such that V ∩ C 6= ∅ for every integral curve C ⊂ DY contained inthe nonfinite locus of f which contains a 1-point, and if U = f−1(V ), f = f | U ,then there exists a commutative diagram

U1f1→ V1

Φ1 ↓ ↓ Ψ1

Uf→ V

such that Φ1 and Ψ1 are products of blow ups of curves which dominate a curve Ccontained in the nonfinite locus of f which are possible centers (for the preimage ofDV = DY ∩ V ) and f1 is toroidal.

Proof. By Lemma 11.6, there exists an open set V ⊂ Y such that V ∩C = ∅if C is a 2-curve or an isolated point contained in the nonfinite locus of f , V ∩C 6= ∅for all curves C contained in the nonfinite locus of f which contain a 1-point, and ifq ∈ C∩V is a 1-point, then there exist permissible parameters u, v, w at q such thatu = w = 0 are local equations of C, and if p1 ∈ f−1(q), then we have permissibleparameters x, y, z in OX,p1 such that p1 is a 1-point:

(11.21)u = xa

v = yw =

∑mj=0 φij (y)x

ij + xn(α+ z)

where i0 < i1 < · · · < im < n, α ∈ k, all φij (y) are nonzero and φij (0) 6= 0 for allj, or p1 is a 2-point:

(11.22)u = (xayb)t

v = zw =

∑mj=0 φij (z)(x

ayb)ij + xcyd

where ad − bc 6= 0, all φij (y) are nonzero, φij (0) 6= 0 for all j, i0 < i1 < · · · < imand ija < c or ijb < d for all j.


Let C be the nonfinite locus of f : U → V . There exists Φ′1 : U ′

1 → U whichis a product of blow ups of 2-curves (which dominate an irreducible component ofC) such that all local forms (11.22) at points p ∈ (f Φ′

1)−1(q) for q ∈ C ∩ V are

such that either m = −1 (so that∑φij (z)(x

ayb)ij = 0), or (xayb)i0 divides xcyd.Suppose that q ∈ C. The set of points p ∈ (f Φ′

1)−1(q) such that ICOU ′

1is

not invertible is a union of points p such that p has permissible parameters x, y, zof the form

(11.23) u = xa, v = y, w = xnz

with n < a or

(11.24) u = (xayb)t, v = z, w = xcyd

with (at− c)(bt− d) < 0.Let Ψ1 : V1 → V be the blow up of C (which has local equations u = w = 0 at

q ∈ C).We can blow up curves above U ′

1 which dominate a component of C to obtainΦ1 : U1 → U ′

1 such that there exists a factorization f1 : U1 → V1, and if q ∈ C,p1 ∈ (f Φ′

1 Φ1)−1(q), then an expression (11.21) or (11.22) holds.Suppose that q ∈ C, with permissible parameters u, v, w as above. Let q1 ∈

Ψ−1

1 (q). q1 has permissible parameters u1, v1, w1 with q1 a 1-point

(11.25) u = u1, v = v1, w = u1(w1 + α)

or q1 a 2-point

(11.26) u = u1v1, v = w1, w = u1.

Suppose that p1 ∈ f−1

1 (q1).

Case 1 Suppose that 0 6= α in (11.25). Then a = i0 and φi0(0) = α (or m = −1,a = n and α = α) if p1 satisfies (11.21), t = i0 and φi0(0) = α if p1 satisfies (11.22).

We thus have that at p1,

(11.27) u1 = xa, v = y, w1 = (φi0(y)− α) +m∑j=1

φij (y)xij−i0 + xn−i0(α+ z)

of the form (11.21) or(11.28)

u1 = (xayb)t, v = z, w1 = (φi0(z)− α) +m∑j=1

φij (z)(xayb)ij−i0 + xc−i0ayd−i0b.

of the form of (11.22).If f1 is not toroidal at p1, we have that u1 = w1− (φi0(v)−α) = 0 are (formal)

local equations of a branch of the nonfinite locus of f1 through q1. After possiblyreplacing V with an open subset of V , for q ∈ C, q is a general point of a componentof C, so the nonfinite locus of f1 through q1 must be the germ of a nonsingularalgebraic curve.

Case 2 Suppose that 0 = α in (11.25). Then i0 > a (or m = −1 and n > a orm = −1, n = a and α = 0) in (11.21), or i0 > t (or m = −1 and (c, d) > t(a, b)


in (11.22)). We have that u1, v, w1 are permissible parameters at q1 for f1 of theform (11.21) or (11.22).

Case 3 (11.26) holds. Then a > i0 (or m = −1, 0 6= α and n < a) in (11.21) ort > i0 (or m = −1 and (c, d) ≤ (at, bt)) in (11.22).

Suppose that (11.21) holds and m ≥ 0. We change variables at p1 to get anexpression

u =m∑i=0

φij (y)xij + xn−i0+a(z + · · · ), v = y, w = xi0

with a = i0 < · · · < im < n− i0 + a. As in the proof of Lemma 11.6, φij (0) 6= 0 forall ij since q is a general point of a component of C. q1 is a 2-point, and we have:

(11.29) u1 = xi0 , v1 =m∑j=0

φij (y)xij−i0 + xn−2i0+a(z + · · · ), w1 = y.

Note that u1 = v1 = 0 are local equations of the nonfinite locus of f1 at q1 if f1

is not toroidal at p1. This is the germ of a 2-curve which dominates a componentof C.

Suppose that (11.22) and (11.26) hold and m ≥ 0. We change variables at p1

to have an expression

u =m∑j=0

φij (z)(xayb)ij + xc−i0a+tayd−i0b+tb, v = z, w = (xayb)i0

with ija < c− i0a+ ta or ijb < d− i0b+ tb for all ij .As in the proof of Lemma 11.6, φij (0) 6= 0 for all ij , since q is a general point

of a component of C. q1 is a 2-point and we have

(11.30) u1 = (xayb)i0 , v1 =m∑j=0

φij (z)(xayb)ij−i0 + xc−2i0a+tayd−2i0b+tb, w1 = z.

u1 = v1 = 0 are local equations of the nonfinite locus of f1 at q1 if f1 is not toroidalat p1. This is the germ of a 2-curve which dominates a component of C.

The nonfinite locus C1 of f1 : U1 → V1 is a (disjoint) union of nonsingular curveswhich dominate components of C. If γ1 is a component of C1 then γ1 consists of1-points or γ1 consists of 2-points. We will construct a commutative diagram

(11.31)U2

f2→ V2

Φ2 ↓ ↓ Ψ2

U1f1→ V1

where Ψ2 : V2 → V1 is the blow up of C1.Suppose that γ1 is a component of C1 and q1 ∈ γ1.First suppose that γ1 consists of 1-points. Then (11.21) or (11.22) holds at

all points p ∈ f−1

1 (q1). The construction of (11.31) above points of γ1 is as in theconstruction of f1 above.


Suppose that γ1 consists of 2-points. Then there exist permissible parametersu1, v1, w1 at q1 such that (11.29) or (11.30) holds at all p ∈ f−1

1 (q1), and u1 = v1 = 0are local equations of γ1 at q1.

If p ∈ f−1

1 (q1) is such that IC1OU2,p is not invertible, then we have permissibleparameters x, y, z at p such that

u1 = xa, v1 = xnz, w1 = y

with n < a oru1 = (xayb)t, v1 = xcyd, w1 = z.

In particular, f1 is toroidal at p.We now blow up curves 2-curves (above U1) which dominate γ1 and are sup-

ported in the locus where U1 → V1 is torodial to obtain the construction ofΦ2 : U2 → U1 above γ1. f2 is toroidal above the torodial locus of f . Letq2 ∈ Ψ

−1

2 (q1). q2 has permissible parameters u2, v2, w2 defined by one of the fol-lowing 3 cases.

q2 is a 1-point

(11.32) u1 = u2, v1 = u2(w2 + α), w1 = v2

with α 6= 0 or q2 is a 2-point

(11.33) u1 = u2, v1 = u2v2, w1 = w2

or q2 is a 2-point

(11.34) u1 = u2v2, v1 = v2, w1 = w2.

As in the case when q2 is a 1-point, we see that if (11.32) holds, then all pointsabove q2 have the form (11.21) or (11.22), and that if (11.33) or (11.34) holds, thenall points p2 above q2 have the form (11.29) or (11.30).

We iterate to construct a commutative diagram

(11.35)

......

↓ ↓

Unfn→ Vn

Φn ↓ ↓ Ψn

Un−1

fn−1→ Vn−1

↓ ↓...

...↓ ↓

U1f1→ V1

↓ ↓

Uf→ V

where each Vr → Vr−1 is the blow up of the nonfinite locus Cr−1 of fr−1, which isa disjoint union of nonsingular curves which dominate components of C.

All points of Un have a form (11.21), (11.22), (11.29) or (11.30). We continuethe construction as long as fn is not toroidal.


Suppose that (11.35) does not converge to a toroidal morphism in a finitenumber of steps. Then there exists a 0-dimensional valuation ν of k(X) withcenter on U such that fn is not toroidal at the center pn of ν on Un for all n. Letqn be the center of ν on Vn.

Suppose that (11.21) holds for p = p0. There exists r(1) such that qr(1) has(formal) permissible parameters ur(1), vr(1), wr(1) defined by

u = uer(1), w = ufr(1)(wr(1) + φi0(0))

where gcd(e, f) = 1 and ef = a

i0.

The germ of f at p factors through Ψr(1), so we have

ur(1) = xae , wr(1) = (φi0(y)− φi0(0)) +

m∑j=1

φij (y)xij−i0 + xn−i0(α+ z).

Set wr(1) = wr(1) − [φi0(v)− φi0(0)].ur(1), v, wr(1) are (formal) regular parameters at qr(1), and ur(1) = wr(1) = 0

are equations of the nonfinite locus at qr(1). We see that at pr(1),

ur(1) = xa(1), v = y, wr(1) =m(1)∑j=0

φ(1)i(1)j(y)xi(1)j + xn(1)(α+ z)

where a(1) = gcd(a, i0), m(1) = m − 1, n(1) = n − i0, i(1)j = ij+1 − i0 for0 ≤ j ≤ m(1), φ(1)i(1)j

(y) = φij+1(y).We iterate to get for k ≤ m+1, r(k) such that qr(k) has permissible parameters

ur(k), v, wr(k) defined by

ur(k−1) = uek

r(k), wr(k−1) = ufk

r(k)(wr(k) + φ(k − 1)i(k−1)0(0))

where gcd(ek, fk) = 1.The rational map U → Vr(k) is a morphism at p = pr(k). Set

wr(k) = wr(k) − [φ(k − 1)i(k−1)0(v)− φ(k − 1)i(k−1)0(0)].

We have an expression

ur(k) = xa(k), v = y, wr(k) =m(k)∑j=0

φ(k)i(k)j(y)xi(k)j + xn(k)(α+ z).

We have (for k ≤ m + 1) a(k) = gcd(a, i0, i1, . . . , ik−1), n(k) = n − ik−1, m(k) =m− k, i(k)j = ij+k − ik−1 for 0 ≤ j ≤ m(k).

We further have

(11.36)ekfk

=a(k − 1)i(k − 1)0

=gcd(a, i0, . . . , ik−2)

ik−1 − ik−2.

ur(k) = wr(k) = 0 are (formal) equations of the nonfinite locus at qr(k).qr(m+1) has permissible parameters ur(m+1), v, wr(m+1) defined by

ur(m+1) = xa(m+1), v = y, wr(m+1) = xn(m+1)(α+ z).

The rational map U → Vr(m+1) is a morphism at p. We have

a(m+ 1) = gcd(a, i0, i1, . . . , im)


and n(m+ 1) = n− im.Finally, we see that there exists r(m + 2) such that fr(m+2) is toroidal at

pr(m+2), a contradiction.A similar argument holds if (11.22) holds at p = p0.We conclude that (11.35) converges after a finite number of iterations in a

diagram which satisfies the conclusions of Lemma 11.7.

Remark 11.8. It follows from a simple variation of the proof of Lemma 11.7that if f : X → Y is prepared, and C ⊂ Y is a 2-curve, then there exists a Zariskiopen subset V of Y such that C ∩ V = ∅ and there exists a commutative diagram

U1f1→ V1

Φ1 ↓ ↓ Ψ1

Uf→ V

as in the conclusions of Lemma 11.7 such that f1 is toroidal.

Definition 11.9. Suppose that f : X → Y is prepared and q ∈ GY (f, τ) is a1-point. Then q is perfect for f if the nonfinite locus of f through q is a germ ofa nonsingular curve γ and if u, v, w are algebraic permissible parameters at q suchthat u = w = 0 are local equations of γ at q then there exist finitely many seriesφi(u, v) ∈ k[[u, v]] such that

1. u, v, wi = w − φi(u, v) are super parameters at q for all i.2. For all p ∈ f−1(q), some wi is weakly good for f at p.

Lemma 11.10. Suppose that f : X → Y is prepared. Let V ⊂ GY (f, τ) be theset of perfect 1-points. Then GY (f, τ)− V is a finite set.

Proof. Suppose that q is a general point of a curve C ⊂ GY (f, τ) (so that q isa 1-point). Let u, v, w be algebraic permissible parameters at q such that u = w = 0are local equations of C.

In a neighborhood of q, we construct a diagram (11.35). (11.35) is finite by theconclusions of Lemma 11.7.

Suppose that p ∈ f−1(q). At p we have permissible parameters x, y, z ∈ OX,psuch that if p is a 1-point:

(11.37)u = xa

v = yw =

∑mj=0 φij (y)x

ij + xn(β + z)

where β ∈ k, i0 < i1 < · · · < im < n, all φij (y) are non zero and φij (0) 6= 0 for allj, or if p is a 2-point:

(11.38)u = (xbyc)a

v = zw =

∑mj=0 φij (z)(x

byc)ij + xdye

where gcd(b, c) = 1, be − cd 6= 0, all φij (z) are non zero, i0 < i1 < · · · < im,(b, c) 6≥ (d, e), and φij (0) 6= 0 for all j. In either case, there exists a largest l ≤ msuch that a | ij if j ≤ l.


If at p there is a form (11.37), set

φp(u, v) =∑j≤l

φij (y)xij =

∑j≤l

φij (v)uija .

If at p there is a form (11.38), set

φp(u, v) =∑j≤l

φij (z)(xbyc)ij =

∑j≤l

φij (v)uija .

In both cases w − φp(u, v) is weakly good for f at p.By Lemma 11.5, there exist finitely many points p1, . . . , pn ∈ X such that if we

set φi(u, v) = φpi(u, v), then for all p ∈ f−1(q), some wi = w − φi(u, v) is weaklygood for f at p.

Since the φij are units in OX,pi, we see that u, v, wi satisfy 5 or 6 of Definition

10.1 of super parameters at pi.Let p = pi for some i, with the notation of (11.37) or (11.38).Suppose that p ∈ f−1(q). We must show that u, v, wi are super parameters at

p.At p we have permissible parameters x, y, z ∈ OX,p such that if p is a 1-point:

(11.39)u = xa

v = y

w =∑mj=0 φij (y)x

ij + xn(β + z)

where β ∈ k, im < n, all φij (y) are non zero,or if p a 2-point:

(11.40)u = (xbyc)a

v = z

w =∑mj=0 φij (z)(x

byc)ij + xdye

where gcd(b, c) = 1, im(b, c) 6≥ (d, e), all φij (z) are non zero.

We know from Lemma 11.6 that all φij and φij are units in OX,p and OX,prespectively.

It will follow that wi are super parameters at p after we have proven that ifthere exists t with t ≤ minl,m and

ija

=ija

and φij (0) = φij (0)

for 0 ≤ j ≤ t, then we have equality of power series in u,

φij (v) = φij (v)

for 0 ≤ j ≤ t, and thus equality of series

t∑j=0

φij (v)uija =

t∑j=0

φij (v)uija .

We will prove this in the case when p = pi satisfies (11.37) and p satisfies(11.39).

The proof of the remaining three cases is similar.


We prove this by induction. First assume that l ≥ 0 and

(11.41)i0a

=i0a

and φi0(0) = φi0(0).Let ν be a valuation of k(X) such that the center of ν on X is p, and identifying

ν with an extension of ν to the quotient field of OX,p which dominates OX,p, wehave

ν(w −k∑j=0

φij (y)xij ) > ν(w −

k−1∑j=0

φij (y)xij )

for 0 ≤ k ≤ m.Let pn be the center of ν on Un, qn be the center of ν on Vn in the commutative

diagram (11.35).With the notation of the proof of Lemma 11.7, we see that the rational map

U → Vr(1) is a morphism at p = pr(1), and fr(1)(p) = qr(1) has (formal) permissibleparameters ur(1), v, wr(1) defined by

(11.42) u = uer(1), w = ufr(1)(wr(1) + φi0(0)), wr(1) = wr(1) − [φi0(v)− φi0(0)]

with gcd(e, f) = 1 and ef = a

i0.

ur(1) = wr(1) = 0

are local equations of (a branch of) the nonfinite locus of fr(1) : U → Vr(1) at qr(1).We see from (11.42), (11.39) and (11.41) that U → Vr(1) is a morphism at p = pr(1),that r(1) = r(1), and fr(1)(p) = qr(1). Further,

ur(1) = wr(1) − [φi0(v)− φi0(0)] = 0

are also (formal) local equations of (a branch of) the nonfinite locus of Ur(1) → Vr(1)at qr(1). Since q is a general point of C, the nonfinite locus of Ur(1) → Vr(1) is anonsingular curve. Thus

φi0(v) = φi0(v).

(11.41) implies that

gcd(a, i0) =a

eand

gcd(a, i0) =a

e.

Suppose that we further have that l ≥ 1, i1a = i1

a and φi1(0) = φi1(0). Then from(11.41) we have

i1 − i0a

=i1 − i0a

.

Thusgcd(a, i0)i1 − i0

=gcd(a, i0)i1 − i0

.

We have (with the notation of (11.36) of the proof of Lemma 11.7) that

e2f2

=gcd(a, i0)i1 − i0

,


the rational map U → Xr(2) is a morphism at p = pr(2), and qr(2) = fr(2)(p) haspermissible parameters ur(2), v, wr(2) defined by

ur(1) = ue2r(2), vr(1) = vr(2), wr(1) = uf2r(2)(wr(2) + φi1(0)),

wr(2) = wr(2) − [φi1(v)− φi1(0)].

We see that fr(2) is a morphism at p, and φi1(0) = φi1(0) implies that fr(2)(p) =qr(2). Let

wr(2) = wr(2) − [φi1(v)− φi1(0)].We have

ur(2) = wr(2) = 0and

ur(2) = wr(2) − [φi1(v)− φi1(0)] = 0

are local equations of (branches of) the nonfinite locus of fr(1) at qr(1). Thus, sincethe nonfinite locus of fr(1) is nonsingular,

φi1(v) = φi1(v).

Assume that l ≥ 2,

i0a

=i0a,i1a

=i1a

andi2a

=i2a

and φi2(0) = φi2(0), as well as φi0(0) = φi0(0) and φi1(0) = φi1(0).Then

gcd(a, i0, i1)a

=gcd(a, i0, i1)

a.

Nowi2 − i1a

=i2 − i1a

impliesgcd(a, i0, i1)i2 − i1

=gcd(a, i0, i1)i2 − i1

.

The rational map U → Vr(3) is a morphism at p and p, fr(3)(p) = fr(3)(p) =qr(3), and

φi2(v) = φi2(v)

since the fundamental locus of fr(3) is nonsingular.Iterating, we see that if j ≤ t,

ija

=ija

andφij (0) = φij (0)

for j ≤ t, then

gcd(a, i0, . . . , ij−1)ij − ij−1

=gcd(a, i0, . . . , ij−1)

ij − ij−1

for j ≤ t andφij (v) = φij (v)

for j ≤ t.


We have verified that a general point of every one dimensional component ofGY (f, τ) is perfect. Thus the conclusions of the lemma hold.

Definition 11.11. Suppose that f : X → Y is prepared. Let V be the largestopen subset of Y on which the conclusions of Lemma 11.7 hold. Let Θ(Y ) = Θ(f, Y )be the set of perfect 1-points in V ∩GY (f, τ).

Remark 11.12. Suppose that f is τ -prepared. Then GY (f, τ) − Θ(f, Y ) is afinite set by Lemma 11.10, Lemma 11.7 and Lemma 11.5.

CHAPTER 12

Relations

In this chapter, we suppose that Y is a nonsingular projective 3-fold withtoroidal structure DY , and f : X → Y is a dominant proper morphism of non-singular 3-folds, with toroidal structures DY and DX = f−1(DY ), such that DX

contains the singular locus of f .

Definition 12.1. A quasi-pre-relation R on Y is an association U from alocally closed subset U(R) ⊂ DY , such that U(R) contains no non trivial opensubsets of 2-curves, U(R) contains no 3-points and dim U(R) ≤ 1.

If q ∈ U(R) is a 2-point,

R(q) = (SR(q), ER,1(q), ER,2(q), wR(q), uR(q), vR(q), eR(q), aR(q), bR(q), λR(q))

with aR(q), bR(q), eR(q) ∈ Z, gcd(aR(q), bR(q), eR(q)) = 1, eR(q) > 1,

uR(q), vR(q), wR(q)

are (possibly formal) permissible parameters at q with uR(q), vR(q) ∈ OY,q, 0 6=λR(q) ∈ k. SR(q) = Spec(OY,q/(wR(q))), ER,1(q) is the divisor uR(q) = 0 on SR(q)and ER,2(q) is the divisor vR(q) = 0 on SR(q).

We will also allow quasi-pre-relations with aR(q) = bR(q) =∞, eR(q) = 1 andλR(q) = 1.

If q ∈ U(R) is a 1-point,

R(q) = (SR(q), ER(q), wR(q), uR(q), vR(q), eR(q), aR(q), λR(q))

with aR(q), eR(q) ∈ Z, gcd(aR(q), eR(q)) = 1, eR(q) > 1, uR(q), vR(q), wR(q) are(possibly formal) permissible parameters at q with uR(q), vR(q) ∈ OY,q, 0 6= λR(q) ∈k. SR(q) = Spec(OY,q/(wR(q))), ER(q) is the divisor uR(q) = 0 on SR(q).

We will also allow quasi-pre-relations with aR(q) =∞, eR(q) = 1 and λR(q) =1.

A restriction R′ of a quasi-pre-relation R is the association from a locally closedsubset U(R′) of U(R) such that R′(q) = R(q) for q ∈ U(R′).

Suppose that R is a quasi-pre-relation and q ∈ U(R) is a 2-point. Let

u = uR(q), v = vR(q), w = wR(q), a = aR(q), b = bR(q), e = eR(q), λ = λR(q).

If a, b 6=∞, then R(q) is determined by the expression

(12.1) we − λuavb.

Depending on the signs of a and b, this expression determines a (formal) germ at qof an (irreducible) surface singularity

(12.2) F = FR(q) = 0

113

114 12. RELATIONS

of one of the following forms:

F = we − λuavb = 0

if a, b ≥ 0 and a+ b > 0,F = weu−a − λvb = 0

if a < 0, b > 0,F = wev−b − λua = 0

if b < 0, a > 0.In the remaining case, a, b ≤ 0,

F = weu−av−b − λ

is a unit in OY,q and F (q) 6= 0.If a, b =∞, and q ∈ U(R) is a 2-point, then R(q) is determined by the expres-

sion

(12.3) F = FR(q) = wR(q) = 0.

Suppose that R is a quasi-pre-relation, and q ∈ U(R) is a 1-point. Let

u = uR(q), v = vR(q), w = wR(q), a = aR(q), e = eR(q), λ = λR(q).

If a 6=∞, then R(q) is determined by the expression

(12.4) we − λua.

This expression determines a (formal) germ at q of an (irreducible) surface singu-larity

(12.5) F = FR(q) = 0

of the formF = we − λua = 0.

if a > 0. In the remaining case, a ≤ 0, so that

F = weu−a − λ

is a unit in OY,q and F (q) 6= 0.If a =∞, and q ∈ U(R) is a 1-point, then R(q) is determined by the expression

(12.6) F = FR(q) = wR(q) = 0.

A quasi-pre-relation R is resolved if FR(q) is a unit in OY,q for all q ∈ U(R)(This includes the case U(R) = ∅).

Definition 12.2. A subvariety G of Y is an admissible center for a quasi-pre-relation R on Y if one of the following holds:

1. G is a 2-point.2. G is a 1-point.3. G is a 2-curve of Y .4. G ⊂ DY is a nonsingular curve which contains a 1-point and makes SNCs

with DY . If q ∈ U(R) ∩G then the (formal) germ of G at q is containedin the germ wR(q) = 0.

12. RELATIONS 115

Observe that admissible centers are possible centers.Suppose that R is a quasi-pre-relation on Y , G is an admissible center for R,

and Ψ : Y1 → Y is the blow up of G.Let W be the union over q ∈ U(R) of points q1 in Ψ−1(q) such that q1 is on the

strict transform of wR(q) = 0. Assume that this is a locally closed subset of DY1 ofdimension ≤ 1 which contains no 2-curves or 3-points (This condition will alwaysbe satisfied when R is a pre-relation, Definition 12.3). The transform R1 of R onY1 is then the quasi-pre-relation on Y1 defined by the condition that U(R1) = W .For such q1, R1(q1) is determined by the following rules:

If q ∈ U(R) ∩G, and

u = uR(q), v = vR(q), w = wR(q),

then G has local equations of one of the following forms at q (corresponding to thecases of Definition 12.2):

1.,2. u = v = w = 0,3. u = v = 0,

4. a) q a 2-point, u = w = 0 or v = w = 04. b) q a 1-point, u = w = 0.If q1 ∈ U(R1) ∩Ψ−1(q), then

u1 = uR1(q1), v1 = vR1(q1), w1 = wR1(q1)

are defined, respectively, by1., 2. u = u1, v = u1(v1+α), w = u1w1 for some α ∈ k, or u = u1v1, v = v1, w =

v1w1,3. u = u1, v = u1(v1 + α), w = w1 for some α ∈ k, or u = u1v1, v = v1, w =w1,

4. a) u = u1, w = u1w1 or v = v1, w = v1w1

4. b) u = u1, v = v1, w = u1w1.

Suppose that q1 ∈ U(R1) ∩Ψ−1(q) and case 1 holds. Suppose that

R(q) = (S,E,w, u, v, e, a, b, λ).

If 0 6= α and a, b 6=∞, we define

aR1(q1) = a+ b− e, eR1(q1) = e, λR1(q1) = λαb.

R1(q1) is thus determined by

we1 − λαbua+b−e1 .

If a, b 6=∞ andu = u1, v = u1v1, w = u1w1,

we define

aR1(q1) = a+ b− e, bR1(q1) = b, eR1(q1) = e, λR1(q1) = λ.

R1(q1) is determined by we1 − λua+b−e1 vb1.If a, b 6=∞ and

u = u1v1, v = v1, w = v1w1,

we define

aR1(q1) = a, bR1(q1) = a+ b− e, eR1(q1) = e, λR1(q1) = λ.

116 12. RELATIONS

R1(q1) is determined by we1 − λua1va+b−e1 .If a = b =∞, we define

aR1(q1) =∞, bR1(q1) =∞, eR1(q1) = 1, λR1(q1) = 1,

and R1(q1) is determined by wR1(q1).In cases 2,3 and 4, we define R1(q1) in an analogous way.Suppose that Ψ1 : Y1 → Y is a sequence of blow ups of admissible centers

for (the transforms of) R, R1 is the transform of R on Y1, q ∈ U(R) and q1 ∈Ψ−1

1 (q) ∩ U(R1). Let

u = uR(q), v = vR(q), w = wR(q).

u1 = uR1(q1), v1 = vR1(q1), w1 = wR1(q1).

u and v are related to u1, v1 birationally. That is, k(u, v) = k(u1, v1). We have oneof the following expressions:

(12.7) u = ua1vb1, v = uc1v

d1 , w = ue1v

f1w1

with ad− bc = ±1,

(12.8) u = ua1γ1(u1, v1), v = ub1γ2(u1, v1), w = uc1γ3(u1, v1)w1

where γ1, γ2, γ3 are unit series,

(12.9) u = (ua1vb1)tγ1(u1, v1), v1 = (ua1v

b1)kγ2(u1, v1), w = ue1v

f1 γ3(u1, v1)w1

where gcd(a, b) = 1, and γ1, γ2, γ3 are unit series.

Definition 12.3. A quasi-pre-relation R is a pre-relation if there exists a non-singular 3-fold Y R with toroidal structure DY , a pre-relation R0 on Y R such thatU(R0) = q is a single point with

R0(q) = (· · · , wR0(q), uR0(q), vR0(q), · · · )

and a sequence of possible blow ups

ΨR : Y = Yn → · · · → Y1 → Y0 = Y R

where each Yi has a quasi-pre-relation Ri which is the restriction of the transformof Ri−1, and Yi+1 → Yi is an admissible blow up for Ri, and R = Rn.

Definition 12.4. A pre-relation R on Y is algebraic if there exists an opensubset V of Y and a nonsingular irreducible closed surface Ω(R) ⊂ V such thatΩ(R) makes SNCs with DY , U(R) ⊂ Ω(R) and SR(q) is the (formal) germ of Ω(R)at q for all q ∈ U(R). Further, if q ∈ Ω(R) ∩ DY , then there exist permissibleparameters uq, vq, wq at q such that wq = 0 is a local equation of Ω(R).

Suppose that R is algebraic, and Ψ : Y1 → Y is an admissible blow up. thenafter possibly replacing Ω(R) with an open subset of Ω(R) (containing U(R)), wehave that the transform R1 of R is algebraic, where Ω(R1) is the strict transformof Ω(R) by Ψ.

Definition 12.5. Suppose that f : X → Y is prepared.A primitive relation R for f is

1. A pre-relation R on Y .

12. RELATIONS 117

2. A locally closed subset T (R) ⊂ f−1(U(R)) ∩ GX(f, τ) such that if p ∈T (R) and f(p) is a 2-point with

R(f(p)) = (S,E1, E2, w, u, v, e, a, b, λ),

then u, v are toroidal forms at p. If a, b 6=∞, then there exist permissibleparameters x, y, z at p for u, v, w such that

(12.10) we = uavbΛ(x, y, z)

where Λ(0, 0, 0) = λ.If a = b = ∞, then u, v, w have a monomial form (Definition 4.4) at

p.If f(p) is a 1-point with

R(f(p)) = (S,E,w, u, v, e, a, λ),

then u, v are toroidal forms at p. If a 6= ∞, then there exist permissibleparameters x, y, z at p for u, v, w such that

(12.11) we = uaΛ(x, y, z)

where Λ(0, 0, 0) = λ.If a =∞, then u, v, w have a monomial form (Definition 4.4) at p.

In all cases, we define R(p) = R(f(p)).A relation R for f is a finite set of pre-relations Ri on Y with associated

primitive relations Ri for f such that the sets T (Ri) are pairwise disjoint.We denote U(R) = ∪iU(Ri) and T (R) = ∪iT (Ri), and define

R(p) = Ri(p)

if p ∈ T (Ri).If U(Ri)∩U(Rj) 6= ∅, then we further require that YRi

= YRj(with the notation

of Definition 12.3) and u(Ri)0(q)= u(Rj)0(q)

, v(Ri)0(q)= v(Rj)0(q)

. This implies that

uRi(q)= uRj(q)

, vRi(q)= vRj(q)

if q ∈ U(Ri) ∩ U(Rj). We will call Ri the pre-relations associated to R. We willsay that R is algebraic if each Ri is algebraic and

(12.12) Ω(Ri) ∩ U(R) = U(Ri)

for all i. We will also denote Ω(Ri) = Ω(Ri). For p ∈ T (Ri) ⊂ T (R), we denote

R(p) =(S = SR(p), E1(p), E2(p), w = wR(p), u = uR(p),v = vR(p), e = eR(p), a = aR(p), b = bR(p), λ = λR(p)

)if f(p) is a 2-point,

R(p) =(S = SR(p), E(p), w = wR(p), u = uR(p),v = vR(p), e = eR(p), a = aR(p), λ = λR(p)

)if f(p) is a 1-point.

A relation R is resolved if T (R) = ∅.

118 12. RELATIONS

Definition 12.6. Suppose that f : X → Y is prepared, R is a relation for fand

X1f1→ Y1

Φ ↓ ↓ Ψ

Xf→ Y

is a commutative diagram such that1. Ψ is a product of blow ups which are admissible for all of the pre-relationsRi associated to R (and their transforms) and Φ is a product of blow upsof possible centers

2. f1 is prepared.3. Let R

1

i be the transforms of the Ri on Y1 and let

Ti = Φ−1(T (Ri)) ∩ f−11 (U(R

1

i )) ∩GX1(f1, τ).

Suppose that if p ∈ Ti then uR

1i (f1(p))

, vR

1i (f1(p))

are toroidal forms atp.

Then the transform R1 of R for f1 is the relation for f1 defined by

T (R1) = ∪Ti,

R1(p) = R1

i (f1(p))for p ∈ Ti.

It is straightforward to verify that R1 satisfies the conditions of Definition 12.5,substituting from (12.10), (12.11) and (12.7) - (12.9) into (4.1).

CHAPTER 13

Well prepared morphisms

Suppose that τ ∈ N and f : X → Y is a dominant, proper, prepared morphismof nonsingular 3-folds with toroidal structures DY and DX = f−1(DY ). Furthersuppose that the singular locus of f is contained in DX . If R is a relation for f withassociated pre-relations Ri, then for p ∈ T (Ri) such that f(p) = q is a 2-point,we have that(13.1)

R(p) =(Si = SR(p), E1 = ER,1(p), E2 = ER,2(p), wi = wR(p), u = uR(p),

v = vR(p), ei = eR(p), ai = aR(p), bi = bR(p), λi = λR(p)

)which we will abbreviate (as in (12.1)) as

(13.2) R(p) = weii − λiu

aivbi ,

with ei > 1, if ai, bi 6=∞, or (as in (12.3))

(13.3) R(p) = wi

if ai, bi =∞. In this case, we require that u, v, wi have a monomial form (Definition4.4) at p.

For p ∈ T (Ri) such that f(p) = q is a 1-point, we have that

(13.4) R(p) =(Si = SR(p), E1 = ER(p), wi = wR(p), u = uR(p),

v = vR(p), ei = eR(p), ai = aR(p), λi = λR(p)

)which we will abbreviate (as in (12.4)) as

(13.5) R(p) = weii − λiu

ai ,

with ei > 1, if ai 6=∞, or (as in (12.5))

(13.6) R(p) = wi

if ai = ∞. In this case, we require that u, v, wi have a monomial form (Definition4.4) at p.

Recall that if p′ ∈ T (R) is such that f(p′) = f(p), then uR(p′) = uR(p) = u andvR(p′) = vR(p) = v. Let I be an index set for the pre-relations Ri associated toR.

Definition 13.1. Suppose that τ ≥ 1. A prepared morphism f : X → Y ispre-τ -quasi-well prepared with relation R if:

1. T (R) = GX(f, τ) ∩ f−1(U(R))2. Suppose that p ∈ T (R). Then τ > 1 implies R(p) has a form (13.2) or

(13.5), τ = 1 implies R(p) has a form (13.3) or (13.6).3. Suppose that τ > 1, q ∈ U(Ri) and p ∈ f−1(q) ∩ T (Ri). Let


, w = wRi(q), e = eRi

(q).

119

120 13. WELL PREPARED MORPHISMS

a) Suppose that q is a 2-point.(i) Suppose that u, v satisfy (4.1) at p, and w = xcγ, where γ is a

unit in OX,p. Then

e = |(aZ + bZ + cZ)/(aZ + bZ)|.(ii) Suppose that u, v satisfy (4.2) at p, and w = xfygγ where γ is

a unit in OX,p. Then

e = |[(a, b)Z + (c, d)Z + (f, g)Z]/[(a, b)Z + (c, d)Z]|.(iii) Suppose that u, v satisfy (4.3) at p and w = (xayb)lγ, where γ

is a unit in OX,p. Then

e = |(kZ + tZ + lZ)/(kZ + tZ)|.(iv) Suppose that u, v satisfy (4.4) at p and w = xgyhziγ where γ

is a unit in OX,p and

Det

a b cd e fg h i

= 0.

Then

e = |[(a, b, c)Z + (d, e, f)Z + (g, h, i)Z]/[(a, b, c)Z + (d, e, f)Z]|b) Suppose that q is a 1-point.

(i) Suppose that u, v satisfy (4.5) at p, and w = xbγ where γ is aunit in OX,p. Then

e = |(aZ + bZ)/aZ|.(ii) Suppose that u, v satisfy (4.6) at p, and w = (xayb)lγ where γ

is a unit in OX,p. Then

e = |(kZ + lZ)/kZ|.

4. If q ∈ U(Ri) ∩ U(Rj), then there exists λij(u, v) ∈ k[[u, v]], with

u = uRi(q)= uRj(q)

, v = vRi(q)= vRj(q)

, wi = wRi(q), wj = wRj(q)

such thatwj = wi + λij(u, v),

and with the notation of Definition 12.5, there exists a series

(λij)0(u(Rj)0(q), v(Rj)0(q)

)

such that

w(Rj)0(q)− w(Ri)0(q)

= (λij)0(u(Rj)0(q), v(Rj)0(q)

),

and λij(u, v) is obtained from λ0ij(u(Rj)0(q)

, v(Rj)0(q)) from the appropriate

expression (12.7) - (12.9).5. Suppose that q ∈ U(Ri), where Ri is a relation associated to R. Thenu = uRi(q)

, v = vRi(q), wi = wRi(q)

are super parameters at q (Definition10.1).

f is τ -quasi-well prepared with relation R if f is pre-τ -quasi-well prepared withT (R) = GX(f, τ).

13. WELL PREPARED MORPHISMS 121

Remark 13.2. All of the equations of 3 of Definition 13.1 follow from thecondition that p ∈ T (Ri)∩f−1(q) except for a) i where (4.1) holds, and a) iii where(4.3) holds.

Remark 13.3. If f : X → Y is τ -quasi-well prepared, then f is τ prepared.

We will allow a prepared morphism without relation (U(R) = ∅) as a type ofpre-τ -quasi-well prepared morphism.

Definition 13.4. f : X → Y is τ -quasi-well prepared with relation R andpre-algebraic structure if f is τ -quasi-well prepared with relation R and

uRi(q), vRi(q)

, wRi(q)∈ OY,q

for all Ri associated to R, and q ∈ U(Ri).

Definition 13.5. f : X → Y is τ -well prepared with relation R if1. f is τ -quasi-well prepared with relation R and pre-algebraic structure.2. The primitive pre-relations Ri associated to R are algebraic, and R is

algebraic (Definition 12.5).3. Suppose that q ∈ U(Ri) ∩ U(Rj). Let wi = wRi(q)

and wj = wRj(q),

u = uRi(q)= uRj(q)

, v = vRi(q)= vRj(q)

.Suppose that q is a 2-point. Then there exists a unit series φij ∈

k[[u, v]] and aij , bij ∈ N (or φij = 0 with aij = bij = −∞) with

(13.7) wj = wi + uaijvbijφij .

Suppose that q is a 1-point. Then there exists a unit series φij ∈k[[u, v]] and cij ∈ N (or φij = 0 with cij = −∞) with

(13.8) wj = wi + ucijφij .

4. For q ∈ U(R) a 2-point, set Iq = i ∈ I | q ∈ U(Ri). Then the set

(13.9) (aij , bij) | i, j ∈ Iqfrom equation (13.7) is totally ordered.

Definition 13.6. Suppose that f : X → Y is pre-τ -quasi-well prepared, q ∈ Yis a 1-point such that q 6∈ U(R). A curve C ⊂ DY such that q ∈ C is called aresolving curve for f and R at q if

1. C makes SNCs with DY .2. C ∩GY (f, τ) ⊂ q.3. If q ∈ C is a 2-point, then there exist super parameters u, v, w at q such

that u = w = 0 are local equations of C at q.4. If q ∈ C is a 1-point, then there exist permissible parameters u, v, w at q

such that u, v are toroidal forms at p for all p ∈ f−1(q), and u = v = 0are local equations of C at q.

Definition 13.7. Suppose that f : X → Y is pre-τ -quasi-well prepared withrelation R.

1. A 2-point q ∈ U(R) is prepared for R.2. A 1-point or 2-point q ∈ Y such that q 6∈ U(R) is prepared for R if there

exist super parameters u, v, w at q.3. A 2-curve C ⊂ Y is prepared for R.4. A 1-point q ∈ U(R) is prepared.


5. A resolving curve C for f and R at a 1-point q 6∈ U(R) is prepared.

If E is a component of DY , Ri is pre-algebraic, and q ∈ U(Ri), we will denoteE · SRi

(q) for the Zariski closure in Y of the curve germ u = wRi(q)= 0 at q, where

u = 0 is a local equation of E.

Definition 13.8. Suppose that f : X → Y is τ -well prepared with relation Rfor f . A nonsingular curve C ⊂ DY which makes SNCs with DY is prepared for Rof type 6 if

1. C = Eα · SRi(qβ) for some component Eα of DY , pre-relation Ri associ-

ated to R and qβ ∈ U(Ri).2. Ω(Ri) contains C.3. If C ′ = Eγ · SRj

(qδ) is such that C ′ ⊂ Ω(Rj), C 6= C ′, and q ∈ C ∩ C ′,

then q ∈ U(Ri) ∩ U(Rj) and C ′ = Eγ · SRj(q).

4. If j 6= i and C = Eγ · SRj(qδ) then C satisfies 1 and 2 and 3 of this

definition (for Rj). (In this case we have by (12.12) that U(Rj) ∩ C =U(Ri) ∩ C).

5. Let

IC = j ∈ I | C = Eγ · SRj(qδ) for some Rj , Eγ , qδ ∈ U(Rj).

Suppose that q ∈ C is a 1-point or a 2-point such that q 6∈ U(R). Thenthere exist u, v ∈ OY,q such that for j ∈ IC there exists wj ∈ OY,q suchthat(a) wj = 0 is a local equation of Ω(Rj) and u, v, wj are permissible

parameters at q such that u = wj = 0 are local equations of C at q.(b) u, v, wj are super parameters at q.(c) If i, j ∈ IC and q is a 1-point, there exist relations

wj = wi + ucijφij(u, v)

where φij is a unit series (or φij = 0 and cij = −∞).(d) If i, j ∈ IC and q is a 2-point (with q 6∈ U(R)) then there exist

relationswj = wi + uaijvbijφij(u, v)

where φij is a unit series (or φij = 0 and aij = bij = −∞), and theset (aij , bij) is totally ordered.

If f : X → Y is τ -well prepared with relation R, and Ri is a pre-relationassociated to R, we will feel free to replace Ω(Ri) with an open subset of Ω(Ri)containing U(Ri), and all curves C = E · SRi

(q) such that E is a component ofDY , q ∈ U(Ri) and C is prepared for R of type 6. This convention will allow somesimplification of the statements of the theorems and proofs.

Definition 13.9. f : X → Y is τ -very-well prepared with relation R if1. f is τ -well prepared with relation R.2. If E is a component of DY and q ∈ U(Ri) ∩ E, then C = E · SRi

(q) isprepared for R of type 6 (Definition 13.8).

3. For all Ri associated to R, let

Vi(Y ) =γ = Eα · SRi

(qγ) | qγ ∈ U(Ri), Eα is a component of DY

.


ThenFi =

∑γ∈Vi(Y )

γ

is a SNC divisor on Ω(Ri) whose intersection graph is a forest (its con-nected components are trees).

If f : X → Y is τ -very-well prepared, we will feel free to replace Ω(Ri) withan open neighborhood of Fi in Ω(Ri). This will allow some simplification of theproofs.

Remark 13.10. Suppose that f : X → Y is τ -very well prepared. Then it fol-lows from Definition 13.9 and (12.12) that Fi∩U(R) = U(Ri) for all Ri associatedto R.

Definition 13.11. Suppose that f : X → Y is pre-τ -quasi-well prepared (orτ -well prepared or τ -very-well prepared) with relation R. Let Ri be the pre-relations associated to R. Suppose that G is a point or nonsingular curve in Ywhich is an admissible center for all of the Ri. Then G is called a permissiblecenter for R if there exists a commutative diagram

(13.10)X1

f1→ Y1

Φ ↓ ↓ Ψ

Xf→ Y

where Ψ is the blow up of G and Φ is a sequence of blow ups

X1 = Xn → · · · → X1 → X

of nonsingular curves and 3-points γi which are possible centers such that1. f1 is prepared and the assumptions of Definition 12.6 hold so that the

transform R1 of R for f1 is defined.2.

τf1(p) ≤ τf (φ(p))

for p ∈ DX1 .3. f1 : X1 → Y1 is pre-τ -quasi-well prepared, (or τ -well prepared or τ -very-

well prepared) with relation R1.

(13.10) is called a pre-τ -quasi-well prepared (or τ -well prepared or τ -very-wellprepared) diagram of R and Ψ.

Definition 13.12. Suppose that f : X → Y is τ -well prepared (or τ -very-wellprepared) with relation R and C ⊂ Y is prepared for R of type 6. Then C is a∗-permissible center for R if there exists a commutative diagram

(13.11)X1

f1→ Y1

Φ ↓ ↓ Ψ

Xf→ Y

such that1. f1 is τ -prepared and the assumptions of Definition 12.6 hold so that the

transform R1 of R for f1 is defined.


2.

τf1(p) ≤ τf (φ(p))

for p ∈ DX1 .3. f1 : X1 → Y1 is τ -well prepared (or τ -very-well prepared).4. (13.11) has a factorization

(13.12)

X1 = Xmf1=fm→ Y m = Y1

↓ ↓...

...↓ ↓

X2f2→ Y 2

Φ2 ↓ ↓ Ψ2

X1f1→ Y 1

Φ1 ↓ ↓ Ψ1

Xf→ Y

where Ψ1 is the blow up of C,

(13.13)X1

f1→ Y 1

Φ1 ↓ ↓ Ψ1

Xf→ Y

is a τ -well prepared diagram of R and Ψ1 of the form (13.10), each Ψi+1 :Y i+1 → Y i for i ≥ 1 is the blow up of a 2-point q ∈ Y i which is preparedfor the transform Ri of R on Xi of type 2 of Definition 13.7, and

Xi+1

fi+1→ Y i+1

Φi+1 ↓ ↓ Ψi+1

Xifi→ Y i

is a τ -well prepared diagram of Ri and Ψi+1 of the form of (13.10).

5. Suppose that E is the strict transform of Ψ−1

1 (C) on Y1. then E ·R1

i (q) isprepared for R1 of type 6 for all primitive relations R

1

i associated to R1,and q ∈ U(R1

i ) ∩ E.6. Suppose that γ ⊂ Y is a curve which is prepared for R of type 6. Then

the strict transform of γ on Y1 is prepared for R1 of type 6.

Definition 13.13. Suppose that f : X → Y is pre-τ -quasi-well prepared (orτ -well prepared or τ -very-well prepared) with relation R. Suppose that

(13.14)X1

f1→ Y1

Φ ↓ ↓ Ψ

Xf→ Y


is a commutative diagram such that there is a factorization

(13.15)

X1 = Xmf1=fm→ Y m = Y1

↓ ↓...

...↓ ↓

X2f2→ Y 2

Φ2 ↓ ↓ Ψ2

X1f1→ Y 1

Φ1 ↓ ↓ Ψ1

Xf→ Y

where each commutative diagram

Xi+1 → Y i+1

Φi+1 ↓ ↓ Ψi+1

Xi → Y i

is either of the form (13.10) or of the form (13.11). Then (13.14) is called a pre-τ -quasi-well prepared (or τ -well prepared or τ -very-well prepared) diagram of R andΨ.

CHAPTER 14

Construction of τ-well prepared diagrams

Lemma 14.1. Suppose that f : X → Y is pre-τ -quasi-well prepared (or τ -wellprepared or τ -very-well prepared) and C ⊂ Y is a 2-curve. Then C is a permissiblecenter for R, and there exists a pre-τ -quasi-well-prepared (or τ -well prepared orτ -very-well prepared) diagram (13.10) of R and the blow up Ψ : Y1 → Y of C suchthat Φ is a product of blow ups of 2-curves. Furthermore,

1. Φ is an isomorphism over f−1(Y − C) and τf1(p1) ≤ τf (Φ(p1)) if p1 ∈DX1 .

2. Suppose that f is τ -well prepared. Then(a) Let E be the exceptional divisor for Ψ. Suppose that q ∈ U(R

1

i )∩E forsome Ri associated to R. Let γi = S

R1i(q) · E. Then γi = Ψ−1(Ψ(q))

is a prepared curve for R1 of type 6.(b) If γ is a prepared curve for R, then the strict transform of γ on Y1

is a prepared curve for R1.3. Suppose that q ∈ C, p ∈ f−1(q), p′ ∈ Φ−1(p), u, v, w are permissible

parameters at q such that u = v = 0 are local equations of C, and w isgood at p. If w is not good at p′, then τf1(p

′) < τf (p).4. If f is τ prepared then f1 is τ prepared.

Proof. By Lemma 10.4, there exists a commutative diagram

X1f1→ Y1

Φ ↓ ↓ Ψ

Xf→ Y

where Φ is a product of blow ups of 2-curves and f1 is prepared, with the propertythat τf1(p1) ≤ τf (Φ(p1)) if p1 ∈ DX1 . We further have that Φ is an isomorphismover f−1(Y − C), and f1 is τ -prepared if f is τ -prepared.

Let Ri be the pre-relations on Y associated to R. C is an admissible centerfor the Ri (Definition 12.2). Let R1

i be the transforms of the Ri on Y1.By consideration of the local forms of the map f1 given in the proof of Lemma

10.4, we see that the conditions of Definition 12.6 hold so that we can define thetransform R1 of R for f1.

We consider one explicit case in detail. Suppose that q1 ∈ U(R1

i ), and p1 ∈f−11 (q1) ∩ Φ−1(T (Ri)) is a 3-point. Let q = Ψ(q1), p = Φ(p1). We have that q

and q1 are 2-points, and p is a 3-point (since Lemma 5.2 is used to construct Φ).There exist permissible parameters u = uRi(p), v = vRi(p), wi = wRi(p) at q suchthat Ri(p) = Ri(q) is determined by

wei − λuavb

127

128 14. CONSTRUCTION OF τ-WELL PREPARED DIAGRAMS

with e = eRi(p), λ = λRi(p) if a = aRi(p), b = bRi(p) 6=∞, and by

wi = 0

if a = aRi(p) = b = bRi(p) = ∞. There exist permissible parameters x, y, z foru, v, wi at p such that an expression (9.1) of Definition 9.1 holds for u, v, wi and wehave a relation

wei = uavbΛ(x, y, z)

where Λ(x, y, z) is a unit series in OX,p with Λ(0, 0, 0) = λ if a, b 6=∞ and u, v, wihave a monomial form in x, y, z if a = b = ∞. After possibly interchanging u andv, we may assume (since q1 is a 2-point) that q1 has permissible parameters u, v, wisuch that

u = u, v = uv.

Since p1 is a 3-point, OX1,p1 has regular parameters x1, y1, z1 such that

x = xa111 ya12

1 za131

y = xa211 ya22

1 za231

z = xa311 ya32

1 za331

with Det(aij) = ±1. Thus u, v, wi has an expression of the form of (9.1) in x1, y1, z1.If a, b 6=∞ we have the relation

wei = ua+bvbΛ,

and R1

i (q1) is determined by

(14.1) wei − ua+bvbλ.

If a = b =∞, u, v, wi have a monomial form in x1, y1, z1.Now we will verify that f1 is pre-τ -quasi-well prepared.We first verify that 1 of Definition 13.1 holds. We have that U(R

1

i ) = Ψ−1(U(Ri))for all i (as Ψ is the blow up of a 2-curve).

T (R1) = ∪i[GX1(f1, τ) ∩ Φ−1(T (Ri)) ∩ f−1

1 (U(R1i ))]

= GX1(f1, τ) ∩[∪i(Φ−1(T (Ri)) ∩ f−1

1 (U(R1i )))

]= GX1(f1, τ) ∩

[∪iΦ−1(T (Ri))

]= GX1(f1, τ) ∩

[Φ−1(T (R))

]= GX1(f1, τ) ∩

[Φ−1(f−1(U(R))

]= GX1(f1, τ) ∩ f−1(U(R)).

From the above and the local calculations of the proof of Lemma 10.4, we see that2, 3 and 4 of Definition 13.1 hold. To prove 3, observe that if p ∈ Φ−1(T (Ri)) ∩f−11 (U(R1

i )) and 3 does not hold at p, then τf1(p) < τf (Φ(p)) implies p 6∈ T (R1i ).

It remains to verify that 5 of Definition 13.1 holds.Suppose that q1 ∈ U(R

1

i ) and p1 ∈ f−11 (q1). Then

u = uRi(q), v = vRi

(q), w = wRi(q)

are super parameters at q = Ψ(q1), and p = Φ(p1) ∈ f−1(q) has permissibleparameters x, y, z for u, v, w such that one of the forms of Definition 10.1 hold foru, v, w and x, y, z. We will verify that 5 of Definition 13.1 holds when q1 and q are2-points. The other cases are similar.

14. CONSTRUCTION OF τ-WELL PREPARED DIAGRAMS 129

After possibly interchanging u and v, we have

(14.2)

uR

1i(q1) = u = u

vR

1i(q1) = v = v

u

wR

1i(q1) = w.

We can verify that there exist permissible parameters x1, y1, z1 at p1 such thatu, v, w have one of the forms 1 - 4 of Definition 10.1 in x1, y1, z1. The most difficultcase to verify is when p1 is a 1-point and p is a 3-point. Then since Lemma 5.2 isused to construct Φ, OX1,p1 has regular parameters x1, y1, z1 defined by

x = xa1(y1 + α)b(z1 + β)c

y = xd1(y1 + α)e(z1 + β)f

w = xg1(y1 + α)h(z1 + β)i

where α, β ∈ k are nonzero and

Det

a b c

d e f

g h i

= ±1.

We substitute intou = xaybzc

v = xdyezf


of 4 of Definition 10.1. Using the fact that

rank

a b cd e fj k l

= 3,

we can make a change of variables in x1, y1, z1 to get permissible parametersx1, y1, z1 at p1 satisfying

(14.3)u = xa1v = xd1(γ + y1)xjykzl = xg1(ε+ z1)

with d > a and 0 6= ε, γ ∈ k.For each of the monomials M in the series xgyhziγ we have a relation

(14.4) M e = uavb

with e, a, b ∈ Z. On substitution of (14.2) and (14.3) into (14.4) we see that

M = xe′

1 φ1(y1)

where φ1 is a unit series. Thus u, v, w have an expansion of the form 1 of Definition10.1 in terms of x1, y1, z1.

The other cases can be verified by a similar but simpler argument to show thatu, v, w are super parameters at q1. Thus 5 of Definition 13.1 holds for f1, so thatf1 is pre-τ -quasi-well prepared.

Suppose that f is τ -well prepared. We will verify that f1 is τ -well prepared. 1and 2 of Definition 13.5 are immediate. We must verify that 3 and 4 of Definition13.5 hold for f1.


Suppose that q1 ∈ U(R1

i ) ∩ U(R1

j ). Let q = Φ(q1) ∈ U(Ri) ∩ U(Rj), and

u = uRi(q) = uRj

(q),v = vRi

(q) = vRj(q),

wi = wRi(q), wj = wRj

(q).

We will assume that q and q1 are 2-points, which is the most difficult case.We have a relation

(14.5) wj = wi + uaijvbijφij(u, v),

from (13.7) for f . After possibly interchanging u and v we have permissible param-eters

u = uR

1i(q1) = u

R1j(q1)

v = vR

1i(q1) = v

R1j(q1)

wi = wR

1i(q1), wj = w

R1j(q1)

at q1, where u = u, v = uv. We have

(14.6) wj = wi + uaij+bijvbijφij(u, uv)

so 3 of Definition 13.5 holds for f1.Since the set (13.9) of Definition 13.5 is totally ordered for q ∈ U(R), it follows

from (14.6) that the corresponding set (13.9) for q1 ∈ U(R1) is totally ordered.Thus 4 of Definition 13.5 holds for f1 and R1 and f1 is τ -well prepared.

We now verify 2 of Lemma 14.1. Let E = Ψ−1(C) be the exceptional divisor ofΨ. Suppose that q ∈ U(R

1

i )∩E. Let q = Ψ(q). There exist permissible parameters


, wi = wRi(q)

at q such that u = v = 0 are local equations of C, and after possibly interchangingu and v,

u = uR

1i (q)

, v = uR

1i (q)

(vR

1i (q)

+ α), wi = wR

1i (q)

for some α ∈ k.Since u is a local equation of E at q, γi = S

R1i(q) · E = Ψ−1(q). Let γ = γi.

We will verify that γi is prepared for R1 of type 6. Since q ∈ Ω(Ri), C intersectsΩ(Ri) transversally at q (and possibly a finite number of other points), and Ω(R

1

i )is the strict transform of Ω(Ri) by Ψ, we have that γi ⊂ Ω(R

1

i ). Thus 2 of Definition13.8 holds. Suppose that for some j, component Eα of DY1 and qβ ∈ U(R

1

j ),

γ′ = Eα · SR1j(qβ)) ⊂ Ω(R

1

j ),

γ 6= γ′ and there exists q ∈ γ∩γ′. Let E1 = Ψ(Eα), a component of DY . Then γ =Ψ(γ′) is a curve on DY through q. Since γ′ ⊂ Ω(R

1

j ), we must have γ ⊂ Ω(Rj)∩E1,so that q ∈ U(R) ∩ γ = U(Rj) ∩ γ, and thus γ = E1 · SRj

(q). We thus have that

Eα is the strict transform of E1, q ∈ U(R1

i ) ∩ U(R1

j ) and γ′ = Eα · SR1j(q). Thus 3

of Definition 13.8 holds.Suppose that γ′ = Eα · SR1

j(qδ) and γ = γ′. Then we must have γ′ = E · S

R1j(q)

and 4 of Definition 13.8 holds.Since for j such that q ∈ U(Rj), U(R

1

j ) contains γj = Ψ−1(q)∩E, 5 of Definition13.8 holds. Thus γ = γi is prepared for R1 of type 6.


Suppose that γ 6= C is a prepared curve for R on Y . The verification that thestrict transform γ′ of γ on Y1 is prepared for f1 follows from a local calculation atpoints of C ∩ γ.

3 of the conclusions of Lemma 14.1 follows from a computation of τf1(p) andτf (Φ(p)) for p ∈ DX1 using local forms of the mappings Φ and Ψ.

Suppose that f is τ -very-well prepared. We have seen that f1 is τ -well prepared,so that 1 of Definition 13.9 holds for f1, and 2 of Definition 13.9 holds for f1 byour verification of 2 of this lemma. Let Ψi : Ω(R

1

i ) → Ω(Ri) be the restriction ofΨ to Ω(R

1

i ). Then Ψi is the blow up of the union of nonsingular points C · Ω(Ri)on the nonsingular surface Ω(Ri). Thus since

Fi =∑

γ∈Vi(Y )

γ

is a SNC divisor on Ω(Ri) whose intersection graph is a forest,

Ψ−1

i (Fi) =∑

γ′∈Vi(Y1)

γ′

is a SNC divisor on Ω(R1

i ) whose intersection graph is a forest and 3 of Definition13.9 holds for f1. Thus f1 is τ -very-well prepared.

The proofs of Remarks 14.2 and 14.3 follow easily from the methods of theproof of Lemma 14.1 and Lemma 10.2.

Remark 14.2. Suppose that f : X → Y is pre-τ -quasi-well prepared (or τ -well prepared or τ -very-well prepared) and C ⊂ DX is a 2-curve or a 3-point. LetΦ1 : X1 → X be the blow up of C, f1 = f Φ1 : X1 → Y . Then

1. f1 is pre-τ -quasi-well prepared (or τ -well prepared or τ -very well prepared).2. Suppose that p1 ∈ X1, p = Φ1(p1), q = f1(p), u, v, w are permissible

parameters at q such that w is good (weakly good) at p for f . If w is notgood (not weakly good) at p1 for f , then τf1(p1) < τf (p).

3. (f Φ1)−1(Θ(f, Y )) ⊂ Θ(f1, Y1).

Remark 14.3. The proof of Lemma 14.1 shows that if f : X → Y is pre-τ -quasi-well prepared (or τ -well prepared or τ -very-well prepared), C ⊂ Y is a2-curve, Ψ : Y1 → Y is the blow up of C and Φ : X1 → X is a sequence of blow upsof 2-curves and 3-points such that the rational map f1 : X1 → Y1 is a morphism,then

X1f1→ Y1

Φ ↓ ↓ Ψ

Xf→ Y

is pre-τ -quasi-well prepared (or τ -well prepared or τ -very-well prepared) for R andΨ. In fact, with the above notation, if f satisfies 1 – 3 of Definition 13.1, then f1satisfies 1 – 3 of Definition 13.1. Further, 2 and 3 of the conclusions of Lemma14.1 hold.


Theorem 14.4. Suppose that f : X → Y is prepared, pre-τ -quasi-well preparedand τ = τf (X) ≥ 1. Then there exists a pre-τ -quasi-well prepared diagram

(14.7)X1

f1→ Y1

Φ ↓ ↓ Ψ

Xf→ Y

such that f1 is pre-τ -well prepared, Φ, Ψ are products of blowups of 2-curves, andGY1(f1, τ) contains no 3-points and no 2-curves, so that f1 is τ -prepared.

Proof. By Lemma 7.1 and Lemma 14.1, there exists a diagram (14.7) suchthat f1 is pre-τ -quasi-well prepared, and GY1(f, τ) contains no 3-points. We maythus assume that GY (f, τ) contains no 3-points.

Suppose that C is a 2-curve of Y such that C ⊂ GY (f, τ). Let H be a generalhyperplane section of Y . H intersects C transversally in general points of C, andf∗(H) is nonsingular and makes SNCs with DX by Bertini’s theorem. Thus ata general point q of C there exist uniformizing parameters u, v, w such that forp ∈ f−1(q), u, v, w have a form 2 (b) or 2 (c) of Definition 4.6 at p. Let Ψ1 : Y1 → Ybe the blow up of C, and

X1f1→ Y1

Φ1 ↓ ↓ Ψ1

Xf→ Y

bet the pre-τ -quasi-well prepared diagram of Lemma 14.1. Φ1 is a product of blowups of 2-curves and from the local forms 2 (b) and 2 (c) of Definition 4.6, we seethat Φ1 is an isomorphism over a general point of C. Since C contains no 3-pointsof Y , we see that GY1(f1, τ) contains no 3-points. We can iterate this constructionto produce a diagram (14.7) satisfying the conclusions of the theorem.

Lemma 14.5. Suppose that f : X → Y is pre-τ -quasi-well prepared (or τ -wellprepared), q ∈ Y is a 1-point such that q 6∈ U(R), and C ⊂ DY is a resolving curvefor f at q. Then there exists a pre-τ -quasi-well prepared (τ -well prepared) diagram

X1f1→ Y1

Φ1 ↓ ↓ Ψ1

Xf→ Y

such that

1. Ψ1 is the blow up of C,2. Φ1 is a sequence of blow ups of 2-curves and 3-points.3. τf1(p1) ≤ τf (Φ1(p1)) for p1 ∈ X1. Thus τf1(p1) < τ if p1 ∈ (Ψ1 f1)−1(C − q).

4. Suppose that C1 ⊂ Y1 is a section over C, q ∈ C is a 1-point, u, v, w arepermissible parameters at q such that u = v = 0 are local equations of C,u, v are toroidal forms at p for all p ∈ f−1(q), and q1 ∈ C1 ∩Ψ−1

1 (q) is a1-point. Then there exist permissible parameters u1, v1, w at q1 such thatu1, v1 are torodial forms at p1 for all p1 ∈ f−1

1 (q1), and u1 = v1 = 0 is alocal equation of C1.


5. Suppose that p ∈ f−1(q), p′ ∈ Φ−11 (p), u, v, w are permissible parameters

at q such that u = v = 0 are local equations of C, and w is good at p forf . If w is not good at p′ for f1, then τf1(p

′) < τf (p).6. If f is τ -prepared then f1 is τ -prepared.

Proof. Suppose that q ∈ C is a 1-point. Then there exist permissible param-eters u, v, w at q such that u = v = 0 are local equations of C, and if p ∈ f−1(q),then u, v are toroidal forms at p. Thus there exist permissible parameters x, y, z atp for u, v, w such that one of the forms (4.5) or (4.6) hold.

If q ∈ C is a 2-point, then there exist super parameters u, v, w at q such thatu = w = 0 are local equations of C. Thus if p ∈ f−1(q), then there exist permissibleparameters x, y, z at p for u, v, w such that one of the forms (10.1) - (10.4) hold.

By Lemma 5.3 and Remark 14.2 (a detailed proof of a similar statement isgiven in the proof of Lemma 14.7), after blowing up 2-curves and 3-points by amorphism Φ0 : X0 → X we obtain that if q ∈ C is a 2-point, u, v, w are theabove permissible parameters at q, and p ∈ (f Φ0)−1(q), then (xayb, xeyf , xgyh)is a principal ideal if (10.2) holds at p, ((xayb)k, (xayb)l, xcyd) is principal if (10.3)holds at p, (xaybzc, xgyhzi, zjykzl) is principal if (10.4) holds at p. We further have,by Lemma 10.2, that f Φ0 is pre-τ -quasi-well prepared (τ -well prepared),

τfΦ0(p) ≤ τf (Φ0(p))

for p ∈ DX0 and C is a resolving curve for f Φ0 at q.We now analyze the points p ∈ X0 where ICOX0 is not principal.First suppose that q ∈ C is a 1-point, p ∈ X0, and f Φ0(p) = q.If p is a 1-point then we have an expression

u = xa, v = y

and u = v = 0 are local equations of C. The non principal locus has local equationsx = y = 0

If p is a 2-point thenu = (xayb)k, v = z

and u = v = 0 are local equations of C. The non principal locus has local equationsx = z = 0 ∪ y = z = 0

Now suppose that q ∈ C is a 2-point and f Φ0(p) = q.If p is a 1-point of the form (10.1), then

u = xa

v = xb(α+ y)w = xcγ(x, y) + xd(z + β).

u = w = 0 are local equations of C. If p is in the non principal locus then wehave (after possibly making a change of variables in z) w = xdz with d < a andx = z = 0 are local equations of the non principal locus.

If p is a 2-point of the form (10.2), then

u = xayb

v = xcyd

w = xeyfγ(x, y) + xgyh(z + β).

u = w = 0 are local equations of C. If p is in the non principal locus then (afterpossibly making a change of variables in z), we have w = xgyhz with (g, h) < (a, b).


Local equations of the non principal locus are x = z = 0 (if g < a, h = b), y = z = 0(if g = a, h < b) and x = z = 0 ∪ y = z = 0 if g < a and h < b.

If p is a 2-point of the form (10.3), then

u = (xayb)k

v = (xayb)t(α+ z)w = (xayb)lγ(xayb, z) + xcyd

u = w = 0 are local equations of C, and p is in the principal locus.If p is a 3-point, of the form (10.4), then

u = xaybzc

v = xdyezf


u = w = 0 are local equations of C, and p is in the principal locus.We see that the non principal locus of ICOX0 is a union of nonsingular curves

which are possible centers and are not 2-curves.Let U0 ⊂ X0 be the largest open set on which the rational map f0 = Ψ−1

1 f Φ0 : X0 → Y1 is a morphism. We will now show that f0 : U0 → Y1 is prepared, andτf0

(p) ≤ τf (Φ0(p)) for p ∈ U0.Suppose that p ∈ U0 ∩ (f Φ0)−1(C). Then (f Φ0)(p) = q is a 2-point, and

there exist super parameters u, v, w at q such that u = w = 0 are local equationsof C, and one of the forms (10.1) - (10.4) hold for u, v, w at p. Let q1 = f0(p). Wehave permissible parameters u1, v, w1 in OY1,q1 such that either

u = u1, w = u1(w1 + δ)

with δ ∈ k, oru = u1w1, w = w1.

The most difficult case to analyze is when p is a 3-point, (so that u, v, w satisfy(10.4)), τfΦ0(p) ≥ 1, and u = u1w1, w = w1. We will work out this case in detail.Since q is a 2-point, we will then have that

τf0(p) ≤ τfΦ0(p) ≤ τf (Φ0(p)) < τ.

In this case q1 is a 3-point. With the notation of (9.1),

u = xaybzc, v = xdyezf , w =∑i≥0

αixaiybizci + xgyhzi

with 0 6= αi ∈ k for all i. Set

γ =∑

αixai−a0ybi−b0zci−c0 + xg−a0yh−a0zi−a0 .

γ ∈ OX,p is a unit series, and

w = xa0yb0zc0γ,

with (a0, b0, c0) < (a, b, c).

Suppose that (a, b, c) and (a0, b0, c0) are linearly independent.After possibly interchanging x, y, z, we may assume that a0b− ab0 6= 0. There

exist λ1, λ2 ∈ Q such that if we set

(14.8) x = xγλ1 , y = yγλ2 ,


thenw = xa0yb0zc0

u = xaybzc

v = xdyezfγλ

for some 0 6= λ ∈ Q.There exists a series P (x, y, z) where the monomials in x, y, z in P with nonzero

coefficients are monomials in xai−a0ybi−b0zci−c0 for i ≥ 0, such that

(14.9) γλ = P + xg−a0yh−b0zi−c0Ω.

Iterating, by substituting (14.8) into successive iterations of (14.9), we see thatthere exists a series P (x, y, z), where the monomials in x, y, z in P with nonzerocoefficients are monomials in xai−a0ybi−b0zci−c0 for i ≥ 0 such that

γλ = P + xg−a0yh−b0zi−c0Ω.

Comparing the Jacobian determinants

Det(∂(u, v, w)∂(x, y, z)

)and

Det(∂(w, u, v)∂(x, y, z)

),

we see that Ω is a unit series.There exist rational numbers β1, β2, β3 such that we can make a formal change

of variables, settingx = xΩ

β1, y = yΩ

β2, z = zΩ

β3

to get an expression

w = xa0 yb0 cc0

u = xaybzc

v = xdyezf(P (x, y, z) + xg−a0 yh−b0 zi−c0

).

Thusw1 = xa0 yb0 cc0

u1 = xa−a0 yb−b0 zc−c0

v = xdyezf(P (x, y, z) + xg−a0 yh−b0 zi−c0

).

is an expression of the form of (9.1).We see that f0 is prepared at p, and

τf0(p) ≤ |((a0, b0, c0)Z + (a− a0, b− b0, c− c0)Z + (d, e, f)Z

+∑i≥1(ai − a0, bi − b0, ci − c0)Z)

/((a0, b0, c0) + (a− a0, b− b0, c− c0)Z + (d, e, f)Z)|= |((a, b, c)Z + (d, e, f)Z +

∑i≥0(ai, bi, ci)Z)

/((a, b, c)Z + (d, e, f)Z + (a0, b0, c0)Z)|= τfΦ0(p) ≤ τf (Φ0(p).

Suppose that (a, b, c) and (a0, b0, c0) are linearly dependent.Then (d, e, f) and (a0, b0, c0) are linearly independent. After possibly inter-

changing x, y, z, we may assume that db0 − ea0 6= 0. There exist λ1, λ2 ∈ Q suchthat if

x = xγλ1 , y = yγλ2 ,


thenw = xa0yb0zc0

v = xdyezf

u = xaybzcγλ

for some 0 6= λ ∈ Q.As in the case when (a, b, c) and (a0, b0, c0) are linearly independent, we can

make a change of variables to get an expression where the monomials in x, y, z inP with nonzero coefficients are monomials in xai−a0 ybi−b0 zci−c0 for i ≥ 0.

w = xa0 yb0 cc0

v = xdyezf

u = xaybzc(P (x, y, z) + xg−a0 yh−b0 zi−c0

).

Thusw1 = xa0 yb0 cc0

v = xdyezf

u1 = xa−a0 yb−b0 zc−c0(P (x, y, z) + xg−a0 yh−b0 zi−c0

).

is an expression of the form of (9.1).We see that f0 is prepared at p, and

τf0(p) ≤ |((a0, b0, c0)Z + (d, e, f)Z

+(a− a0, b− b0, c− c0)Z +∑i≥1(ai − a0, bi − b0, ci − c0)Z)

/((a0, b0, c0) + (d, e, f)Z + (a− a0, b− b0, c− c0)Z)|= τfΦ0(p) ≤ τf (Φ0(p)).

We conclude that f0 : U0 → Y1 is prepared, and τf0(p) ≤ τf (Φ0(p)) for p ∈ U0.

Thus f0 is pre-τ -quasi-well prepared (τ -well prepared), as C ∩ U(R) = ∅.Let Z0 = X0.We now construct a sequence of morphisms Λi : Zi → Zi−1 which are the blow

up of a possible curve Ci−1 contained in the locus where ICOZi−1 is not invertible.Let hi : Zi → Y1 be the rational map hi = f0 Λ1 · · · Λi. We will verify by

induction that:A. For all p1 in the locus where hi is a morphism, hi is τ -prepared and

τhi(p1) ≤ τf (Φ0 Λ1 Λ2 · · · Λi(p1)).

B. Suppose that p1 ∈ Zi is a point where ICOZi,p1 is not principal. Letp = Λ1· · ·Λi(p1), q = fΦ0(p). Then there exist permissible parametersu, v, w at q, permissible parameters x, y, z in OZ0,p for u, v and regularparameters x1, y1, z1 in OZi,p1 such that we have one of the followingforms:

1. q a 1-point, p a 1-point, p1 a 1-point. We have an expression

u = xa, v = z, w =∑

i<n,aij 6=0

aijxizj + xn(y + β)

in OZ0,p, where u = v = 0 are local equations of C.We have permissible parameters x1, y1, z1 at p1 such that

x = x1, y = y1, z = xb1z1

with b < a, and


(14.10)u = xa1v = xb1z1w =

∑aijx

i+bj1 zj1 + xn1 (y1 + β).

2. q a 1-point, p a 2-point. We have an expression

u = (xayb)k, v = z, w =∑i,l≥0

ailzl(xayb)i + xeyf

in OZ0,p, where the sum is over i, l such that (ia, ib) 6≥ (e, f), af−eb 6=0 and u = v = 0 are local equations of C.We have permissible parameters x1, y1, z1 at p1 such that

x = x1, y = y1, z = xc1yd1z1

with (c, d) < (ak, bk), and

(14.11)u = (xa1y

b1)k

v = xc1yd1z1

w =∑i,l ailz

l1xai+cl1 ybi+dl1 + xe1y

f1 .

3. q a 2-point, p a 1-point, p1 a 1-point. We have an expression

(14.12)u = xa1v = xb1(β + y1)w = xd1z1

in OZi,pi , with d < a, u = w = 0 a local equation of C4. q a 2-point, p a 2-point, p1 a 2-point. We have an expression

(14.13)u = xa1y

b1

v = xc1yd1

w = xg1yh1 z1

with (g, h) < (a, b), u = w = 0 are local equation of C.

We have verified that A and B are true for h0 = f0. Suppose that A and Bare true for hi. We will verify that A and B are true for hi+1.

We may suppose that p1 ∈ Ci (recall that Ci is the curve blown up by Λi+1).Then Ci has local equations x1 = z1 = 0 if p1 satisfies (14.10). Ci has localequations x1 = z1 = 0 (or y1 = z1 = 0) in (14.11). Ci has local equations x1 =z1 = 0 in (14.12). Ci has local equations x1 = z1 = 0 (or y1 = z1 = 0) in (14.13).

After possibly interchanging x1 and y1, we may assume that x1 = z1 = 0 is alocal equation of Ci.

Suppose that p2 ∈ Λ−1i+1(p1). Then OZi+1,p2 has regular parameters x2, y2, z2

such that

(14.14) x1 = x2, y1 = y2, z1 = x2(z2 + α)

with α ∈ k, or

(14.15) x1 = x2z2, y1 = y2, z1 = z2.


Suppose that (14.10) holds. Then

τfΦ0(p) = |[∑aij 6=0

iZ + aZ]/aZ|.

Suppose that (14.10) and (14.14) hold. Then

(14.16)u = xa2v = xb+1

2 (z2 + α)w =

∑aijx

i+(b+1)j2 (z2 + α)j + xn2 (y2 + β).

Suppose that a = b+ 1 in (14.16). Then Zi+1 → Y1 is a morphism at p2. Letq1 = hi+1(p2). There exist permissible parameters u1, v1, w at q1 defined by

u = u1, v = u1(v1 + α).


(14.17)u1 = xa2v1 = z2w =

∑aijx

i+aj2 (z2 + α)j + xn2 (y2 + β),

of type (4.5) so that hi+1 is prepared at p2. We have by (9.11) that

τhi+1(p2) ≤ |(aZ +∑aij 6=0

[i+ aj]Z)/aZ| = τfΦ0(p) ≤ τf (Φ0(p).

Suppose that b+ 1 < a and α 6= 0 in (14.16). Then Zi+1 → Y1 is a morphismat p2. Let q1 = hi+1(p2). There exist permissible parameters u1, v1, w at q1 definedby

u = u1v1, v = v1.

There exist regular parameters x2, y2, z2 in OZi+1,p2 such that

(14.18)

u1 = xa−b−12 (z2 + α)−1 = xa−b−1

2

v1 = xb+12 (z2 + α) = xb+1

2 (z2 + α)w =

∑s x

s2(∑i+(b+1)j=s aij(z2 + α)j) + xn2 (y2 + β)

=∑s x

s2(z2 + α)

sa

[∑j aij(z2 + α)j(

a−b−1a )

]+ xn2 (y2 + β).

of type (4.1), so that hi+1 is prepared at p2.We observe that ∑

i+(b+1)j=s

aij(z2 + α)j 6= 0

if and only if some aij 6= 0 with i+ (b+ 1)j = s. To show this, observe that

xs[∑

i+(b+1)j=s

aij

( y

xb+1

)j] =

∑i+(b+1)j=s

aijxiyj .

We calculate from (9.9),


τhi+1(p2) = |[(a− b− 1)Z + (b+ 1)Z+∑aij 6=0,i+(b+1)j 6≥n(i+ (b+ 1)j)Z]/[(a− b− 1)Z + (b+ 1)Z]|

≤ |[∑aij 6=0 iZ + aZ + (b+ 1)Z]/[aZ + (b+ 1)Z]|

≤ τfΦ0(p) ≤ τf (Φ0(p))

Suppose that b+1 < a, 0 = α in (14.16). Then we are back in the form (14.10)with a decrease in a− b.


u = xa2za2

v = xb2zb+12

w =∑aijx

i+bj2 z

i+(b+1)j2 + xn2 z

n2 (y2 + β).

Zi+1 → Y1 is a morphism at p2. Let q1 = hi+1(p2). There exist permissibleparameters u1, v1, w at q1 defined by

u = u1v1, v = v1.


(14.19)u1 = xa−b2 za−b−1

2

v1 = xb2zb+12

w =∑aijx

i+bj2 z

i+(b+1)j2 + xn2 z

n2 (y2 + β)

of type (4.2), so that hi+1 is prepared at p2.From (9.3), we have

τhi+1(p2) = |((a− b, a− b− 1)Z + (b, b+ 1)Z+∑aij 6=0,(i+bj,i+(b+1)j) 6≥(n,n)(i+ bj, i+ (b+ 1)j)Z)

/((a− b, a− b− 1)Z + (b, b+ 1)Z)|≤ |((a, a)Z +

∑aij 6=0(i, i)Z)/(a, a)Z)|

= τfΦ0(p) ≤ τf (Φ0(p)).


τfΦ0(p) = |(kZ +∑ail 6=0

iZ)/kZ|.

Suppose that (14.11) and (14.14) hold. Then we have

(14.20)u = (xa2y

b2)k

v = xc+12 yd2(z2 + α)

w =∑i,l ail(z2 + α)lxai+(c+1)l

2 ybi+dl2 + xe2yf2 .

Suppose that (ak, bk) = (c+1, d) in (14.20). Then Zi+1 → Y1 is a morphism at p2.Let q1 = hi+1(p2). There exist permissible parameters u1, v1, w at q1 defined by

u = u1

v = u1(v1 + α).



(14.21)

u1 = (xa2yb2)k

v1 = z2

w =∑i,l ail(z2 + α)lxai+(c+1)l

2 ybi+dl2 + xe2yf2

=∑i,l ail(z2 + α)l(xa2y

b2)i+lk + xe2y

f2

of type (4.6), so that hi+1 is prepared at p2. From (9.6), we have

τhi+1(p2) ≤ |(kZ +∑

ail 6=0,(i+lk)(a,b) 6≥(e,f)

(i+ lk)Z/kZ| ≤ τfΦ0(p) ≤ τf (Φ0(p)).

Suppose that (ak, bk) > (c + 1, d) and 0 6= α in (14.20). Then Zi+1 → Y1 is amorphism at p2. Let q1 = hi+1(p2). There exist permissible parameters u1, v1, wat q1 defined by

u = u1v1, v = v1.

Suppose that ad− b(c+ 1) 6= 0 in (14.20). Then there exist regular parametersx2, y2, z2 in OZi+1,p2 defined by

x2 = x2(z2 + α)−bkh

y2 = y2(z2 + α)akh

z2 = (z2 + α)ebk−fak

h − α

with h = adk − (c+ 1)bk and α = αebk−fak

h , such that

(14.22)u1 = xak−c−1

2 ybk−d2 (z2 + α)−1 = xak−c−12 ybk−d2

v1 = xc+12 yd2(z2 + α) = xc+1

2 yd2w =

∑i,l ailx

ai+(c+1)l2 ybi+dl2 + xe2y

f2 (z2 + α)

with 0 6= α ∈ k, which is an expression of type (4.2), so that hi+1 is prepared atp2. From (9.3), we see that

(14.23)

τhi+1(p2) = |(ak − c− 1, bk − d)Z + (c+ 1, d)Z+∑ail 6=0,i(a,b)+l(c+1,d) 6≥(e,f)[i(a, b) + l(c+ 1, d)]Z]/

[(ak − c− 1, bk − d)Z + (c+ 1, d)Z]|≤ |(k(a, b)Z +

∑ail 6=0 i(a, b)Z)/k(a, b)Z|

= |kZ +∑ail 6=0 iZ/kZ| = τfΦ0(p) ≤ τf (Φ0(p)).

Suppose that ad − b(c+ 1) = 0 in (14.20), (and (ak, bk) > (c+ 1, d), 0 6= α in(14.20)). There exist positive integers a, b, t, k such that gcd(a, b) = 1, and

(ak − c− 1, bk − d) = t(a, b), (c+ 1, d) = k(a, b).

From k(a, b) = (t+k)(a, b) and gcd(a, b) = 1, gcd(a, b) = 1, we see that (a, b) = (a, b)and t+ k = k.

There exist regular parameters x2, y2, z2 in OZi+1,p2 , defined by

x2 = x2(z2 + α)−f

ht

y2 = y2(z2 + α)e

ht

z2 = (z2 + α)1+kt − α


where h = fa− eb, α = e1+kt , such that

(14.24)u1 = xak−c−1

2 ybk−d2 (z2 + α)−1 = (xa2yb2)t

v1 = xc+12 yd2(z2 + α) = (xa2y

b2)k(z2 + α)

w =∑i,l ail(z2 + α)l+

ik (xa2y

b2)i+kl + xe2y

f2 ,

which is an expression of type (4.3), so that hi+1 is prepared at p2.From (9.4), we have

(14.25)

τhi+1(p2) = |(tZ + kZ +∑ail 6=0,(i+kl)(a,b) 6≥(e,f)(lk + i)Z)/(tZ + kZ)|

≤ |(kZ + kZ +∑ail 6=0 iZ)/(kZ + kZ)|

≤ |(kZ +∑ail 6=0 iZ)/kZ|

= τfΦ0(p) ≤ τf (Φ0(p)).

Suppose that (ak, bk) > (c + 1, d) and 0 = α in (14.20). Then we are back inthe form (14.11), with a decrease in (ak − c) + (bk − d).


(14.26)u = xak2 ybk2 z

ak2

v = xc2yd2zc+12

w =∑i,l ailx

ai+cl2 ybi+dl2 zl+ai+cl2 + xe2y

f2 z

e2.

Then Zi+1 → Y1 is a morphism at p2. Let q1 = hi+1(p2). There exist permis-sible parameters u1, v1, w at q1 defined by

u = u1v1, v = v1.


(14.27)u1 = xak−c2 ybk−d2 zak−c−1

2

v1 = xc2yd2zc+12

w =∑i,l ailx

ai+cl2 ybi+dl2 zl+ai+cl2 + xe2y

f2 z

e2.

of type (4.4), so that hi+1 is prepared at p2. We calculate τhi+1(p2) from (9.1).Set

G = ((ak, bk, ak)Z + (c, d, c+ 1)Z+∑ail 6=0(ai+ cl, bi+ dl, l + ai+ cl)Z)/

((ak, bk, ak)Z + (c, d, c+ 1)Z)∼= ((ak, bk, ak)Z +

∑ail 6=0(ai, bi, ai)Z)/(ak, bk, ak)Z).

τhi+1(p2) = |(ak − c, bk − d, ak − c− 1)Z + (c, d, c+ 1)Z+∑ail 6=0,(ai+cl,bi+dl,l+ai+cl) 6>(e,f,e)(ai+ cl, bi+ dl, l + ai+ cl)Z)/

((ak − c, bk − d, ak − c− 1)Z + (c, d, c+ 1)Z)|≤ |G|.

The homomorphism Z → Z3 defined by x 7→ (xa, xb, xa) induces an isomor-phism

(14.28) (kZ +∑ail 6=0

iZ)/kZ→ H.

Thus τhi+1(p2) ≤ τfΦ0(p) ≤ τf (Φ0(p)).



(14.29)u = xa2v = xb2(β + y2)w = xd+1

2 (z2 + α).

Suppose that d + 1 = a in (14.29). Then Zi+1 → Y1 is a morphism at p2. Letq1 = hi+1(p2). There exist permissible parameters u1, v, w1 at q1 defined by

u = u1, w = u1(w1 + α).


u1 = xa2 , v = xb2(β + y2), w1 = z2

which is toroidal, so that hi+1 is prepared at p2, and

τhi+1(p2) = −∞ ≤ τf (Φ0(p)).

Suppose that d+ 1 < a and 0 6= α in (14.29). Then Zi+1 → Y1 is a morphismat p2. Let q1 = hi+1(p1). There exist permissible parameters u1, v, w1 at q1 definedby

u = u1w1, w = w1,

so that q1 is a 3-point.There exist regular parameters x2, y2, z2 in OZi+1,p2 and 0 6= α, β ∈ k such that

u1 = xa−d−12 (z2 + α)−1 = xa−d−1

2

v = xb2(β + y2)w1 = xd+1

2 (z2 + α)


τhi+1(p2) = −∞ ≤ τf (Φ0(p)).

Suppose that d + 1 < a and 0 = α in (14.29). Then we are back in the form(14.12) with a decrease in a− d.


u = xa2za2

v = xb2zb2(α+ y2)

w = xd2zd+12 .

Thus Zi+1 → Y1 is a morphism at p2. Let q1 = hi+1(p2). There exist permissibleparameters u1, v, w1 at q1 defined by

u = u1w1, w = w1.

We have a 2-point mapping to a 3-point, and an expression

u1 = xa−d2 za−d−12

v = xb2zb2(α+ y2)

w1 = xd2zd+12

which is a toroidal form, so that hi+1 is prepared at p2, and

τhi+1(p2) = −∞ ≤ τf (Φ0(p)).



(14.30)u = xa2y

b2

v = xc2yd2

w = xg+12 yh2 (z2 + α).

Suppose that (a, b) = (g + 1, h) in (14.30). Then Zi+1 → Y1 is a morphism at p2.Let q1 = hi+1(p2). There exist permissible parameters u1, v, w1 at q1 defined by

u = u1, w = u1(w1 + α).We have an expression

u1 = xa2yb2

v = xc2yd2

w1 = z2which is toroidal, so that hi+1 is prepared at p2, and

τhi+1(p2) = −∞ ≤ τf (Φ0(p)).

Suppose that (g + 1, h) < (a, b) and 0 6= α in (14.30). Then Zi+1 → Y1 is amorphism at p2. Let q1 = hi+1(p2). There exist permissible parameters u1, v, w1

at q1 defined by

u = u1w1, w = w1.


u1 = xa−g−12 yb−h2 (z2 + α)−1

v = xc2yd2

w1 = xg+12 yh2 (z2 + α)

which is toroidal after a change of variable in OZi+1,p2 , so that hi+1 is prepared atp2, and

τhi+1(p2) = −∞ ≤ τf (Φ0(p)).Suppose that (g + 1, h) < (a, b) and α = 0 in (14.30). Then we are back in the

form (14.13), with a decrease in (a− g) + (b− h).Suppose that (14.13) and (14.15) hold. Then

u = xa2yb2za2

v = xc2yd2zc2

w = xg2yh2 z

g+12 .

Thus Zi+1 → Y1 is a morphism at p2. Let q1 = hi+1(p2). There exist permissibleparameters u1, v, w1 at q1 defined by

u = u1w1, w = w1.

We have a 3-point mapping to a 3-point, and an expression

u1 = xa−g2 yb−h2 za−g−12

v = xc2yd2zc2

w1 = xg2yh2 z

g+12


τhi+1(p2) = −∞ ≤ τf (Φ0(p)).

We have thus verified A and B for all hi.


Suppose that p ∈ Zi is in the locus Σi where ICOZi is not locally principal.Define

C(p) =

a− b if (14.10) holds at pak − c+ bk − d if (14.11) holds at pa− d if (14.12) holds at pa− g + b− h if (14.13) holds at p.

LetC(Zi) = maxC(p) | p ∈ Σi.

We have shown in the above analysis that

0 ≤ C(Zi+1) < C(Zi)

if the rational map hi is not a morphism. Thus there exists a finite n such thatf1 = hn, Φ1 = Φ0 Λ1 · · · Λn, X1 = Zn satisfy 1 - 3 of the conclusions of Lemma14.5.

We now prove 4. If p ∈ f−1(q), then we have permissible parameters x, y, z foru, v, w in OX,p such that there is one of the forms:

p a 1-point

(14.31) u = xa, v = y, w = g(x, y) + xn(z + β)

or p a 2-point

(14.32) u = (xayb)k, v = z, w = h(xayb, z) + xcyd.

By assumption, u = v = 0 are local equations of C.Since q1 is a 1-point, at q1 we have permissible parameters u1, v1, w1 defined

byu = u1, v = u1(v1 + α), w = w1

for some α ∈ k. Since C1 is a section over C, there exists a series λ(v1, w) suchthat u1 = λ(v1, w1) = 0 are local equations of C1 (in OY1,q1), and

k[[u, v, w]]/(u, v)→ k[[u1, v1, w]]/(u1, λ)

is an isomorphism, which implies that λ = (v1 − φ(w))γ where φ is a series and γ

is a unit series in OY1,q1 . Set

u1 = u1, v1 = v1 − φ(w).

u1, v1, w are permissible parameters at q1, and u1 = v1 = 0 are local equations ofC1 at q1.

Suppose that p1 ∈ f−11 (q1). First suppose that p = Φ1(p1) has the form (14.31).

Then OX1,p1 has regular parameters x1, y1, z1 with x = x1, y = xa1(y1 + α), sinceX0 → X is an isomorphism over p. p1 is a 1-point, and substituting into (14.31),we have

u1 = xa1 , v1 = y1, w = g(x1, xa1(y1 + α)) + xn1 (z + β)

and thus u1, v1 are toroidal forms at p1.Now suppose that p = Φ1(p1) has the form (14.32).Let

p′ = Λ1 · · · Λn(p) ∈ X0.

There exist regular parameters x1, y1, z1 in OX1,p′ such that

(14.33) x = xa1yb1, y = xc1y

d1 , z = z1


or

(14.34) x = xa1(y1 + α)b, y = xc1(y1 + α)d, z = z1

with 0 6= α ∈ k. If (14.33) holds, then we have

(14.35)u = (xa

′

1 yb′

1 )k

v = z1w = h(xa

′

1 yb′

1 , z1) + xc′

1 yd′

1 .

If (14.34) holds, then we have 0 6= α′ ∈ k and permissible parameters x1, y1, z1in OX1,p′ such that

(14.36)u = xa

′k1

v = z1w = h(xa

′

1 , z1) + xb′

1 (y1 + α′)

with α′ ∈ k.Suppose that a form (14.35) holds at p′. Then OX1,p1 has regular parameters

x2, y2, z2 withx1 = x2, y1 = y2, z1 = xa

′k2 yb

′k2 (z2 + α).

p1 is a 2-point, and substituting into (14.35), we have

u1 = (xa′

2 yb′

2 )k, v1 = z2, w = h(xa′

2 yb′

2 , xa′k2 yb

′k2 (z2 + α)) + xc

′

2 yd′

2 .

Thus u1, v1 are toroidal forms at p1.Suppose that a form (14.36) holds at p′. Then OX1,p1 has regular parameters

x2, y2, z2 withx1 = x2, y1 = y2, z1 = xa

′k2 (z2 + α).

p1 is a 1-point, and substituting into (14.36), we have

u1 = xa′k

2 , v1 = z2, w = h(xa′

2 , xa′k2 (z2 + α)) + xc

′

2 yd′

2 .

Thus u1, v1 are toroidal forms at p1.We now prove 5. We may assume that τf1(p

′) = τf (p). There exists a smallesti such that X1 → Zi+1 is an isomorphism at p′. Let p2 be the image of p′ in Zi+1.Let p be the image of p2 in Z0.

If Z0 → X is not an isomorphism at p, then p is a 2-point, and we havepermissible parameters x, y, z at p such that

(14.37)u = (xayb)k

v = zw =

∑i,l≥0 ailz

l(xayb)i + xeyf ,

where gcd(a, b) = 1 and the sum is over i such that (ia, ib) 6≥ (e, f). Since we areassuming w is good at p for f , we have ail = 0 if k divides i.

Since Z0 → X is a sequence of blow ups of 2-curves above p, we have regularparameters x1, y1, z in OX0,p such that

(14.38) x = xa1yb1, y = xc1y

d1

with ad− bc = ±1, (and gcd(aa+ bc, ab+ bd) = 1) or

x = xa1(y1 + α)b, y = xc1(y1 + α)d

where a, c ∈ N, b, d ∈ Q, 0 6= α ∈ k are such that

(14.39) xayb = xaa+cb1 , xeyf = xae+cf1 (y1 + α).


If either (14.38) or (14.39) holds, we see from substitution into (14.37) that wis good at p for f Φ0.

Since q is a 1-point, f0 = Ψ−11 f Φ0 is not a morphism at p. Let p1 be the

image of p2 in Zi. At p1 we have an expression (14.10) or (14.11).First suppose that (14.10) holds at p1. Since w is good at p for f Φ0, we have

aij = 0 if a divides i.Suppose that (14.10) holds at p1 and at p2 and there is an expression (14.17).Suppose that i + aj ∈ aZ for some i. Then i ∈ aZ, and since w is good at p

for f Φ0, we have that aij = 0. Thus w is good at p2 for hi+1 (and w is good atp′ for f1).

Suppose that (14.10) holds at p1 and at p2 there is an expression (14.18).If w is not good at p2 for hi+1, there exists aij 6= 0 with i + (b + 1)j < n and

i+ (b+ 1)j ∈ aZ + (b+ 1)Z. Thus i ∈ aZ + (b+ 1)Z which implies that i is in thekernel of the surjective projection homomorphism

[∑aij 6=0

iZ + aZ]/aZ]→ [∑aij 6=0

iZ + aZ + (b+ 1)Z]/[aZ + (b+ 1)Z].

Thusτf1(p

′) = τhi+1(p2) < τfΦ0(p) ≤ τf (p),a contradiction.

Suppose that (14.10) holds at p1 and at p2 there is an expression (14.19).Suppose that w is not good at p2 for hi+1. Then there exists aij 6= 0 with

(i+ bj, i+ (b+ 1)j) 6≥ (n, n)

and

(i+ bj, i+ (b+ 1)j) ∈ (a− b, a− b− 1)Z + (b, b+ 1)Z = (a, a)Z + (b, b+ 1)Z,

which implies that (i, i) ∈ (a, a)Z + (b, b+ 1)Z. Thus i ∈ aZ, a contradiction, andwe conclude that w is good at p2 for hi+1.

Now suppose that (14.11) holds at p1. Since w is good at p for f Φ0, we haveail = 0 if k divides i.

Suppose that (14.11) holds at p1 and at p2 there is an expression (14.21).Suppose that (i + lk)(a, b) 6≥ (e, f) and i + lk ∈ kZ. then i ∈ kZ, and since w isgood at p for f Φ0, we have that ail = 0. Thus w is good at p2 for hi+1.

Suppose that (14.11) holds at p1 and at p2 there is an expression (14.22). Thenby a similar calculation to the above analysis of (14.18), (14.23) is a strict inequalityif w is not good at p2 for hi+1.

Suppose that (14.11) holds at p1 and at p2 there is an expression (14.24). Then(14.25) is a strict inequality if w is not good at p2 for hi+1.

Suppose that (14.11) holds aat p1 and p2 has an expression (14.27). The ho-momorphism (14.28) has a nontrivial kernel if w is not good at p2 for hi+1.

In all of these cases, we have a contradiction to our assumption that τf1(p′) =

τf (p).

Lemma 14.6. Suppose that f : X → Y is pre-τ -quasi-well prepared with relationR, q ∈ Y − U(R) is a 2-point such that q ∈ GY (f, τ), and u, v, w are permissibleparameters at q. Let E be the 2-curve of Y containing q. Then there exists a


pre-τ -quasi-well diagram

X1f1→ Y1

Φ ↓ ↓ Ψ

Xf→ Y

such that Φ and Ψ are products of blow ups of 2-curves, f1 is pre-τ -quasi-well pre-pared, Φ is an isomorphism over f−1(Y −E), and if q1 ∈ Ψ−1(q) (with permissibleparameters u1, v1, w), then if p1 ∈ f−1

1 (q1) is such that τf1(p1) = τ , then thereexists a series φp1(u1, v1) such that w−φp1(u1, v1) is good at p1. If f is τ -prepared,then f1 is τ -prepared.

Proof. Let u, v, w be permissible parameters at q. By Lemma 10.5 andLemma 14.1, there exist sequences of blow ups of 2-curves Ψ0 : Y0 → Y andΦ0 : X0 → X making a pre-τ -quasi-well diagram

X0f0→ Y0

Φ0 ↓ ↓ Ψ0

Xf→ Y

such that if q0 ∈ Ψ−10 (q) is a 2-point, then there are permissible parameters

(u0, v0, w0) at q0 such thatu = ua0v

b0, v = uc0, v

d0

with ad− bc = ±1, and there are no points of the form 2 (b) or 2 (c) of Definition4.6 for f0 above q0.

We construct an infinite commutative diagram of morphisms

(14.40)

......

↓ ↓Xn


......

Φ2 ↓ ↓ Ψ2

X1f1→ Y1

Φ1 ↓ ↓ Ψ1

X0f0→ Y0

as follows. Order the 2-curves of Y0 which intersect Ψ−10 (q), and let Ψ1 : Y1 → Y0

be the blow up of the 2-curve C of smallest order. Then construct (by Lemma 14.1)a pre-τ -quasi-well diagram

(14.41)X1

f1→ Y1

Φ1 ↓ ↓ Ψ1

X0f0→ Y0

where Ψ1 is a product of blow up of 2-curves and Φ1 is an isomorphism abovef−10 (Y0 − C). Order the 2-curves of Y1 which intersect (Ψ0 Ψ1)−1(q) so that the

2-curves contained in the exceptional divisor of Ψ1 have larger order than the orderof the (strict transforms of the) 2-curves of Y0.


Let Ψ2 : Y2 → Y1 be the blow up of the 2-curve C1 on Y1 of smallest order, andconstruct a pre-τ -quasi-well diagram

X2f2→ Y2

Φ2 ↓ ↓ Ψ2

X1f1→ Y1

as in (14.41). We now iterate to construct (14.40). Let

Ψn = Ψ0 Ψ1 · Ψn : Yn → Y,

Φn = Φ0 Φ1 · · · Φn : Xn → X.

For all qn ∈ Ψ−1

n (q) there exist permissible parameters un, vn, w at qn such that

u = uan(vn + α)b, v = ucn(vn + α)d

with ad− bc 6= 0 and α ∈ k. qn is a 2-point if α = 0, and a 1-point if α 6= 0Suppose that pn ∈ (Ψn fn)−1(q). We will say that pn is good for qn = fn(pn)

if τfn(pn) < τ or if τfn(pn) = τ and there exists a series φpn(un, vn) such thatw − φpn(un, vn) is good for fn at pn.

We first observe that if pn is good for qn and pn+1 ∈ Φ−1n+1(pn), then pn+1 is

good for qn+1. (This follows from 3 of Lemma 14.1).Let ν be a zero-dimensional valuation of k(X) whose center on Y is q, and let

pn be the center of ν on Xn, qn = fn(pn).We will show that there exists n0 such that n ≥ n0 implies pn is good for qn.Once we have established this, it will follow from compactness of the Zariski-

Riemann manifold of X [Z] that there exists n′ such that all p ∈ (Ψn′ fn′)−1(q)are good for q′ = fn′(p), so that the conclusions of the theorem hold.

We may identify ν with an extension of ν to the quotient field of OXn,pn whichdominates OXn,pn .

If ν(u) and ν(v) are rationally dependent, then there exists n0 such that qn0 isa 1-point, which implies that pn0 is good (by Remark 11.4), and thus pn is goodfor all n ≥ n0.

So we may assume that ν(u) and ν(v) are rationally independent. We thenhave that

u = uann vbn

n , v = ucnn v

dnn

with andn − bncn = ±1 for all n. We thus have (for n >> 0) that qn is a 2-point,and pn has one of the forms (14.42) or (14.43) below:

pn is a 2-point

(14.42)un = xany

bn

vn = xcnydn

w = γn + xenyfn(zn + β)

withγn =

∑αiMi,

β, αi ∈ k,Mi = xai

1 ybi1 and Mei

i = uki1 v

li1

with ki, li, ei ∈ Z, ei > 0 and ai, bi ∈ N, or


pn is a 3-point

(14.43)un = xany

bnzcn

vn = xdnyenzfn

w = γn +N

with N = xgnn y

hnn zinn , rank(un, vn, N) = 3,

γn =∑

αiMi,

αi ∈ k,Mi = xai

1 ybi1 z

ci1 and Mei

i = uki1 v

li1

with ki, li, ei ∈ Z, ei > 0 and ai, bi, ci ∈ N.Further, for n >> 0, either all points pn have the form (14.42) or all pn have

the form (14.43).It is shown in the proof of Lemma 10.6 that if (14.43) holds, then for n >> 0,

we have ki, li ∈ N for all i, which implies there exists a good form for pn at qn.Essentially the same argument shows that the same statement holds if (14.42) holdsfor n >> 0.

Lemma 14.7. Suppose that f : X → Y is pre-τ -quasi-well prepared (or τ -wellprepared or τ -very-well prepared) and q ∈ U(R) is a 2-point (prepared of type 1 inDefinition 13.7). Then q is a permissible center for R, and there exists a pre-τ -quasi-well prepared (or τ -well prepared or τ -very-well prepared) diagram (13.10) ofR and the blow up Ψ : Y1 → Y of q such that:

1. Φ is an isomorphism over f−1(Y − Σ(Y )).2. Suppose that f is τ -well prepared. Then

(a) Let E be the exceptional divisor of Ψ. Suppose that q1 ∈ U(R1

i ) ∩E.Let γi = S

R1i(q1) · E. Then γi is a prepared curve for R1 of type 6.

Suppose that q′ ∈ U(R1

j ) ∩ E. Let γj = SR

1j(q′) · E. Then either

(i) γi = γj or(ii) γi, γj intersect transversally at a 2 point on E (their tangent

spaces have distinct directions at this point) and γi, γj are oth-erwise disjoint.

(b) If γ is a prepared curve on Y then the strict transform of γ is aprepared curve on Y1.

3. If f is τ -prepared then f1 is τ -prepared.

Proof. There exists a pre-relation Ri associated to R such that q ∈ U(Ri) ⊂U(R). Fix such an i. Let

u = uRi(q), v = vRi

(q), wi = wRi(q).

u, v, wi are super parameters at q, and wi = 0 is a local equation of SRi(q). Let

mq ⊂ OY,q be the maximal ideal.By Lemma 5.1, there exists a morphism Φ0 : X0 → X which is a sequence of

blow ups of 2-curves such that (u, v)OX,p is invertible for all p ∈ (f Φ0)−1(q), Φ0

is an isomorphism over f−1(Y − Σ(Y )), f Φ0 is prepared and u, v, wi are superparameters for f Φ0 at q (by Lemma 10.2).

We will next show that there exists a sequence of blow ups of 2-curves and 3-points Φ1 : X1 → X0 such that (u, v)OX1,p is invertible at all p ∈ (f Φ0 Φ1)−1(q)


and if mqOX1,p is not invertible, then we have permissible parameters x, y, z foru, v, w at p of one of the following forms:

p is a 1-point

(14.44) u = xa, v = xb(α+ y), wi = xdz

with α 6= 0 and d < mina, b or,p is a 2-point of the type of (4.2) of Definition 4.1

(14.45) u = xayb, v = xcyd, wi = xgyhz

with ad− bc 6= 0, and (g, h) < min(a, b), (c, d).Suppose that p ∈ (f Φ0)−1(q) is a 3-point, so that u, v, wi have a form 4 of

Definition 10.1 at p. There exist x, y, z ∈ OX0,p and series λ1, λ2, λ3 ∈ OX0,p suchthat x = xλ1, y = yλ2, z = zλ3. Let Ip ⊂ OX0,p be the ideal

Ip = (u, v, xgyhzi, xjykzl).

Observe that OX0,p/Ip is supported on (f Φ0)−1(q). By Lemma 5.3, there exists

a sequence of blow ups of 2-curves and 3-points Φ1 : X1 → X0 such that Φ0 Φ1

is an isomorphism above f−1(Y −Σ(Y )) and IpOX1,p1 is invertible for all 3-pointsp ∈ (f Φ0)−1(q) and p1 ∈ Φ−1

1 (p). Thus if p ∈ (f Φ0)−1(q) is a 3-point andp1 ∈ (f Φ0 Φ1)−1(p), then mqOX1,p1 is invertible. f Φ0 Φ1 is prepared, andu, v, wi are super parameters for f Φ0 Φ1 at q by Lemma 10.2.

We construct an infinite sequence of morphisms

(14.46) · · · → XnΦn→ · · · Φ3→ X2

Φ2→ X1

as follows. Order the 2-curves C of X1 such that q ∈ (f Φ0 Φ1)(C) ⊂ Σ(Y ). LetΦ2 : X2 → X1 be the blow up of the 2-curve C1 on X1 of smallest order. Order the2-curves C ′ of X2 such that q ∈ (f Φ0 Φ1 Φ2)(C ′) ⊂ Σ(Y ) so that the 2-curvescontained in the exceptional divisor of Φ2 have order larger than the order of the(strict transform of the) 2-curves C of X1 such that q ∈ f Φ0 Φ1(C) ⊂ Σ(Y ).Let Φ3 : Y3 → Y2 be the blow ups of the 2-curve C2 on Y3 of smallest order. LetΦn = Φ2 · · ·Φn : Xn → X1 be constructed by iterating this procedure. Themorphisms f Φ0 Φ1 Φn are prepared, and u, v, wi are super parameters forf Φ0 Φ1 Φn.

Suppose that ν is a 0-dimensional valuation of k(X). Let pn be the center ofν on Xn. Say that ν is resolved on Xn if (at least) one of the following holds:

1. mqOXn,pn is invertible.2. pn is a 1-point of the form (14.44).3. pn is a 2-point of the form (14.45).

The condition of begin resolved on Xn is an open condition on the Zariski-Riemannmanifold of X. Further, if ν is resolved on Xn and n′ > n, then Xn′ is also resolvedat ν.

Suppose that ν is a 0-dimensional valuation of k(X). Suppose that the centerof ν on Xn is a 3-point, then the center of ν on X1 is also a 3-point, so mqOXn,pn

is invertible.Suppose that the center of ν on Xn is a 1-point. Then u, v, wi have a form

1 of Definition 10.1 at pn, and thus have a form (14.44) at pn if mqOXn,pn is notinvertible.

Suppose that the center p of ν on X1 is a 2-point such that u, v, wi have a form2 of Definition 10.1 at p. There exist x, y ∈ OX1,p and series λ1, λ2 ∈ OX1,p such


that x = xλ1, y = yλ2. Let Ip ⊂ OX1,p be the ideal (u, v, xeyf , xgyh). There existsan n such that IpOXn,p1 is invertible for all p1 ∈ Φ

−1

n (p). Let p1 be the center of νon Xn. u, v, wi are super parameters for f Φ0 Φ1 Φn at q. Then p1 is either a1-point (so that ν is resolved on Xn) or a 2-point of the form 2 of Definition 10.1.If p1 is a 2-point and mqOXn,p1 is not invertible, then u, v, wi have a form (14.45).Thus ν is resolved on Xn.

Suppose that the center p of ν on X1 is a 2-point such that u, v, wi have a form3 of Definition 10.1 at p. Then there exist x, y ∈ OX,p and series λ1, λ2 ∈ OX,psuch that x = xλ1, y = yλ2. Let Ip ⊂ OX,p be the ideal

Ip = (u, v, (xayb)l, xcyd).

There exists an n such that IpOXn,p1 is invertible for all p1 ∈ Φ−1

n (p). Let p1 bethe center of ν on Xn. Then mqOXn,p1 is invertible. Thus ν is resolved on Xn.

By compactness of the Zariski-Riemann manifold [Z], there exists an n suchthat every valuation ν of k(X) is resolved on Xn. Rename X1 = Xn and

f1 = f Φ0 Φ1 Φn : X1 → Y.

If mqOX1,p1 is not invertible at some p1 ∈ f−11 (q), then one of the forms (14.44) or

(14.45) must hold at p1.The locus of points p on X1 where mqOX1,p is not invertible is a (possibly

not irreducible) curve E which makes SNCs with the toroidal structure of X. Eis supported at points of the form (14.44) (with d < min(a, b)) and (14.45) (with(g, h) < min(a, b), (c, d)). x = z = 0 is a local equation of E in (14.44). x = z = 0,y = z = 0 or xy = z = 0 are the possible local equations of E in (14.45).

For an irreducible component C of E, define an invariant

A(C) = mina, b − d > 0

computed at a 1-point p ∈ C (which has an expression (14.44)). Let C be acomponent of E such that A(C) = maxC⊂EA(C). C is nonsingular and makesSNCs with DX1 . Let Φ2 : X2 → X1 be the blow up of C.

Suppose that p ∈ C, so that u, v, wi have the form (14.44) or (14.45) at p.We may assume that x = z = 0 are local equations of C at p. Suppose thatp1 ∈ Φ−1

2 (p). p1 has (formal) regular parameters x1, y1, z1 defined by

(14.47) x = x1, y = y1, z = x1(z1 + β)

with β ∈ k or

(14.48) x = x1z1, z = z1.

Suppose that p ∈ C is a 1-point, so that (14.44) holds for u, v, wi at p. Under(14.47) we have mqOX2,p1 and thus mqOX2,p1 is invertible except possibly if β = 0.If mqOX2,p1 is not invertible we have

u = xa1 , v = xb1(α+ y), wi = xd+11 z1

withd+ 1 < mina, b.

Then the curve C1 with local equations x1 = z1 is a component of the locus wheremqOX2 is not invertible. We have

A(C1) = mina, b − (d+ 1) < A(C).


Under (14.48) we have a 2-point

u = (x1z1)a, v = (x1z1)b(α+ y), wi = xd1zd+11

and a local equation of the toroidal structure DX2 is x1z1 = 0. Since

d+ 1 ≤ mina, b,

mqOX2,p1 is invertible.Suppose that p ∈ C is a 2-point, so that (14.45) holds at p. Since we are

assuming that x = z = 0 is a local equation of C at p, we have that g < mina, c.Let p1 ∈ Φ−1

2 (p). p1 has regular parameters x1, y1, z1 defined by (14.47) or (14.48).Under the substitution (14.47), mqOX2,p1 is invertible, unless we have β = 0

andu = xa1y

b1, v = xc1y

d1 , wi = xg+1

1 yh1 z1

which is back in the form (14.45).Under the substitution (14.48),

(14.49) u = xa1yb1za1 , v = xc1y

d1zc1, wi = xg1y

h1 z

g+11

so that p1 is a 3-point, and since

(g + 1, h) ≤ min(a, b), (c, d),

mqOX2,p1 is invertible.Observe that u, v, wi are super parameters for f1 Φ2.By descending induction on maxC⊂EA(C), we construct a sequence of blow

ups Φ4 : X4 → X2 such that for f4 = f1 Φ2 Φ4 : X4 → Y , u, v, wi are superparameters at q for f4, and mqOX4 is invertible. Thus f4 : X4 → Y factors throughthe blow up Ψ : Y1 → Y of q. Let f4 : X4 → Y1, Φ : X4 → X be the resultingmaps.

Let q1 ∈ Ψ−1(q). We obtain permissible parameters u, v, w at q1 of one of thefollowing forms:

1. q1 a 1-point

u = u, v = u(v + α), wi = u(w + β)

with α, β ∈ k, α 6= 0.2. q1 a 2-point

(14.50) u = u, v = uv,wi = u(wi + α)

with α ∈ k, or

(14.51) u = uv, v = v, wi = v(wi + α)

with α ∈ k, or3. q1 a 3-point

(14.52) u = uwi, v = vwi, wi = wi.

If q1 has the form (14.50) or (14.51) and p ∈ f−14 (q1) then u, v are toroidal

forms at p.Suppose that q1 has the form (14.52). Let p ∈ f−1

4 (q1). u, v, wi have one ofthe forms 1 - 4 of Definition 10.1 at p. Since wi must divide u and v, we certainlyhave that u, v, wi are monomials in the local equations of the toroidal structure atp, times unit series in OX4,p.


Suppose that u, v, wi have a form 1 of Definition 10.1 at p. We have an expres-sion

u = xa

v = xb(α+ y)wi = xc(γ(x, y) + xdz)

where 0 6= α ∈ k, c < a, c < b, γ is a unit series and d ≥ 0. Set x = x(γ + xdz)1c .

We have expansionswi = xc

u = xa(γ + xdz)−ac

v = xb(γ + xdz)−bc (α+ y)

at p.If d > 0 and ∂γ

∂y (0, 0) = 0, then there exist y, z ∈ OX4,p such that x, y, z are

regular parameters in OX4,p and

wi = xc

v = xb(α+ y)u = xaγ(x, y, z)

where 0 6= α ∈ k and γ is a unit series. Then

wi = xc

v = xb−c(α+ y)u = xa−cγ(x, y, z)

and wi, v are toroidal forms at p.If d = 0 or ∂γ

∂y (0, 0) 6= 0, then there exist y, z ∈ OX4,p such that x, y, z are

regular parameters in OX4,p and

wi = xc

u = xa(α+ y)v = xbγ(x, y, z)

where 0 6= α ∈ k and γ is a unit series. Then

wi = xc

u = xa−c(α+ y)v = xb−cγ

and wi, u are toroidal forms at p.Suppose that u, v, wi have a form 3 of Definition 10.1 at p. There are two cases.

Either

(14.53)u = (xayb)k

v = (xayb)t(α+ z)wi = xcyd

with 0 6= α ∈ k and ad− bc 6= 0, or

(14.54)u = (xayb)k

v = (xayb)t(α+ z)wi = (xayb)lγ

with 0 6= α ∈ k, l ≤ mink, t and γ is a unit series.If (14.53) holds then u,wi are toroidal forms at p of the form of (4.2) of Defi-

nition 4.1.


Assume that (14.54) holds. We then have

u = (xayb)k−lγ−1

v = (xayb)t−lγ−1(α+ z)wi = (xayb)lγ.

If ∂γ∂z (0, 0, 0) 6= 0, then there exist regular parameters x, y, z at p such that

u = (xayb)k−l

wi = (xayb)l(β + z)v = (xayb)t−lγ(x, y, z)

where 0 6= β ∈ k and γ is a unit series. Thus u,wi are toroidal forms at pIf ∂γ

∂z (0, 0, 0) = 0 then there exist regular parameters x, y, z at p such that

u = (xayb)k−l

v = (xayb)t−l(β + z)wi = (xayb)lγ(x, y, z)

where 0 6= β ∈ k and γ is a unit series. Thus u, v are toroidal forms at p.If u, v, wi have a form 2 or 4 of Definition 10.1 at p, then a simpler analysis

shows that two of u, v, wi are toroidal forms at p, and u, v, wi are monomials inlocal equations of the toroidal structure at p times unit series.

We conclude that the morphism f4 is prepared.We have that for p ∈ f−1

4 (q), τf4(p) = −∞ unless Φ factors as a sequence ofblow ups of 2-curves and 3-points at p.

If Φ factors as a sequence of blowups of 2-curves and 3-points at p, we calculatefrom the above local analysis that

(14.55) τf4(p) ≤ τf (Φ(p)).

Suppose that f is τ -prepared. We will verify that f4 is τ -prepared. Let q1 ∈Ψ−1(q) be the 3-point. By upper semi-continuity of τf4 (Lemma 9.3) it suffices toshow that τf4(p) < τ for all p ∈ f−1

4 (q1).Suppose that p ∈ f−1

4 (q1). Then Φ(p) ∈ T (Ri) for some i. Let

u = uR

1i (q1)

, v = vR

1i (q1)

, wi = wR

1i (q1)

.

u, v, wi are defined by (14.52), and we have permissible parameters x, y, z inOX4,p such that one of the expressions following (14.52) hold.

We will verify that τf4(p) < τ if u, v, wi have the form 1 of Definition 10.1 interms of x, y, z and τ > 1. We leave the other cases to the reader.

We may assume that τf4(p) = τ . We have an expression

u = xa

v = xb(α+ y)wi =

∑mj=0 αij (y)x

ij + xn(z + β)= xi0(γ(x, y) + xn−i0(z + β))

where im < n, αij (y) 6= 0 for all j and γ is a unit in OX4,p.

τf4(p) = |(aZ + bZ +

m∑j=0

ijZ)/(aZ + bZ)| = τ.

We further assume that we are in the case n− i0 > 0 and ∂γ∂y (0, 0) = 0.


Let

x = x(m∑j=0

αij (y)xij−i0 + xn−i0(z + β))

1i0 ,

y = (m∑j=0

αij (y)xij−i0 + xn−i0(z + β))−

bi0 (α+ y)− α.


u = xa(P1(y0, xi1−i0 , . . . , xim−i0) + xn−i0Ω1)

v = xb(α+ y)wi = xi0

where P1 is a series, Ω1 ∈ OX4,p.By iteration, substituting

x− x(m∑j=0

αij (y)xij−i0 + xn−i0(z + β))−

1i0

and

y = (m∑j=0

αij (y)xij−i0 + xn−i0(z + β))

bi0 (y + α)− α

into P1(y, xi1−i0 , . . . , xim−i0), we see that there exists a unit series P (y, xi1−i0 , . . . , xim−i0)and a series Ω2 ∈ OX4,p such that

u = xa(P (y, xi1−i0 , . . . , xim−i0) + xn−i0Ω2.

Now comparing the jacobian determinants

Det(∂(u, v, wi)∂(x, y, z)

)and

Det(∂(u, v, wi)∂(x, y, z)

),

we see that if we set β = Ω2(0, 0, 0) and z = Ω2 − β, then x, y, z are regularparameters in OX4,p and we have an expression

u = xa(P (y, xi1−i0 , . . . , xim−i0) + xn−i0(z + β)

)v = xb(α+ y)wi = xi0 .

Thus we have an expression

wi = xi0

v = xb−i0(α+ y)u = xa−i0

(P (y, xi1−i0 , . . . , xim−i0) + xn−i0(z + β)

).

We compute (since q1 is a 3-point)

τf4(p) ≤ | (i0Z + (b− i0)Z +∑

(a− ij)Z) / (i0Z + (b− i0)Z + (a− i0)Z) |= | (aZ + bZ +

∑ijZ) / (aZ + bZ + i0Z) |

< | (aZ + bZ +∑ijZ) / (aZ + bZ) |

= τf4(p) = τ


since we have| (aZ + bZ + i0Z) / (aZ + bZ) | = eRi

(q) > 1.We conclude that f4 is τ -prepared.

For j such that q ∈ U(Rj), let wj = wRj(q). We will analyze our construction

of X4 → X to show that u, v, wj are super parameters for f4.Since f is pre-τ -quasi-well prepared, for j such that q ∈ U(Rj), there exists a

series λij(u, v) such that wj = wi + λij(u, v).The morphism Φ0 Φ1 : X1 → X which we constructed is a product of blow

ups of 2-curves and 3-points. Since u, v, wj are super parameters for f , u, v, wj arethus also super parameters for f Φ0 Φ1.

X4 → X1 is a sequence of blow ups of curves C such that A(C) > 0. Supposethat C is such a curve, and p ∈ C. It suffices to analyze the blow up Φ2 : X2 → X1

of C in our construction. We saw that u, v, wi have a form (14.44) at p withd < mina, b or a form (14.45) with (g, h) ≤ min(a, b), (c, d) and g < mina, c.In either case x = z = 0 are local equations of C. We will show that for all j suchthat q ∈ U(Rj), u, v, wj are super parameters for f Φ0 Φ1 Φ2.

First assume that p ∈ C is a 1-point. Then, we have

u = xa, v = xb(α+ y), wi = xdz

with 0 6= α ∈ k and d < mina, b. Thus, since

wj = wi + λij(u, v),

wj = xdz where z = z+xΩ(x, y) for some series Ω ∈ OX2,p. It follows that x = z = 0are local equations of C at p, and u, v, wj have a form (10.1) of Definition 10.1 atp.

Now assume that p ∈ C is a 2-point. Then we have

u = xayb, v = xcyd, wi = xgyhz

where after possibly interchanging u and v, we have (g, h) < (a, b) ≤ (c, d) (recallthat (u, v)OX1,p is invertible). Since wj − wi ∈ k[[u, v]], we have an expression

(14.56) wj = xgyhz

at p where x = z = 0 are local equations of C at p. it follows that u, v, wj have aform (10.2) of Definition 10.1 at p.

By induction, u, v, wj are super parameters for f4 = f Φ0 Φ1 Φ2 Φ4.q is an admissible center for all 2-point relations Rj associated to R (Definition

12.2). For all j, let R1

j be the transform of Rj on Y1.

Suppose that q1 ∈ U(R1

j ) ∩Ψ−1(q).

Since u, v, wj are super parameters at all p ∈ f−1

4 (q), we have that u, v aretoroidal forms at all points p ∈ f−1

4 (q). Let

u = uR

1j (q1)

, v = vR

1j (q1)

, wj = wR

1j (q1)

after possibly interchanging u and v, we have an expression

u = u, v = u(v + α), wj = uwj ,

with α ∈ k. Thus u, v are toroidal forms at all p ∈ f−14 (q1).

It follows (Definition 12.6) that the transform R1 of R for f4 is defined, and 2of Definition 13.1 holds.


To verify 1 of Definition 13.1 for f4, we must show that p1 ∈ GX4(f4, τ)∩f−1

4 (q)implies p1 ∈ T (R1

j ) for some j. We will verify that this is the case when p1 is a3-point. We leave the remaining cases to the reader. Let q1 = f4(p1).

Suppose that p1 ∈ f−1

4 (q) is a 3-point and p1 6∈ T (R1j ) for any j. We will verify

that τf4(p1) < τ .If Φ(p1) 6∈ T (Rj) for any j, then by (14.55),

τf4(p1) ≤ τf (Φ(p)) < τ.

Now suppose that p1 ∈ f−1

4 (q)∩Φ−1

(T (Rj)). IfX4 → X1 is not an isomorphismat p1, then our construction shows that f4 is toroidal at p1. Thus X4 → X1 is anisomorphism at p1, and Φ is a sequence of blow ups of 2-curves and 3-points at p1.p = Φ(p1) ∈ T (Rj) is then a 3-point with permissible parameters x, y, z such that

(14.57)u = xaybzc

v = xdyezf

wj = M0γ

where rank(u, v) = 2, γ(x, y, z) is a unit series, M0 is a monomial in x, y, z andwj = wRj

(q). Let wej

j = λjuajvbj define Rj(q) if τ > 1, wj = 0 define Rj(q) if

τ = 1. If τ > 1, then

(14.58) Mej

0 = uajvbj and γej (0, 0, 0) = λj ,

and rank(u, v,M0) = 3 if τ = 1. As we have just observed, Φ is a sequence ofblow ups of 2-curves and 3-points at p1. Since p1 is a 3-point, we have permissibleparameters x1, y1, z1 at p1 such that

x = xa111 ya12

1 za131

y = xa211 ya22

1 za231

z = xa311 ya32

1 za331

and Det(aij) = ±1. On substitution into (14.57) we see that an expression

(14.59)u = xa1y

b1zc1

v = xd1ye1zf1

wj = M0γ

where rank(x1,y1,z1)(u, v) = 2, holds at p1 for u, v, wj , and the relation (14.58)holds at p1 if τ > 1, and rank(x1,y1,z1)(u, v,M0) = 3 if τ = 1. We further haveτf4

(p1) = τf (Φ(p1)) = τ .

Continuing to suppose that p1 ∈ f−1

4 (q)∩Φ−1

(T (Rj)) is a 3-point, further sup-pose that p1 6∈ f−1

4 (U(R1

j )). Then f4(p1) = q1 where (after possibly interchangingu and v) q1 has permissible parameters

(14.60)u = uv = u(v + α)wj = u(wj + β)

α, β ∈ k and β is non zero, or

(14.61)u = uwjv = vwjwj = wj .


Suppose that (14.60) holds at q1. Since rank(u, v) = 2 in (14.59), we musthave α = 0 (and 0 6= β). But then M0 = u, a contradiction to the assumptionthat ej > 1 (and gcd(aj , bj , ej) = 1) in (14.58) if τ > 1, or to the assumption thatrank(u, v,M0) = 3 if τ = 1.

Suppose that (14.61) holds at q1. Then q1 = f4(p) is a 3-point. From equations(14.61) and (14.59) we have (in the notation of Definition 9.1) that τf4(p) = −∞ ifτ = 1 and if τ > 1 then

Hf4,p = Hf4,p= Hf,Φ(p),

Af4,p = Af,Φ(p) + (a0, b0, c0)Z

since q = f4(p) = f(Φ(p)) is a 2-point. Thus, since ej > 1 in (14.58), we have

τf4(p) = |Hf4,p/Af4,p| < |Hf,Φ(p)/Af,Φ(p)| = τ.

We verify by a local calculation that 3 of Definition 13.1 holds, since we wouldhave τf4(p) < τ if one of the equations of 3 of Definition 13.1 were to fail atp ∈ T (R1).

Now we verify 4 of Definition 13.1. Suppose that q1 ∈ U(R1). We will verify4 when q1 is a 2-point. Let q = Ψ(q1) ∈ U(R). If q1 ∈ U(R

1

i ) ∩ U(R1

j ) then (afterpossibly interchanging u and v) we have

u = uv = uvwi = uwiwj = uwj

where

u = uRi(q) = uRj

(q), v = vRi(q) = vRj

(q), wi = wRi(q), wj = wRj

(q)

and

u = uR

1i(q1) = u

R1j(q1), v = v

R1i(q1) = u

R1j(q1), wi = w

R1i(q1), wj = w

R1j(q1).

Since f is pre-τ -quasi-well prepared, there exists a series λij(u, v) such that

wj = wi + λij(u, v).

Since λij(0, 0) = 0, u divides λij(u, uv), and

wj = wi +λij(u, uv)

u.

Thus 4 of Definition 13.1 holds for f4.Earlier in the proof we verified that if q ∈ U(Rj), for some Rj associated to R,

thenu = uRj

(q), v = vRj(q), wj = wRj

(q)

are super parameters for f4 at q. If q1 ∈ U(R1

j ) ∩ Ψ−1(q), then (after possiblyinterchanging u and v) we have permissible parameters at q1

u = uR

1j(q1), v = v

R1j(q1), wj = w

R1j(q1)

such thatu = u, v = u(v + α), wj = uwj

with α ∈ k.


Substituting into the forms of Definition 10.1 for f4, we see that u, v, wj aresuper parameters for f4 at q1. Thus 5 of Definition 13.1 holds for f4 and we haveverified that f4 is pre-τ -quasi-well prepared.

Now suppose that f is τ -well prepared. We will verify that f4 is τ -well prepared.1 and 2 of Definition 13.5 are immediate.

We will verify that 3 of Definition 13.5 holds for f4. Suppose that q1 ∈ U(R1

i )∩U(R

1

j ) andu = u

R1i(q1) = u

R1j(q1), v = v

R1i(q1) = v

R1j(q1),

wi = wR

1i(q1), wj = w

R1j(q1).

Let q = Ψ(q1),u = uRi

(q) = uRj(q), v = vRi

(q) = vRj(q),

wi = wRi(q), wj = wRj

(q).

We will verify 3 in the case when q and q1 are 2-points. The remaining cases aresimilar. Since f is τ -well prepared, there exist unit series φij(u, v) such that

wj = wi + uaijvbijφij

(or φij = 0 and aij = bij = −∞). After possibly interchanging u and v, we mayassume that

u = u, v = uv,wi = uwi, wj = uwj .

Letφij(u, v) = φij(u, uv).

Then we have

(14.62) wj = wi + uaijvbijφij

where

(14.63) aij = aij + bij − 1, bij = bij .

Thus 3 of Definition 13.5 holds for f4.Since the set (13.9) associated to q and R is totally ordered, the set (13.9)

associated to q1 and R1 is also totally ordered. Thus 4 of Definition 13.5 holds forf4, and we see that f4 is τ -well prepared.

We now verify 2 of Lemma 14.7. Suppose that q1 ∈ U(R1

i ) ∩ E for some R1

i

associated to R1. Continuing with the notation we used in the verification that f4is τ -well prepared, let γi = S

R1i(q1) · E. γi is covered by two affine charts, with

uniformizing parameters u, v, wi and u, v, wi defined by

(14.64) u = u, v = uv,wi = uwi

and

(14.65) u = uv, v = v, wi = vwi.

In the chart (14.64), u = 0 is a local equation for E, and v = 0 is a localequation for the strict transform of the component E2 of DY with local equationv = 0 at q. In (14.65), v = 0 is a local equation of E and u = 0 is a local equationof the strict transform of the component E1 of DY with local equation u = 0 atq. In the chart defined by (14.64) u = wi = 0 are local equations of γi and in thechart defined by (14.65) v = wi = 0 are local equations of γi. Thus γi makes SNCswith DY1 , and γi is a line on E ∼= P2.


If q ∈ U(Rj) for some j 6= i, then we see from (14.62) and (14.63) that γj =SR

1j(q′) · E (where q′ ∈ Ψ−1(q) ∩ U(R

1

j )) has local equations u = wj = 0 in the

chart (14.64) wherewj = wi + uaij+bij−1vbijφij .

In the chart (14.65), γj has local equations v = wj = 0 where wj = wi +uaij vaij+bij−1φij . If aij + bij > 1, then γi = γj so that 2 (a) (i) holds in thestatement of Lemma 14.7. If aij+bij = 1 then 2 (a) (ii) of the statement of Lemma14.7 holds.

γi is a prepared curve for R1 of type 6 since γi ⊂ U(R1

i ) for all i.We now verify 2 (b) of Lemma 14.7. Suppose that γ is prepared for R, q ∈ γ,

and γ is prepared for R of type 6. Then γ = E2 · SRi(q) for some Ri and component

E2 of DY containing q and γ ⊂ Ω(Ri). Let u = uRi(q), v = vRi

(q), wi = wRi(q).

We may assume that v = 0 is a local equation of E2. Then v = wi = 0 are localequations at q of γ. Let γ′ be the strict transform of γ on Y1. q1 = γ′ · E haspermissible parameters

u = uR

1i(q1), v = v

R1i(q1), wi = w

R1i(q1),

whereu = u, v = uv,wi = uwi,

and v = wi = 0 are local equations of γ′. Let E2 be the strict transform of E2.Since q1 ∈ U(R

1

i ), we have γ′ = E2 · SR1i(q1) and γ′ ⊂ Ω(R

1

i ). Thus the conditionsof Definition 13.8 hold for γ′, and γ′ is prepared for R1 of type 6. If γ is a 2-curveon Y through q, then the strict transform γ′ of γ on Y1 is a 2-curve so γ′ is preparedfor R1.

Finally, suppose that f is τ -very-well prepared. We have shown that 1 and 2of Definition 13.9 hold for f4. Since whenever Ri is a pre-relation associated to Rcontaining q, Ω(R

1

i ) → Ω(Ri) is the blow up of a point on a nonsingular surface,and Vi(Y ) satisfies 3 of Definition 13.9, 3 of Definition 13.9 holds for Vi(Y1). Thusf4 is τ -very-well prepared.

Lemma 14.8. Suppose that f : X → Y is pre-τ -quasi-well prepared (or τ -well prepared or τ -very-well prepared), q ∈ U(R) is a 1-point (which is prepared oftype 4 of Definition 13.7). Then there exists a pre-τ -quasi-well prepared (or τ -wellprepared or τ -very-well prepared) diagram

(14.66)X1

f1→ Y1

Φ ↓ ↓ Ψ

Xf→ Y

where Ψ is the blow up of q such that1. Φ is a sequence of blow ups of 2-curves over f−1(Y − q).2. Suppose that f is τ -well prepared.

(a) Let E be the exceptional divisor of Ψ. Suppose that q1 ∈ U(R1

i ) ∩E.Let γi = S

R1i(q1) · E. Then γi is a prepared curve for R1 of type 6.

Suppose that q′ ∈ U(R1

j ) ∩ E. Let γj = SR

1j(q′) · E. Then either

(i) γi = γj or


(ii) γi, γj intersect transversally at a 2 point on E (their tangentspaces have distinct directions at this point) and γi, γj are oth-erwise disjoint.

(b) If γ is a prepared curve on Y then the strict transform of γ is aprepared curve on Y1.

3. If f is τ -prepared then f1 is τ prepared.

The proof of Lemma 14.8 is a variant of the proof of Lemma 14.7, keeping inmind the simpler forms 5 and 6 of Definition 10.1 of super parameters above a1-point, and the simpler from (13.8) of 3 of Definition 13.5 at q.

Lemma 14.9. Suppose that f : X → Y is pre-τ -quasi-well prepared (or τ -wellprepared or τ -very-well prepared) with relation R. Suppose that q ∈ Y is a 1-pointor a 2-point such that q 6∈ U(R) and q is prepared of type 2 of Definition 13.7 forR. Then q is a permissible center for R and there exists a pre-τ -quasi-well prepared(or τ -well prepared or τ -very-well prepared) diagram (13.10) of R and the blow upΨ : Y1 → Y of q such that:

1. Φ is an isomorphism over f−1(Y − Σ(Y )) if q is a 2-point.2. Suppose that f is τ -well prepared. If γ is a prepared curve on Y then the

strict transform of γ is a prepared curve on Y1.

The proof of Lemma 14.9 is a simplification of the proofs of Lemma 14.7 andLemma 14.8.

Lemma 14.10. Suppose that f : X → Y is pre-τ -quasi-well prepared, q ∈ Y isa 1-point such that q 6∈ U(R), and C ⊂ DY is an integral curve such that q ∈ C.Suppose that C satisfies 2 and 4 of Definition 13.6 of a resolving curve. Then thereexists a pre-τ -quasi-well prepared diagram

X1f1→ Y1

Φ1 ↓ ↓ Ψ1

Xf→ Y

such that

1. Ψ1 is a product of blow ups of 2-curves and 2-points, Φ1 is a product ofpossible blow ups, Φ1 is an isomorphism over f−1(Y − Σ(Y )).

2. Let C be the strict transform of C on Y1. Then C is a resolving curve forf1 and R1 at q.

3. If f is τ -prepared, then f1 is τ -prepared.

Proof. There exists a sequence of blow ups of 2-curves Ψ1 : Y1 → Y such thatthe strict transform C1 of C on Y1 contains no 3-points. Let

(14.67)X1

f1→ Y1

Φ1 ↓ ↓ Ψ1

Xf→ Y

be the pre-τ -quasi-well prepared diagram obtained by iterating the construction ofLemma 14.1.

Now by Lemma 14.1 and Lemma 10.7, there exists a pre-τ -quasi-well prepareddiagram


(14.68)X2

f2→ Y2

Φ2 ↓ ↓ Ψ2

X1f1→ Y1

obtained by iterating the construction of Lemma 14.1 such that Ψ2 and Ψ2 areproducts of blow ups of 2-curves, and for all 2-points q on the strict transform C2

of C on Y2, there exist super parameters u, v, w at q.Let Ψ3 : Y3 → Y2 be the blow up of the 2-points on C2. By our construction,

these points are disjoint from GY2(f2, τ). By Lemma 14.9, there exists a pre-τ -quasi-well prepared diagram

(14.69)X3

f3→ Y3

Φ3 ↓ ↓ Ψ3

X2f2→ Y2.

By iterating the above construction, (and by embedded resolution of planecurve singularities, c.f. Section 3.4, Exercise 3.13 [C6]) we eventually construct apre-τ -quasi-well prepared diagram

X ′ f ′→ Y ′

Φ′ ↓ ↓ Ψ′

Xf→ Y

such that the strict transform C ′ of C on Y ′ is nonsingular and makes SNCs withDY ′ . If q ∈ C ′ is a 2-point, let u, v, w be permissible parameters at q such thatu = w = 0 are local equations of C ′.

By Lemma 14.1 and Lemma 10.7, there exists a pre-τ -quasi-well prepared dia-gram

X ′′ f ′′→ Y ′′

Φ′′ ↓ ↓ Ψ′′

X ′ f ′→ Y ′

such that 3 of Definition 13.6 holds for the strict transform of C on Y ′′. Thus theconclusions of the lemma hold.

Lemma 14.11. Suppose that f : X → Y is τ -very-well prepared and C ⊂ Y is aprepared curve of type 6 (of Definition 13.8). Further suppose that qδ ∈ C ∩U(Rj)for some Rj associated to R implies C = E · SRj

(qδ) for some component E ofDX . Then C is a *-permissible center for R, and there exists a τ -very-well prepareddiagram

X1f1→ Y1

Φ1 ↓ ↓ Ψ1

Xf→ Y

of R of the form of (13.11).

Proof. Let C = Eα · SRi(qβ). For q ∈ C, we have permissible parameters

(14.70) u, v, wi


such that u = wi = 0 are local equations at q for C, with the notation of 5 ofDefinition 13.8, if q 6∈ U(Ri), and u = uRi

(q), v = vRi(q), wi = wRi

(q) if q ∈ U(Ri).As in the proof of Lemma 14.7, after blowing up 2-curves and 3-points above X, bya morphism Φ0Φ1 : X1 → X, with associated morphism f1 = f Φ0Φ1 : X1 → Y ,we have that the following holds.

1. If q ∈ C is a 2-point, and p ∈ f−11 (q) and ICOX1,p is not invertible, then

one of the forms (14.44) or (14.45) hold at p ∈ f−11 (q) (with d < a in

(14.44), (g, h) < (a, b) in (14.45)).2. If q ∈ C is a 1 point then a form (14.71) below holds at p ∈ f−1

1 (q) ifICOX1,p is not invertible.

(14.71)u = xa

v = ywi = xdz

where d < a and p is a 1-point.As in the proof of Lemma 14.7, the locus of points in X1 where ICOX1 is not

invertible is a (possibly reducible) curve E which makes SNCs with the toroidalstructure of X1. As in the proof of Lemma 14.7, we can construct a sequence ofblow ups of sections over components of E, X4 → X1, such that the resulting mapf4 : X4 → Y factors through the blow up Ψ1 : Y1 → Y of C. Let Φ : X4 → X bethe composite map. By our construction, if u, v, wi are our permissible parametersat q ∈ C of (14.70), then u, v, wi are super parameters for f4 at q.

Let f4 : X4 → Y1 be the resulting morphism. As in the proof of Lemma 14.7,we see that f4 is prepared.

Suppose that q ∈ C ∩ U(Rj) for some j. Then q ∈ U(Ri), since C is preparedof type 6. By the hypothesis of this lemma, the germ of C at q is contained inSRj

(q). Since C ⊂ Eα, the germ of C at q is Eα · SRj(q).

Suppose that q is a 2-point. The case where q is a 1-point is simpler. Then inthe forms of (13.7) of Definition 13.5 for R (after possibly interchanging u and v),we have ajk > 0 for all j, k ∈ Iq.

Ψ−11 (q) is covered by 2 affine charts. The first chart has uniformizing parameters

u, v, wi defined by

(14.72) u = u, v = v, wi = uwi.

The second chart has uniformizing parameters u′, v′, w′i defined by

(14.73) u = u′w′i, v = v′, wi = w′i.

For j ∈ Iq we have a relation

wj = wi + uaijvbijφij(u, v)

with aij > 0 (where wj = wRj(q)). As in the proof of Lemma 14.7, it follows that

f4 is τ -prepared, the transform R1 of R for f4 is defined, and f4 is pre-τ -quasi-wellprepared.

We now verify that f4 is τ -well prepared. 1 of Definition 13.5 is immediate.Suppose that C ∩ U(Rj) 6= ∅ for some j. Then C = Eα ·Rj(qβ) ⊂ Ω(Rj).

Since Ω(Rj) is nonsingular and makes SNCs with DY , Ω(R1

j ) is nonsingular, makes

SNCs with DY1 , contains U(R1

j ), and Ω(R1

j ) ∩ U(R1) = U(R1

j ). If C ∩ U(Rj) = ∅,then after possibly replacing Ω(Rj) with a neighborhood of Fj in Ω(Rj) (with the


notation of Definition 13.9, and following our convention on Ω(Rj) stated afterDefinition 13.9), we have that Ω(Rj) ∩ C = ∅. Thus 2 of Definition 13.5 holds forf4.

We now verify 3 and 4 of Definition 13.5 for f4. Suppose that q ∈ C ∩ U(Rj)is a 2-point. The case where q is a 1-point is simpler. Then qj = U(R

1

j )∩ f−14 (q) is

a 2-point in the chart (14.72), and

(14.74) wR

1j(qj) = wj =

wju

= wi + uaij−1vbijφij(u, v).

Thus qj ∈ U(R1

i ) if and only if aij − 1 + bij > 0. Let qi = U(R1

i ) ∩ f−14 (q),

Iqi = j | qi ∈ U(R1

j ).

j ∈ Iqi if and only if j ∈ Iq and aij + bij > 1.From equation (14.74) we see that the set (13.9) of Definition 13.5 corresponding

to qi and R1 is totally ordered, since the set (13.9) of Definition 13.5 correspondingto q and R is totally ordered. In particular, we see that 3 and 4 of Definition 13.5hold for f4. We have completed the verification that f4 is τ -well prepared.

Let E = Ψ−11 (C) and if C∩U(Rj) 6= ∅, let γj = Ω(R

1

j )·E. γj is also nonsingular

and makes SNCs with DY1 . We have that γj = E · SR

1j(q1) for all q1 ∈ E ∩ U(R

1

j ),and γj is a section of E over C. In particular, 1 and 2 of Definition 13.8 hold forγj .

Suppose that for some Rk associated to R, there exists γ′ ⊂ Ω(R1

k), withγ′ = Eβ · SR1

k(qδ), and γ′ ∩ γj 6= ∅.

First suppose that γ′ ⊂ E. Then U(R1

k) ∩ E 6= ∅ and γ′ = γk = E · SR

1k(q2)

for all q2 ∈ U(R1

k) ∩ E. Let F = Eα be the component of DY containing C.Since C is prepared of type 6 for R and Ψ1(γj) = Ψ1(γk) = C, we have thatC ∩ U(Rj) = C ∩ U(Rk) and C = F · SRj

(q) = F · SRk(q) for q ∈ C ∩ U(Rj).

Suppose that q2 ∈ γj ∩ γk. Let q = Ψ1(q2) ∈ C. Let u, v, wi and u, v, wk be thepermissible parameters at q of (14.70).

Suppose that q is a 1-point. We have

wj = wk + uckjφkj(u, v),

where φkj is a unit series (or φkj = 0) by 5 (d) of Definition 13.8, since C is preparedfor R of type 6. We have that q2 has permissible parameters u, v, wj where

(14.75) u = u, v = v, wj = wju.

Define wk by wk = wku. We have that

wj = wk + uckj−1φkj(u, v).

u = wj = 0 are local equations of γj at q2 and u = wk = 0 are local equations ofγk at q2. Thus q2 is a 1-point with cjk − 1 > 0 and γj = γk.

Now suppose that q2 ∈ γj ∩ γk and q = Ψ1(q2) ∈ C is a 2-point. First supposethat q 6∈ C ∩ U(Rj) = C ∩ U(Rk). We have a relation

wj = wk + uakjvbkjφkj(u, v)


where φjk is a unit series (or φkj = 0) by 5 (d) of Definition 13.8 for C. We haveakj ≥ 1. q2 has permissible parameters u, v, wj where

(14.76) u = u, v = v, wj = wju.

Define wk by wk = wku. We then have

(14.77) wj = wk + uakj−1vbkjφkj(u, v).

Thus q2 is a 2-point (and q2 6∈ U(R1)).Finally, suppose that q2 ∈ γj ∩ γk and q = Ψ1(q2) ∈ C ∩ U(Rj) = C ∩ U(Rk).

Then q2 ∈ U(R1

j ) ∩ U(R1

k), and we have equations (14.76) and (14.77).Now suppose that γ′ = Eβ · SR1

k(qδ) 6⊂ E, and γ′ ∩ γj 6= ∅. Then there exists a

component G of DY such that γ = Ψ1(γ′) ⊂ G, and Eβ is the strict transform ofG. Suppose that q2 ∈ γ′ ∩ γj . q = Ψ1(q2) ∈ γ ∩ C implies q ∈ U(Rj) ∩ U(Rk) andγ = G · SRk

(q) by 3 of Definition 13.8 for R. Thus q2 ∈ U(R1

j ) ∩ U(R1

k).Suppose that q2 ∈ γj , q = Ψ1(q2) ∈ C and p ∈ f−1

4 (q2). Let u, v, wj be thepermissible parameters at q of (14.70). Further suppose that q2 is a 1-point. Letu, v, wj be the permissible parameters at q2 of (14.75). u, v, wj satisfy 5 (b) ofDefinition 13.8 at p. Substituting (14.75) into these forms, we see that u, v, wjsatisfy 5 (b) of Definition 13.8 at p. Now suppose that q2 ∈ γj is a 2-point, butq2 6∈ U(R

1

j ). Let u, v, wj be the permissible parameters at q2 of (14.76). u, v, wjsatisfy 5 (b) of Definition 13.8 at p. Substituting (14.76) into these forms, we seethat u, v, wj satisfy 5 (b) of Definition 13.8 at p. Thus 5 of Definition 13.8 holdsfor γj .

We have seen that the curves γj only fail to be prepared of type 6 for f4 at afinite set of 2-points T1 ⊂ E, where condition 3 of Definition 13.8 fails. If q ∈ T1,then q 6∈ U(R1) and there exist γj and γk such that γj 6= γk and q ∈ γj ∩ γk. Forq ∈ T1, let

Jq = j | q ∈ γj = E · SR

1j(qε) for some qε ∈ U(R

1

j ).Observe that:

1. For q ∈ T1, j ∈ Jq and p ∈ f−14 (q), the permissible parameters u, v, wj at

q defined by (14.72) have a form 5 (b) of Definition 13.8.2. For i, j ∈ Jq there exists a relation of the form 5 (d) of Definition 13.8 forwi and wj , and the set (aij , bij) is totally ordered.

In particular, the points in T1 are prepared for R1 of type 2 of Definition 13.7. LetΨ2 : Y2 → Y1 be the blow up of T1, and let

X5f5→ Y2

↓ ↓ Ψ2

X4f4→ Y1

be the τ -well prepared diagram of Lemma 14.9. Let

T2 =

q ∈ Ψ−12 (T1) such that q ∈ γ2

i ∩ γ2k

where γ2i , γ

2j are the strict transforms of some

γi, γk such that γi 6= γk and γi ∩ γk 6= ∅

.

The points of T2 must again be prepared for the transform R2 of R on X5 of type 2of Definition 13.7, and the points of T2 satisfy the corresponding statements 1 and2 above that the points of T1 and Jq satisfy.


We can iterate this process a finite number of times to produce a τ -well prepareddiagram

Xmfm→ Y m

↓ ↓X1 = X4

f4→ Y 1 = Y1

↓ ↓X

f→ Y

of the form of (13.12) such that the strict transform of the γi are disjoint on Y mabove T1. It follows that the strict transforms of the γi are prepared of type 6 forthe transform Rm of R on Xm.

To show that fm is τ -very-well prepared, it only remains to verify that thestrict transform γ′ of a curve γ = Eβ · SRi

(q) on Y (with γ 6= C and γ ∩ C 6= ∅) isprepared on Y m. Since γ is prepared of type 6, we have that γ ∩ C ⊂ U(Ri). Byour previous analysis, we then know that γ′ ∩ E ⊂ U(R1), so that Y m → Y1 is anisomorphism in a neighborhood of γ′. It thus suffices to check that γ′ is preparedof type 6 on Y1. This follows by a local analysis.

Remark 14.12. 1. Suppose that f : X → Y is pre-τ -quasi-well preparedand C ⊂ DY is a nonsingular (integral) curve which makes SNCs withDY and contains a 1-point such that(a) q ∈ C ∩ U(Ri) for some pre-relation Ri associated to R implies the

(formal) germ of C at q is contained in SRi(q), and

(b) q ∈ C − U(R) implies that one of the following holds(i) there exist super parameters u, v, w at q such that u = w = 0

are local equations of C at q.(ii) q is a 1-point with τf (p) < τ for all p ∈ f−1(q) and there exist

permissible parameters u, v, w at q such that u, v are toroidalforms at p for all p ∈ f−1(q) and u = v = 0 are local equationsof C at q.

Then there exists a pre-τ -quasi-well prepared diagram

X1f1→ Y 1

Φ1 ↓ ↓ Ψ1

Xf→ Y

where Ψ1 is the blow up of C. If C 6⊂ GY (f, τ) and f is τ -prepared thenf1 is τ -prepared.

2. Further suppose that f : X → Y is τ -well prepared, and if γ = Eβ ·Rk(qα)is prepared for R of type 6, then either C = γ or q ∈ C ∩ γ impliesq ∈ U(Rk) and the germ of C at q is contained in SRk

(q). Then thereexists a τ -well prepared diagram

X1f1→ Y1

Φ ↓ ↓ Ψ

Xf→ Y


where Ψ is the blow up Ψ1 of C, possibly followed by blow ups of 2-pointswhich are prepared for the transform of R (of type 2 of Definition 13.7) ifC is prepared of type 6 for R, such that(a) If γ ⊂ Y is prepared for f , (and γ 6= C) then the strict transform of

γ is prepared for f1.(b) If C ⊂ Y is prepared for f (of type 6) and q ∈ U(Ri)∩C for some i,

then E · SR

1i(q′) is prepared for f1 (of type 6) for all q′ ∈ Ψ−1(q) ∩

U(R1

i ), where E is the component of DY1 dominating C.

The proof of Remark 14.12 is a straight forward generalization of Lemma 14.11,Lemma 14.10 and Lemma 14.5.

CHAPTER 15

Construction of a τ-very well prepared morphism

Suppose that f : X → Y is a dominant, proper morphism of nonsingular 3-folds, with toroidal structures DY and DX = f−1(DY ), such that DX contains thelocus where f is not smooth.

Theorem 15.1. Suppose that τ ≥ 1, f : X → Y is τ -prepared, q ∈ Y is a1-point or a 2-point, and u, v, w are permissible parameters at q such that u, v aretoroidal forms at p for all p ∈ f−1(q) and u, v ∈ OY,q. Further suppose that theZariski closure of u = v = 0 in Y intersects GY (f, τ) in a finite set of points. Thenthere exists an affine neighborhood V of q in Y and a commutative diagram

(15.1)X1

f1→ Y1

Φ1 ↓ ↓ Ψ1

f−1(V )f→ V

such that u = v = 0 are local equations of a nonsingular curve γ0 in V , Ψ1 is aproduct of possible blow ups of 2-curves and resolving curves which are sections overγ0, Φ1 is a product of possible blow ups, f1 is τ -prepared, τf1(p1) ≤ τf (Φ1(p1)) forp1 ∈ X1, and there exist permissible parameters uq′ , vq′ , w at all q′ ∈ Ψ−1

1 (q) suchthat u, v are related to uq′ , vq′ birationally and uq′ , vq′ , w are super parameters atq′. Further, τf1(p) < τ if p ∈ (Ψ1 f1)−1(γ0)− (Ψ1 f1)−1(q).

Proof. Let V be an affine neighborhood of q with uniformizing parametersu, v, w such that u, v are toroidal forms at all p′ above u = v = 0 ∩ V , theintersection of the nonfinite locus of f with the curve u = w = 0 on V is q, andu = v = 0 ∩ GY (f, τ) ∩ V ⊂ q. Let W = f−1(V ) and f = f | W . We have asmooth morphism π0 : V → S0 = spec(k[u, v]). Give S0 the toroidal structure

DS0 =uv = 0 if q is a 2-pointu = 0 if q is a 1-point.

Let q = π0(q). The morphism π0 f : W → S0 is toroidal with respect to DS0 and(π0 f)−1(DS0). We construct a commutative diagram

169

170 15. CONSTRUCTION OF A τ-VERY WELL PREPARED MORPHISM

(15.2)

......

...↓ ↓ ↓

Wnfn→ Vn

πn→ SnΦn ↓ Ψn ↓ Λn ↓

......

...Φ2 ↓ Ψ2 ↓ Λ2 ↓

W1f1→ V1

π1→ S1

Φ1 ↓ Ψ1 ↓ Λ1 ↓

Wf→ V

π0→ S0.

Here Λ1 : S1 → S0 is the blow up of q, Ψ1 is the blow up of γ0 = π−10 (q), which

is the curve with equations u = v = 0 in V . If q is a 1-point then γ0 is a resolvingcurve for f at q. Φ1 is the morphism of Lemma 14.1 (if q is a 2-point) or Lemma14.5 (if q is a 1-point). π1 f1 : W1 → S1 is toroidal.

Suppose that q1 ∈ Λ−11 (q). Then OS1,q1 has regular parameters u1, v1 defined

by

(15.3) u = u1, v = u1(v1 + α)

for some α ∈ k, or

(15.4) u = u1v1, v = v1.

Λ2 : S2 → S1 is the blow up of the finitely many points q1 = π1(q1) above qsuch that q1 in Φ−1

1 (q) and u1, v1, w are not super parameters at q1. Ψ2 : V2 → V1

is the blow up of the (disjoint) curves γ1 = π−11 (q1), and Φ2 : W2 → W1 is the

morphism of Lemma 14.1 or Lemma 14.5. π2 f2 : W2 → S2 is toroidal.We continue in this way to construct (15.2) as long as fn : Wn → Vn does not

satisfy the conclusions of the theorem.Suppose that the algorithm never ends.Let ν be a 0-dimensional valuation of k(X) whose center on Y is q, and let pi

be the center of ν on Wi, qi the center of ν on Vi. Let qi = πi(qi). We may supposethat Ψi is not an isomorphism at the center of ν for all i. There exist permissibleparameters ui, vi, wi at qi for all i such that ui, vi are regular parameters in OSi,qi

,obtained by iteration of (15.3) and (15.4), as determined by ν. We will show thatthere exists j0 such that uj , vj , w are super parameters at pj for all j ≥ j0. Ifui, vi, w are super parameters at pi for some i, we have that uj , vj , w are superparameters at pj for all j ≥ i. Suppose that ui, vi, w are not super parameters atpi for all i.

We may identify ν with an extension of ν to the quotient field of OWi,piwhich

dominates OWi,pi .There exist permissible parameters xi, yi, zi in OWi,pi such that we have one of

the following forms:pi a 1-point, qi (and qi) a 1-point

(15.5) ui = xaii , vi = yi

15. CONSTRUCTION OF A τ-VERY WELL PREPARED MORPHISM 171

or pi a 2-point, qi (and qi) a 1-point

(15.6) ui = (xaii y

bii )k, vi = zi

pi a 1-point, qi (and qi) a 2-point

(15.7) ui = xaii , vi = xbi

i (yi + αi)

with 0 6= αi or pi a 2-point, qi (and qi) a 2-point

(15.8) ui = xaii y

bii , vi = xci

i ydii

pi a 2-point, qi (and qi) a 2-point

(15.9) ui = (xaii y

bii )ki , vi = (xai

i ybii )ti(zi + αi)

with αi 6= 0 or pi a 3-point, qi (and qi) a 2-point

(15.10) ui = xaii y

bii z

cii , vi = xdi

i yeii z

fi

i .

Suppose that f ∈ OS0,q. Then (by embedded resolution of plane curve singu-larities, c.f. Section 3.4, Exercise 3.13 [C6]) there exists an n0 such that uvf = 0is a SNC divisor in OSn,qn

for all n ≥ n0.We further have that if g ∈ OW,p is a fractional series in u and v (a series in

fractional rational powers of u and v), then g ∈ OWn,pn is a fractional series in unand vn.

First suppose that qi is a 2-point for all i. Then ν(u) and ν(v) are rationallyindependent, and we have that (15.8) or (15.10) hold for all i. By the algorithm ofLemma 14.1, we see that Φi+1 is a sequence of blow ups of 2-curves above pi for alli.

We either have that pi is a 3-point for all i (and a form (15.10) holds for all i)or some pi is a 2-point, and thus (15.8) holds for all i sufficiently large.

Suppose that pi is a 2-point for some i. We may then assume that pi is a2-point for all i, and (15.8) holds for all i. Then xi and yi are monomials in xi+1

and yi+1 for all i, and zi+1 = zi for all i.We have an expression

u = xayb

v = xcyd

w = f1(x, y) + xlymz

in OW,p. There exists n ∈ N and a, b, c, d ∈ Z such that

xn = uavb, yn = ucvd.

ν(uavb) = ν(xn) > 0, ν(ucvd) = ν(yn) > 0 imply that for i >> 0,

uavb = uaii v

bii , u

cvd = ucii v

dii

with ai, bi, ci, di ∈ N (by Theorem 2.7 [C2]).We have

ui = xaii y

bii

vi = xcii y

dii

w = f1(x, y) + (xeii y

fi

i )l(xgi

i yhii )mz.

f1(x, y) = f1((uavb)1n , (ucvd)

1n ) = g1(u

1ni , v

1ni ) ∈ k[[u

1ni , v

1ni ]].


Let ω ∈ k be a primitive n-th of unity. Set

f =n∏

i,j=1

g1(ωiu1ni , ω

jv1ni ) ∈ k[[ui, vi]].

Recall that for i ≥ n0, fuv = 0 is a SNC divisor in OSi,qi. Since qi is a 2-point for

all i,fuv = uai v

biγ

where a, b > 0, γ is a unit series in OSi,qi. As g1 | f in OWi,pi

, we have that ui, vi, ware super parameters at pi.

Suppose that pi is a 3-point for all i. Then xi, yi, zi are monomials in xi+1, yi+1, zi+1

for all i. By Lemma 10.6, we obtain an expression of the form (10.19) and (10.20)for ui, vi, w in terms of xi, yi, zi for i 0. By a variant on the argument for thecase when pi is a 2-point for all i, we obtain that ui, vi, wi are super parameters atpi for i 0.

The final case is when qi is a 1-point for some i.Suppose that qi is a 1-point for some i. Without loss of generality, we may

assume that qi = q and pi = p.If p is a 1-point (and q is a 1-point), we have permissible parameters x, y, z in

OW,p such thatu = xa, v = y, w = f1(x, y) + xrz

of the form (15.5).Set

f =a∏i=1

f1(ωiu1a , v) ∈ k[[u, v]]

where ω ∈ k is a primitive a-th root of unity.If p is a 2-point (and q is a 1-point), we have permissible parameters x, y, z ∈

OW,p such thatu = (xayb)k, v = z, w = f1(xayb, z) + xcyd

of the form (15.6). Set

f =k∏i=1

f1(ωiu1k , v) ∈ k[[u, v]]

where ω ∈ k is a primitive k-th root of unity.Recall that fuv = 0 is a SNC divisor in OSi,qi

for i ≥ n0.We will now show that if i ≥ n0 and qi is a 2-point, then ui, vi, w are super

parameters at qi. In these cases qi is a 2-point, uivi = 0 is a local equation of DVi

and u = 0 is a local equation of DVi . Thus fuv = umi vni γ, where m,n > 0 and

γ ∈ OSi,qiis a unit series. Since f1 | f in OXi,pi , and f1 is a fractional series in ui

and vi, we have the desired conclusion.We have reduced to the case where qi is a 1-point for all i.Since fuv = 0 is a SNC divisor for i ≥ n0, the only cases where ui, vi, w are

not super parameters at pi are if ui, vi satisfy (15.5) at pi, and

(15.11) ui = xai , vi = yi, w = xsi (yi − φ(xai ))rγ(xi, yi) + xci (zi + α)

where γ is a unit series, ord(φ) > 0, α ∈ k, or pi satisfies (15.6), and

(15.12) ui = (xai ybi )k, vi = zi, w = (xai y

bi )s(z1 − φ((xai y

bi )k))rγ(xai y

bi , zi) + xciy

di


where gcd(a, b) = 1, ord φ > 0 and γ is a unit series.We consider the case when (15.12) holds at pi. The case (15.11) is similar.By the constructions of Lemma 14.1 and Lemma 14.5, we have a factorization

of Wi+1 →Wi, by blow ups of possible centers.

Wi+1 = ZmΩm→ Zm−1

Ωm−1→ · · · Ω2→ Z1Ω1→ Z0 = Wi

Let aj be the center of ν on Zj . We may assume that each morphism Zl → Zl−1 isnot an isomorphism at the center of ν.

If j < m, we have permissible parameters xij , yij , zij in OZj ,ajsuch that aj is

a 2-point and

(15.13)ui = (xaijy

bij)

k,

vi = zijxeiijy

fi

ij ,

w = (xaijybij)

s(zijxeiijy

fi

ij − φ((xaijybij)

k)rγ(xaijybij , zijx

eiijy

fi

ij ) + xcijydij

with (ei, fi) < (ak, bk).After possibly interchanging xij and yij , we may assume that ei < ak and there

are regular parameters xi,j+1, yi,j+1, zi,j+1 in OZj+1,aj+1 and α ∈ k, defined by

(15.14) xij = xi,j+1, zij = xi,j+1(zi,j+1 + α).

or

(15.15) xij = xi,j+1zi,j+1, zij = zi,j+1.


(15.16)

ui = (xai,j+1ybi,j+1)

k

vi = (zi,j+1 + α)xei+1i,j+1y

fi

i,j+1

w = (xai,j+1ybi,j+1)

s[xei+1i,j+1y

fi

i,j+1(zi,j+1 + α)− φ((xai,j+1ybi,j+1)

k)]r

γ(xai,j+1ybi,j+1, x

ei+1i,j+1y

fi

i,j+1(zi,j+1 + α)) + xci,j+1ydi,j+1.

Suppose that 0 6= α, (ei + 1, fi) < (ak, bk) and afi − b(ei + 1) 6= 0 in (15.16).Then the rational map Zj+1 → Vi+1 is a morphism near aj+1 = pi+1.

We make a change of variables, to get permissible parameters x, y, z ∈ OWi+1,pi+1

and 0 6= β ∈ k such that

ui+1 = ui

vi= x

ak−(ei+1)i,j+1 ybk−fi

i,j+1 (zi,j+1 + α)−1 = xak−(ei+1)ybk−fi

vi+1 = vi = (zi,j+1 + α)xei+1i,j+1y

fi

i,j+1 = xei+1yfi

w = (xayb)s[xei+1yfi − φ((xayb)k)]rγ(xayb, xei+1yfi) + xcyd(z + β)= xas+r(ei+1)ybs+rfi(1− φ((xayb)k)

xei+1yfi)rγ(xayb, xei+1yfi) + xcyd(z + β)

and ui+1, vi+1, w are super parameters at pi+1.Suppose that (15.14) holds, 0 6= α, (ei+1, fi) < (ak, bk) and (ei+1, fi) = ti(a, b)

for some integer ti. Then we make a change of variable to get permissible parameters


x, y, z ∈ OWi+1,pi+1 such that

ui+1 = ui

vi= (xayb)k−ti

vi+1 = vi = (xayb)ti(z + α)w = (xayb)s(z + α)

sk [(xayb)ti(α+ z)− φ((xayb)k(z + α))]r

γ(xayb(z + α)1k , (xayb)ti(z + α)) + xcyd

= (xayb)s+tir(z + α)sk (α+ z − φ((xayb)k(z+α))

(xayb)ti)r

γ(xayb(z + α)1k , (xayb)ti(z + α)) + xcyd

as ti < k, and ui+1, vi+1, w are thus super parameters at pi+1.Suppose that (15.14) holds, and (ei + 1, fi) = (ak, bk). Then the rational map

Zj+1 → Vi+1 is a morphism near aj+1 = pi+1, and the 1-point qi+1 has permissibleparameters defined by

ui = ui+1, vi = ui+1(vi+1 + α).

Substituting into (15.16), we see that

ui+1 = (xai,j+1ybi,j+1)

k

vi+1 = zi,j+1

w = (xai,j+1ybi,j+1)

s+kr[zi,j+1 + α− φ((xai,j+1y

bi,j+1)

k)

(xai,j+1y

bi,j+1)

k ]rγ + xci,j+1ydi,j+1.

Thus ui+1, vi+1, w are super parameters at pi+1, or we have a form (15.12), with

(15.17) (c− as, d− bs) > (c− (a(s+ kr), d− b(s+ kr)).

In fact, (c− (a(s+ kr), d− b(s+ kr)) must decrease from (c−as, d− bs) by at least(1, 1).

If we have (15.14) with (ei + 1, fi) < (ak, bk) with α = 0, then (15.16) is backin the form (15.13) (with a decrease in (ak − ei) + (bk − fi).

Suppose that (15.15) holds. The rational map Zj+1 → Vi+1 is a morphismnear aj+1 = pi+1, and the 2-point qi+1 has permissible parameters defined byui = ui+1vi+1, vi = vi+1. Substituting into (15.13), we have

ui+1 = xak−eii,j+1 y

bk−fi

i,j+1 zak−ei−1i,j+1

vi+1 = xeii,j+1y

fi

i,j+1zei+1i,j+1

w = (xai,j+1ybi,j+1z

ai,j+1)

s(xeii,j+1y

fi

i,j+1zei+1i,j+1 − φ((xai,j+1y

bi,j+1z

ai,j+1)

k))rγ+xci,j+1y

di,j+1z

ci,j+1

= (xai,j+1ybi,j+1z

ai,j+1)

s(xeii,j+1y

fi

i,j+1zei+1i,j+1)

r(1− φ((xai,j+1y

bi,j+1z

ai,j+1)

k)

xeii,j+1y

fii,j+1z

ei+1i,j+1

)rγ

+xci,j+1ydi,j+1z

ci,j+1

Thus ui+1, vi+1, w are super parameters at the 3-point pi+1.We thus have that ui, vi, w are super parameters at pi for i >> 0 unless each

pi has a form (15.12), and by (15.17) we eventually get

(c− as, d− bs) < (0, 0).

We thus have that w = xciydi γ where γ is a unit series. Thus ui, vi, w are super

parameters at pi.We have shown that for any 0-dimensional valuation ν of k(X) whose center

is q on Y , there exists j1 such that uj , vj , w are super parameters at pj for j ≥ j1.By compactness of the Zariski Riemann manifold [Z], it follows that the sequence


(15.2) must terminate after a finite number of steps, in Wn → Vn satisfying theconclusions of the theorem.

Theorem 15.2. Suppose that τ ≥ 1, f : X → Y is pre-τ -quasi-well prepared(or τ -well prepared) with relation R, f is τ -prepared and C ⊂ Y is a reduced (butpossibly not irreducible) curve consisting of components of the nonfinite locus off which contain a 1-point of Y . Then there exist sequences of possible blow upsΦ1 : X1 → X and Ψ1 : Y1 → Y such that there is a commutative diagram

(15.18)X1

f1→ Y1

Φ1 ↓ ↓ Ψ1

Xf→ Y

satisfying1.

τf1(p1) ≤ τf (Φ1(p1))for p1 ∈ DX1 .

2. f1 is pre-τ -quasi-well prepared with respect to a relation R1 which extendsthe transform of R.

3. Φ−11 (T (R)) ∩GX1(f1, τ) ⊂ T (R1).

4. If f is τ -quasi-well prepared (τ -well prepared) then (15.18) is a τ -quasi-well prepared (τ -well prepared) diagram.

5. The strict transform C of C on Y1 is nonsingular and makes SNCs withDY1 .

6. If Cj is an irreducible component of C and q ∈ U(R1

i ) for some R1

i asso-ciated to R1 is such that q ∈ Cj then the germ of Cj at q is contained inSR

1i(q).

7. If f is τ -well prepared, Cj is an irreducible component of C, E is a com-ponent of DY1 , qα ∈ U(R1

k) ∩ E, and γ = E ·R1k(qα) is prepared for R1

of type 6 (of Definition 13.8), then either Cj = γ or q ∈ Cj ∩ γ impliesq ∈ U(R

1

k) (and thus the germ of Cj at q is contained in SRk(q) by 6).

8. The components Cj of C are permissible centers (or *-permissible centersif f is τ -well prepared and Cj is prepared of type 6) for R1.

9. ψ1 is a sequence of blow ups of prepared 1-points, prepared 2-points, 2-curves and resolving curves.

There is a pre-τ -quasi-well prepared (or τ -well prepared) diagram

X2f2→ Y2

Φ2 ↓ ↓ Ψ2

X1f1→ Y1

where Ψ2 is the blow up of C, possibly followed by blow ups of 2-points which areprepared of type 2 of Definition 13.7 for the transform of R if f is τ -well preparedand C contains a component which is prepared of type 6 for R.

Proof. Let q1, . . . , qm be the 1-points of C which are not contained in U(R),and for which there do not exist super parameters u, v, w ∈ OY,q such that u =w = 0 are local equations of C at q. This set is finite by Lemma 11.6.


Step 1. Let γ1 be a general curve through q1 on DY .If q ∈ γ1 is a 1-point, then there exist permissible parameters u, v, w ∈ OY,q,

such that u, v have a toroidal form at p for all p ∈ f−1(q), u = v = 0 are local equa-tions of γ1 at q. This follows from (the proof of) Lemma 4.9 since γ1 intersects thenonfinite locus of f transversally at general points of one dimensional componentsof the nonfinite locus.

Since γ1 is a general curve through q1, (γ1 − q1) ∩ GY (f, τ) ⊂ Θ(f, Y ) byRemark 11.12. Suppose that q ∈ (γ1 − q1) ∩ (GY (f, τ)− U(R)).

Since q is perfect for f and by Lemma 11.5, there exist locally closed subsetsV 1, . . . , V n of X which are a partition of GX(f, τ) ∩ f−1(q) and series

φ1(u, v), . . . , φn(u, v)

such that u, v, wi = w − φi are super parameters for f at q and wi is weakly goodat p for p ∈ V i. Here u, v, w ∈ OY,q are permissible parameters at the 1-point qsuch that u, v have a toroidal form at p, u = v = 0 are local equations of γ1 at q.

If p ∈ f−1(q) is a 1-point and τf (p) > 1, then wi = 0 is supported on DX at p,since u, v, wi are super parameters at q.

By blowing up 2-curves above X, by a map Φ1 : X1 → X, with induced mapf1 = f Φ1 : X1 → Y , we obtain by Lemma 10.3 and Lemma 14.1 that wi = 0is supported on DX1 at p1 for all p1 ∈ f−1

1 (q) such that τf1(p1) > 1. u, v, wp aresuper parameters for f1 at q.

By Lemma 10.3 and Remark 14.2, there exist locally closed subsets V1, . . . , Vnof X1 which are a partition of GX1(f1, τ) ∩ f−1

1 (q) such that

1. u, v, wi = w − φi(u, v) are super parameters for f1 at q2. wi is weakly good at p for p ∈ Vi3. wi = 0 is supported on DX1 at p for p ∈ Vi if τ > 1.

Suppose that τ > 1. Then there exists an expression weii − uaiΛi = 0 for

p ∈ Vi ∩DX1 , where Λi is a unit on Vi, ei, ai ∈ N, ei > 1 and gcd(ei, ai) = 1. Sinceu, v, wi are super parameters at the 1-point q, we see from (10.5) and (10.6) thatthere exists λiq ∈ k such that Λi(p) = λiq for p ∈ Vi.

Suppose that τ = 1. Then wi = 0 is a monomial form at p for p ∈ Vi byRemark 11.3.

We may now define relations Rq,i for f1 by T (Rq,i) = Vi, U(Rq,i) = q, anddefine Rq,i(p) for p ∈ T (Rq,i) by Rq,i = wei

i − λiquai if τ > 1 and by Rq,i = wi ifτ = 1.

We extend the transform R1 of R on X1 by adding in the new relations Rq,ifor all q ∈ (γ1 − q1) ∩ (GY (f, τ) − U(R)). f1 : X1 → Y is then τ -prepared andpre-τ -quasi-well prepared for R1. We have that

(γ1 − q1) ∩GY (f1, τ) ⊂ U(R1).

By Lemma 14.7 and Lemma 14.8, and since for q ∈ γ1 ∩U(R1i ), the germ of γ1

at q is not contained in SR1i(q) for any relation R1

i associated to R1 (γ1 is general),there exists a pre-τ -quasi-well prepared (τ -well prepared) diagram

X2f2→ Y2

Φ2 ↓ ↓ Ψ2

X1f1→ Y


such that f2 is τ -prepared, Ψ2 is a product of blow ups of prepared 1-points oftype 4 (of Definition 13.7) and prepared 2-points of type 1 (of Definition 13.7) suchthat the strict transform γ2

1 of γ1 on Y2 satisfies 1,2 and 4 of Definition 13.6 of aresolving curve for f2 at q1 (we may identify q1 with a point of Y2 since Ψ2 is anisomorphism above q1). Further, if q ∈ γ2

1 is a 2-point, we have that τf2(p) < τ

for p ∈ f−12 (q). By Lemma 10.7 and Remark 14.3 there exists a pre-τ -quasi-well

prepared (τ -well prepared) diagram

X3f3→ Y3

Φ3 ↓ ↓ Ψ3

X2f2→ Y2

where Φ3 is a product of blow ups of 2-curves and 3-points and Ψ3 is a productof blow ups of 2-curves such that if q ∈ γ3

1 is a 2-point, where γ31 is the strict

transform of γ1 on Y3, then there exist super parameters u, v, w for f3 at q suchthat u = w = 0 are local equations of γ3

1 at q. Thus γ31 is a resolving curve for f3

at q.Let

X4f4→ Y4

Φ4 ↓ ↓ Ψ4

X3f1→ Y3

be the pre-τ -quasi-well prepared (τ -well prepared) diagram of Lemma 14.5 whereΨ4 is the blow up of γ3

1 . The strict transform C4 of C on Y4 intersects Ψ−14 (q1) in

a 2-point. Y4 → Y is an isomorphism over a neighborhood of q2, . . . , qm. f4 isτ -prepared.

Step 2. Iterate the construction of Step 1 for the points q2, . . . , qm. We obtain acommutative diagram

Xf→ Y

Φ ↓ ↓ Ψ

Xf→ Y

such that f is τ -prepared, pre-τ -quasi-well prepared (τ -well prepared) with respectto a relation R which extends the transform of R. If C is the strict transform of Con Y , and if q ∈ C − U(R) is a 1-point, then there exist super parameters u, v, wat q such that u = w = 0 are local equations of C at q.

Step 3. By Lemma 10.7 and Remark 14.3, there exists a pre-τ -quasi-well prepared(τ -well prepared) diagram

X1f1→ Y1

Φ1 ↓ ↓ Ψ1

Xf→ Y

where Φ1 is a product of blow ups of 2-curves and 3-points, and Ψ1 is a product ofblow ups of 2-curves such that if C1 is the strict transform of C on Y1, and q ∈ C1

is a 2-point not in U(R1), then there exist super parameters u, v, w at q, so that q


is prepared of type 2 of Definition 13.7. Let

X2f2→ Y2

Φ2 ↓ ↓ Ψ2

X1f1→ Y1

be the pre-τ -quasi-well prepared (τ -well prepared) diagram obtained by blowing upthe 2-points on C1 − U(R1) (by Lemma 14.9).

Step 4. By embedded resolution of plane curve singularities (c.f. Section 3.4 andExercise 3.13 [C6]), we can iterate Step 3 to construct a pre-τ -quasi-well prepared(τ -well prepared) diagram

X3f3→ Y3

Φ3 ↓ ↓ Ψ3

X2f2→ Y2

such that if C3 is the strict transform of C on Y3 and q ∈ C3 − U(R3) is a 2-point, then there exist super parameters u, v, w at q such that u = w = 0 are localequations of C3.

Step 5. By embedded resolution of plane curve singularities (c.f. Section 3.4 andExercise 3.13 [C6]), Lemma 14.1 (for 3-points in C3), Step 3, Lemma 14.7 andLemma 14.8, there exists a pre-τ -quasi-well prepared (τ -well prepared) diagram

X4f4→ Y4

Φ4 ↓ ↓ Ψ4

X3f3→ Y3

such that the strict transform C4 of C on Y4 satisfies the hypotheses of 1 of Remark14.12.

Step 6. If f is τ -well-prepared, then we must perform a final sequence of blowups.Assume that f (and thus f4) is τ -well prepared (for the transform R4 of R). Let

Σ =q ∈ C4 such that q ∈ γ and q 6∈ U(R4)

where γ is a curve which is prepared of type 6 for R4.

.

Σ is a finite set of points which are prepared of type 2 of Definition 13.7 (by 5b) of Definition 13.8).

By Lemma 14.9, and embedded resolution of plane curve singularities, we canconstruct a τ -well prepared diagram

X5f5→ Y5

Φ5 ↓ ↓ Ψ5

X4f4→ Y4

such that the hypotheses of 2 of Remark 14.12 hold, and thus the conclusions ofthe theorem hold.


Theorem 15.3. Suppose that f : X → Y is τ -prepared and pre-τ -quasi-well-prepared (τ -well prepared) with relation R and R′ is a restriction of R with U(R′) =GY (f, τ)−Θ(f, Y ). Then there exists a pre-τ -quasi-well prepared (τ -well prepared)diagram of R

X1f1→ Y1

Φ1 ↓ ↓ Ψ1

Xf→ Y

such that Φ1, Ψ1 are products of blow ups of possible centers and f1 is τ -quasi-well prepared (τ -well prepared) with respect to the transform (R1)′ of R′. Further,there exists a finite set of points Ω = q1, . . . , qr ⊂ Y such that f1 is toroidal onf−11 (Y1 −Ψ−1

1 (Ω)), and Ψ1 f1(T ((R1)′)) ⊂ U(R′).

Proof. Let C ⊂ Y be the union of the one dimensional components in thenonfinite locus of f which contain a 1-point. By Lemma 11.7, there exists a Zariskiopen subset Y of Y such that Y ∩GY (f, τ) = Θ(f, Y ), Y contains a generic pointof each component of C, and there exists a commutative diagram

(15.19)

Xnfn→ Y n

Φn ↓ ↓ Ψn

Xn−1

fn−1→ Y n−1

Φn−1 ↓ ↓ Ψn−1

......

↓ ↓

X1f1→ Y 1

Φ1 ↓ ↓ Ψ1

f−1

(Y ) = Xf→ Y

where fn is toroidal, each Ψi is the blow up of a nonsingular curve γi (in thefundamental locus of f i−1) dominating a component of C, and Φi is a sequence ofblow ups of nonsingular curves dominating γi. Further, we can choose Y so thatfn is τ -quasi-well prepared (τ -well prepared) for the transform of the restriction ofR to Y .

We have that q ∈ (C −Θ(f, Y )) ∩GY (f, τ) implies q ∈ U(R′).We apply Theorem 15.2 to C. We construct a pre-τ -quasi-well prepared (τ -well

prepared) diagram of R. Let R1 be the transform of R on X1, and let (R1)′ be thetransform of R′ on X1 (which is a restriction of R1). Let the diagram be

(15.20)X1

f1→ Y1

Φ1 ↓ ↓ Ψ1

Xf→ Y,

which, after possibly replacing Y with a proper open subset of Y , restricts to

(15.21)X1

f1→ Y 1

Φ1 ↓ ↓ Ψ1

Xf→ Y


over Y . By Theorem 15.2, Remark 14.3 and Lemma 14.1, we may modify thediagrams (15.20) and (15.21) by blowing up 2-curves and 3-points above Y1 and X1

to achieve the condition that f1 is τ -prepared. We may insert blow ups of 2-curvesin the diagram (15.19) without changing the conclusion that fn is toroidal (no 3-points are in (15.19)). We have that GY1(f1, τ) ⊂ U((R1)′) ∪ Φ−1

1 (f−1(Θ(f, Y ))).We iterate this construction for the Zariski closure of γi in Yi, using Theorem 15.2if γi contains a 1-point, and Lemma 14.1 if γi is a 2-curve, for 2 ≤ i ≤ n, to achievethe conclusions of the theorem.

Theorem 15.4. Suppose that f : X → Y is τ -prepared. Then there exists acommutative diagram

X1f1→ Y1

Φ ↓ ↓ Ψ

Xf→ Y

such that Φ and Ψ are products of possible blow ups, and there exists a relation R1

for f1 such that f1 is τ -quasi-well prepared with relation R1.

Proof. We will give an inductive construction. The set GY (f, τ)−Θ(f, Y ) isfinite by Remark 11.12. Suppose that there exists a relation R on X such that Ω =U(R) ⊂ GY (f, τ)−Θ(f, Y ). (Initially, Ω = ∅). Let Λ(Y ) = GY (f, τ)−Θ(f, Y )−Ω.

Remark 11.4, Lemma 14.6 , Lemma 10.5 and Lemma 14.1 imply there exists apre-τ -quasi-well prepared diagram (for R)

X1f1→ Y1

Φ1 ↓ ↓ Ψ1

Xf→ Y

obtained by blowing up 2-curves such that if q1 ∈ Ψ−11 (Λ(Y )), then there exist alge-

braic permissible parameters u1, v1, w1 at q1 such that if p1 ∈ f−11 (q1)∩GX1(f1, τ),

then there exist good parameters u1, v1, w1−φp1(u1, v1) at p1 for f1. Further, if q1is a 2-point, then u1, v1 are toroidal forms at all p ∈ f−1

1 (q1), and if C is the 2-curvecontaining q1, then all points q′ ∈ C have permissible parameters uq′ , vq′ , wq′ at q′

such that uq′ , vq′ are toroidal forms at all p ∈ f−11 (q′). In addition, we have that

f1 is τ -prepared.We have Ψ−1

1 (Θ(f, Y )) ∩ GY1(f1, τ) ⊂ Θ(f1, Y1). Let R1 be the transform ofR on X1. We restrict R1 by removing from U(R1) the points of Θ(f1, Y1), sothat U(R1) ∩ Θ(f1, Y1) = ∅. Let Ω1 = U(R1), a finite set of points (by Remark11.12). We also have that a general point of each curve contained in GY1(f1, τ) isin Θ(f1, Y1), since f1 is τ -prepared. Let

Λ(Y1) = GY1(f1, τ)−Θ(f1, Y1)− Ω1 ⊂ Ψ−11 (Λ(Y )).

Λ(Y1) is a finite set of points. By Lemma 11.5, and our construction of f1, for eachq ∈ Λ(Y1), we can associate algebraic permissible parameters

(15.22) uq, vq, wq

at q, locally closed subsets A1, . . . , An(q) of f−11 (q) ∩GX1(f1, τ) such that

A1, . . . , An(q)


is a partition of f−11 (q) ∩GX1(f1, τ), and permissible parameters

uq, vq, wqi = wq − φi(uq, vq)

at q for 1 ≤ i ≤ n(q) such that for p1 ∈ Ai, wqi is good at p1 for f1. We can assumethat uq = vq = 0 are local equations of a general curve through q on DY if q is a1-point (by Bertini’s theorem and Remark 11.4).

Now fix q ∈ Λ(Y1). By Theorem 15.1, applied to uq, vq, wqi for 1 ≤ i ≤ n(q),there exists an affine neighborhood Vq of q and a commutative diagram

(15.23)X

f→ YΦ ↓ ↓ Ψ

f−11 (Vq) → Vq

satisfying the conclusions of Theorem 15.1 for all wqi. If q ∈ Ψ−1

(q), then thereexist u, v ∈ OY ,q such that u, v, wqi are permissible parameters at q, u, v, wqi aresuper parameters at q for all i, and by 3 of Lemma 14.1 and 5 of Lemma 14.5, for

p ∈ f−1(q) ∩GX(f, τ) ∩ Φ

−1(Ai),

wqi is good at p.Observe that we can modify the construction of (15.1) in Theorem 15.1, by a

diagram (15.2), by performing an arbitrary sequence of blow ups of 2-curves aboveeach Vi and Wi before constructing Vi+1 and Wi+1.

Suppose that our fixed q ∈ Λ(Y1) is 1-point. Recall that we have chosenuq, vq, wqi so that uq = vq = 0 are local equations of a general curve C on DY1

through q, so that C makes SNCs with DY1 , C intersects 2-curves of Y1 at generalpoints, and C − q intersects the nonfinite locus γ of f1 transversally at generalpoints of irreducible 1-dimensional components of γ. Thus we have GY1(f1, τ) ∩(C − q) ⊂ Θ(f1, Y1) and C intersects Θ(f1, Y1) transversally at general points ofcurves in GY1(f1, τ). For q ∈ GY1(f1, τ)∩(C−q), there exist algebraic permissibleparameters

uq, vq, wq

at the 1-point q such that uq = vq = 0 are local equations of C, and since qis perfect for f , there exist finitely many series φi(uq, vq) ∈ k[[uq, vq]] such thatuq, vq, wqi = wq − φi are super parameters at q for all i, and for p ∈ f−1

1 (q), somewqi is weakly good at p.

Now by Lemma 11.5, for q ∈ GY1(f1, τ) ∩ (C − q), there exist locally closedsubsets Vi ⊂ X1 for 1 ≤ i ≤ n(q) such that f−1

1 (q)∩GX1(f1, τ) is the disjoint unionof V1, . . . , Vn(q), and for p ∈ Vi, wqi is weakly good for f1 at p.

By Lemma 10.3 and Remark 14.2, there exists a sequence of blow ups of 2-curvesΦ : X → X1 with induced pre-τ -quasi-well prepared morphism f = f Φ : X → Y1

such that for all q ∈ GY1(f1, τ) ∩ (C − q), if p ∈ f−1(q), then for all i, uq, vq, wqiare super parameters at p and if τf (p) > 1, then wqi = 0 is a local equation ofa divisor supported on DX at p. Further, if p ∈ Φ−1(Vi) ∩ GX(f , τ), then wqi isweakly good for f at p. f is τ -prepared.

We define new primitive relations R1q,i for the finitely many

q ∈ GY1(f , τ) ∩ (C − q) ⊂ GY1(f1, τ) ∩ (C − q)

and 1 ≤ i ≤ n(q).


Let U(R1q,i) = q, T (R1

q,i) = Φ−1(Vi) ∩GX(f , τ).Suppose that τ > 1.If p ∈ Φ−1(Vi) is a 1-point, then we have an expression

uq = xa, vq = y, wqi = xcγ

where x, y, z are regular parameters in OX,p, γ ∈ OX,p is a unit series and a 6 | c.Let d = gcd(a, c) < a, e = a

d > 1 and c = cd .

We define R1q,i(p) = weqi − γ(0, 0, 0)eucq.

If p ∈ Φ−1(Vi) is a 2-point, then we have an expression

uq = (xayb)k, vq = z, wqi = (xayb)lγ

where x, y, z are regular parameters in OX,p, γ ∈ OX,p is a unit series and k 6 | l.Let d = gcd(k, l) < k, e = k

d > 1 and c = ld . We define R1

q,i(p) = weqi−γ(0, 0, 0)eucq.Suppose that τ = 1. We then define R1

q,i(p) by the relation wqi = 0 forp ∈ Φ−1(Vi) (by Remark 11.3).

We extend the transform R1 of R1 on X to include the primitive relations R1q,i

for q ∈ GY1(f , τ) ∩ (C − q) which we have just defined. Observe that we nowhave

(C − q) ∩GY1(f , τ) ⊂ U(R1).

By embedded resolution of plane curve singularities (c.f. Section 3.4 and Exer-cise 3.13 [C6]), and Lemmas 14.7 and 14.8, there exists a pre-τ -quasi-well prepareddiagram

Xf→ Y

Φ ↓ ↓ Ψ

Xf→ Y1

such that Ψ is a sequence of blow ups of prepared 1-points and 2-points (of types4 and 1 of Definition 13.7) such that if C is the strict transform of C on Y , thenC ∩ GY (f, τ) = q (since the germ of C at q ∈ GY1(f1, τ) ∩ (C − q) is notcontained in the surface germ wqi = 0 for any i). Further, f is τ -prepared. ThusC satisfies 1 and 2 of Definition 13.6 of a resolving curve. 4 of the definition holdssince f1 is τ -prepared and C intersects the nonfinite locus of f transversally atgeneral points.

By Lemma 10.7 and Lemma 14.1, there exists a pre-τ -quasi-well prepared dia-gram

X2f2→ Y2

Φ2 ↓ ↓ Ψ2

Xf→ Y

where Ψ2 and Φ2 are products of blow ups of 2-curves such that the strict transformC2 of C satisfies 3 of Definition 13.6. Observe that Ψ Ψ2 is an isomorphism overq, so that we may identify q with a point of Y2. X2 → X1 is a sequence of blow upsof 2-curves over f−1

1 (q). Thus C2 is a resolving curve for f2 at q. We further havethat f2 is τ -prepared.


Let Φ′ : X2 → X1 be our morphism Φ′ = Φ Φ Φ2. Let Ai = (Φ′)−1(Ai) ∩GX2

(f2, τ) for 1 ≤ i ≤ n(q). We have that uq, vq, wqi are permissible parameters atq such that wqi is good for f2 at all p ∈ Ai (by Remark 14.2).

After possibly replacing the neighborhood Vq of q in (15.23) with a smallerneighborhood of q, we may identify Vq with a neighborhood of q in Y2.

LetX2

f2→ Y2

Φ2 ↓ ↓ Ψ2

X2f2→ Y2

be the pre-τ -quasi-well prepared diagram of the conclusions of Lemma 14.5,where Ψ2 is the blow up of C2. f2 is τ -prepared.

As (f1 Φ′)−1(Vq) → f−11 (Vq) is a sequence of blow ups of 2-curves, as com-

mented after the construction of (15.23), we can assume that if we restrict thediagram

X2 → Y2

Φ′′ ↓ ↓ Ψ′′

X1 → Y1

that we have constructed to Vq, we obtain the diagram

W1 → V1

Φ1 ↓ ↓ Ψ1

f−11 (Vq) = W → V = Vq

of (15.2) constructed in the proof of Theorem 15.1.Now suppose that our fixed q ∈ Λ(Y1) is a 2-point. Then, by Lemma 14.1 we

construct a τ -quasi-well prepared diagram for R1 and the blow up Ψ′′ of C

X2f2→ Y2

Φ′′ ↓ ↓ Ψ′′

X1 → Y1

where Φ′′ is a sequence of blow ups of 2-curves. f2 is τ -prepared.We can assume that if we restrict the diagram

X2 → Y2

↓ ↓X1 → Y1

to Vq, we obtain the diagram

W1 → V1

Φ1 ↓ ↓ Ψ1

f−11 (Vq) = W → V = Vq

of (15.2) constructed in the proof of Theorem 15.1.Let R2 be our relation on X2. We have (by Remark 14.2)

(Ψ′′)−1(Θ(f1, Y1)− C) ⊂ Θ(f2, Y2).

We restrict R2 if necessary, so that U(R2) ∩Θ(f2, Y2) = ∅. Let Ω2 = U(R2)With the notation of the proof of Theorem 15.1, let C2 be the Zariski closure

of γ1, the curve blown up in V2 → V1, in Y2. C2 is a section over C. Either C2 is a2-curve or C2 contains a 1-point.


By our assumptions on C, at all points q′ ∈ C, there exist permissible param-eters uq′ , vq′ , wq′ at q′ such that uq′ = vq′ = 0 are local equations of C and uq′ , vq′are toroidal forms at q′. Thus if q1 ∈ (Ψ′′)−1(q′) ∩C2, then there exist permissibleparameters u1, v1, w such that u1 = v1 = 0 are local equations of C2 at q1, andu1, v1 are toroidal forms at all p ∈ f−1

2 (q1). In particular, if q1 is a 1-point, thenC2 satisfies 2 and 4 of Definition 13.6 of a resolving curve for f2 at q1.

Suppose that C2 contains a 1-point. Then q1 = (Ψ′′)−1(q) ∩ C2 is a 1-point.We can apply Lemma 14.10 to construct a pre-τ -quasi-well prepared diagram

(15.24)X3

f3→ Y3

↓ ↓X2

f2→ Y2

satisfying the conclusions of Lemma 14.10, so that the strict transform of C2 is aresolving curve for f3 at q1. We have that f3 is τ -prepared.

The vertical arrows of (15.24) are products of blow ups of 2-curves above Vq.Let

X4f4→ Y4

Φ4 ↓ ↓ Ψ4

X3f3→ Y3

be the pre-τ -quasi-well prepared diagram of the conclusions of Lemma 14.5, whereΨ4 is the blow up of the strict transform of C2. f4 is τ -prepared.

Suppose that C2 is a 2-curve. Then we construct (from Lemma 14.1)

(15.25)X4 → Y4

↓ ↓X2 → Y2

as a τ -quasi-well prepared diagram for R2 and the blow up of C2.As commented after the construction of (15.23), we may assume that the dia-

gram (15.25) restricts to the diagram

W2 → V2

↓ ↓W1 → V1

of (15.2) above Vq.Since (15.2) is finite, after finitely many iterations, we achieve a pre-τ -quasi-well

prepared diagram

X5f5→ Y5

Φ5 ↓ ↓ Ψ5

X1 → Y1

which restricts to the diagram (15.23) above Vq, Ψ5 is an isomorphism over Λ(Y1)−q and Φ5 is a sequence of blow ups of 2-curves over Λ(Y1) − q. Further, f5 isτ -prepared.

By Lemma 10.3, and Remark 14.2, 3 of Lemma 14.1 and 5 of Lemma 14.5, thereexists a sequence of blow ups of 2-curves Φ6 : X6 → X5 such that f6 : X6 → Y5

is pre-τ -quasi-well prepared, τ -prepared, and (with the notation introduced with(15.23)), there exist algebraic permissible parameters uq, vq, wq ∈ OY1,q and wqi =


wq − φi(uq, vq) such that for q in the finite set

Σ = (Ψ5 Ψ6)−1(q) ∩GY5(f6, τ)−Θ(f6, Y5),

we have algebraic permissible parameters uq, vq, wq at q, and series

φi(uq, vq) = φi(uq, vq)

such that uq, vq, wqi = wq − φi(uq, vq) are super parameters for all i, and for

p ∈ Vqi = f−16 (q) ∩ (Φ5 Φ6)−1(Ai) ∩GX6(f6, τ),

wqi = wq − φi(uq, vq) is good for f6 at p, and wqi = 0 is supported on DX6 at p ifτ > 1.

We define new primitive relations R6q,i for q ∈ Σ and 1 ≤ i ≤ n(q).

Let U(R6q,i) = q, T (R6

q,i) = Vqi.Suppose that τ > 1.If p ∈ Vqi is a 1-point, then we have an expression

uq = xa, vq = y, wqi = xcγ

where x, y, z are regular parameters in OX6,p, γ ∈ OX6,p is a unit series and a 6 | c.Let

a′ =a

gcd(a, c)> 1, c′ =

c

gcd(a, c).

We define R6q,i(p) = wa

′

qi − γ(0, 0, 0)a′uc

′

q .If p ∈ Vi is a 2-point, then we have an expression

uq = (xayb)k, vq = z, wqi = (xayb)lγ

where x, y, z are regular parameters in OX6,p, γ ∈ OX6,p is a unit series and k 6 | l.Let

k′ =k

gcd(k, l)> 1, l′ =

l

gcd(k, l).

We define R6q,i(p) = wk

′

qi − γ(0, 0, 0)k′ul

′

q .Suppose that τ = 1. We then define R6

q,i(p) = wqi. By Remark 11.3, wqi is amonomial form at p.

We can extend the transform R6 of R5 on X6 to include these new primitiverelations R6

q,i.We now restrict R6 to remove the points of Θ(f6, Y5) from U(R6). Let

Λ(Y6) = GY5(f6, τ)−Θ(f6, Y5)− U(R6).

By our construction, Λ(Y6) ⊂ Ψ−15 (Λ(Y1)−q), and over Λ(Y1)−q, the vertical

arrows of the pre-τ -quasi-well prepared diagram

X6f6→ Y5

Φ5 Φ6 ↓ ↓ Ψ5

X1f1→ Y1

are blow ups of 2-curves.By induction on |Λ(Y1)|, we may iterate the above procedure to construct a

commutative diagram

X7f7→ Y7

↓ ↓X → Y


such that f7 is pre-τ -quasi-well prepared with U(R7) = GY7(f7, τ)−Θ(f7, Y7), andf7 is τ -prepared.

Now by Theorem 15.3 there exists a pre-τ -quasi-well diagram for R7

X8f8→ Y8

↓ ↓X7 → Y7

such that f8 is τ -quasi-well prepared for the transform of R7.

Lemma 15.5. Suppose that τ ≥ 1, f : X → Y is τ -quasi-well prepared withrelation R. Further suppose there exists a τ -quasi-well prepared diagram for R

(15.26)X

f→ Y

Φ ↓ ↓ Ψ

Xf→ Y,

where R is the transform of R on X, such that if q1 ∈ U(R) is on a component Eof DY such that Ψ(E) is not a point, then T (R) ∩ f−1(q1) = ∅. Then there existsa commutative diagram

X1f1→ Y1

Φ ↓ ↓ Ψ

Xf→ Y

such that Φ, Ψ are products of blow ups of possible centers, and f1 is τ -quasi-wellprepared with relation R1 and pre-algebraic structure. Further, R1 is algebraic. (R1

will in general not be the transform of R.)

Proof. Given a diagram (15.26), we will define a new relation R′ on X for f .This is accomplished as follows. We have that |U(R) − Θ(f , Y )| < ∞ by Remark11.12. Suppose that q1 ∈ U(R)−Θ(f , Y ) is such that f−1(q1) ∩ T (R) 6= ∅. Let

Jq1 = i | T (Ri) ∩ f−1(q1) 6= ∅.

Let q = Ψ(q1). For j ∈ Jq1 , let

u = uRj(q), v = vRj(q)

, wj = wRj(q).

Letu1 = u

Rj(q1), v1 = v

Rj(q1), wj,1 = w

Rj(q1).

Since Ψ is a composition of admissible blow ups for the transforms of the pre-relations Ri on Y , by the description of admissible blow ups (12.7) - (12.9) followingDefinition 12.2, we have in OY1,q, one of the following relations: q1 a 2-point

(15.27) u = ua1vb1, v = uc1v

d1 , wj = ue1v

f1wj,1

with ad− bc = ±1, or q1 a 1-point,

(15.28) u = u1, v = uc1v1, wj = ue1wj,1

or q1 a 1-point

(15.29) u = ua1γ1(u1, v1), v = ub1γ2(u1, v1), wj = uc1γ3(u1, v1)wj,1


where γ1, γ2, γ3 are unit series and γ1, γ2, γ3 ∈ OY1,q or q1 a 2-point

(15.30) u = (ua1vb1)tγ1(u1, v1), v1 = (uc1v

d1)kγ2(u1, v1), wj = ue1v

f1 γ3(u1, v1)wj,1

where γ1, γ2, γ3 are unit series, and γ1, γ2, γ3 ∈ OY1,q, a, b > 0. Since (15.26) isa τ -quasi-well prepared diagram, the exponents appearing in (15.27) - (15.30) areindependent of j ∈ Jq1 .

By assumption, if q1 is on a component E of DY we must have Ψ(E) is a point.Thus in (15.27) we have a, b, c, d, e, f all nonzero. If (15.28) holds, we have c, e > 0.In (15.29) we have a, b, c > 0. In (15.30) we have t, k, e, f > 0.

Suppose that p1 ∈ T (Rj)∩ f−1(q1) and τ > 1. Let p = Φ(p1) ∈ T (Rj). On X1,wj,1 = 0 is a divisor supported on DX1 at p1. In (15.27), (15.29), (15.30) we seethat u = 0 is a local equation of DX1 at p1, and v = 0 is a local equation of DX1

at p1. In (15.28), we have that u = 0 is a local equation of DX1 at p1, wj = 0 is alocal equation of DX1 at p1, and v = 0 is a local equation of a divisor that containsDX1 at p1. Thus in all cases, there exists a natural number r such that wj dividesur and vr in OX,p1 .

Define

η = η(q1) =

max2r2, e, f if (15.27) or (15.30) holdsmax2r2, c if (15.29) holdsmax2r2, e if (15.28) holds

where the maximum is over j ∈ Jq1 and p1 ∈ T (Rj) ∩ f−1(q1).Fix j ∈ Jq1 . There exists σ(u, v, wj) ∈ k[[u, v, wj ]] = OY,q such that the order

of the series σ is greater than η and wj + σ ∈ OY,q. Let

w∗j = wj + σ(u, v, wj).

For p1 ∈ T (Rj) ∩ f−1(q1), we have

w∗j = wjγp1j

where γp1j ∈ OX1,p1 is a unit series. If (15.27) holds at q1, set

(15.31) wq1,j =w∗j

ue1vf1

= wj,1 +σ(ua1v

b1, u

c1vd1 , u

e1vf1wj,1)

ue1vf1

∈ OY ,q1 ∩ k(Y ) = OY ,q1 ,

since u1, v1 ∈ OY ,q1 (this is part of the definition of a relation).We further have

wq1,j = wj,1γp1,j .

There is a similar argument if (15.28), (15.29) or (15.30) holds.For k ∈ Jq1 , there exists λjk(u, v) ∈ k[[u, v]] such that wk = wj + λjk(u, v).

Writeλjk(u, v) = αk(u, v) + hk(u, v)

where αk(u, v) is a polynomial, and hk(u, v) is a series of order greater than η. Set

w∗k = wj + σ(u, v, wj) + λjk(u, v)− hk(u, v) ∈ OY,q.

w∗k = wk + σ(u, v, wk − λjk(u, v))− hk(u, v)= wk + σk(u, v, wk)

where σk is a series of order greater than η.


Suppose that (15.27) holds at q1. Set

wq1,k =w∗k

ue1vf1

.

From (15.27) we see that u1, v1, wq1,k are permissible parameters at q1 and wq1,k ∈OY ,q1 . We further have that

(15.32) wq1,k = wk,1γp1k

for some unit series γp1k ∈ OX1,p1.

We have that for k ∈ Jq1 ,

(15.33) wq1,k = wk,1 +σk(ua1v

b1, u

c1vd1 , u

e1vf1wk,1)

ue1vf1

withσk

ue1vf1

∈ OY ,q1 .

We further have

wq1,k − wq1,j = w∗k−w∗j

ue1v

f1

= λjk(u,v)−hk(u,v)

ue1v

f1

∈ k((u1, v1)) ∩ k[[u1, v1, wj,1]] = k[[u1, v1]].

There is a similar argument if (15.28), (15.29) or (15.30) holds. In these cases(15.31) becomes

wq1j =w∗jue1, wq1j =

w∗juc1, wq1j =

w∗j

ue1vf1

respectively.In all these cases, an equation (15.32) holds.In case (15.28), a variant of equation (15.33) holds.Suppose that (15.30) holds and k ∈ Jq1 . Then there exist series σk(u, v, wk)

such that ord σk > η and such that

w∗k = wk + σk(u, v, wk).

Thus(15.34)

wq1,k = γ3(u1, v1)wk,1+σk((ua1v

b1)tγ1(u1, v1), (uc1v

d1)kγ2(u1, v1), ue1v

f1 γ3(u1, v1)wj,1)

ue1vf1

withσk

ue1vf1

∈ OY ,q1 .

There is a similar expression if (15.29) holds.Now suppose that τ = 1. We make the same argument as the τ > 1 case if

wj,1 satisfies a form 3 or 4 of Definition 4.4 or a form 1, 2, 5 with β 6= 0. A similar,but slightly different argument is required if wj1 satisfies 1, 2 or 5 of Definition 4.4,with β = 0.

We now define the new relation R′ on X for f . Set T (R′) = T (R)−f−1(Θ(f , Y )),U(R′) = f(T (R))−Θ(f , Y ). U(R′) is a finite set by Remark 11.12. For q1 ∈ U(R′),


we define primitive relations Rq1,k as follows. Set U(Rq1,k) = q1, T (Rq1,k) =T (Rk) ∩ f−1(q1).

For p1 ∈ T (Rq1,k) define

uRq1,k(p1) = uRk(p1), vRq1,k(p1) = vRk(p1)

, wRq1,k(p1) = wq1,k.

If τ > 1 and q1 is a 2-point (from 15.32), we define Rq1,k(p1) by

aRq1,k(p1) = aRk

(p1), bRq1,k(p1) = bRk

(p1), eRq1,k(p1) = eRk

(p1)

andλRq1,k

(p1) = λRk(p1)γp1k(0, 0, 0)eRk

(p1).

If τ = 1, and q1 is a 2-point, we define

aRq1,k(p1) = bRq1,k

(p1) =∞.

If q1 is a 1-point, we define Rq1,k(p1) in an analogous way.From the above calculations, we see that f : X → Y with the relation R′

satisfies 1 - 4 of the conditions of Definition 13.1 of a pre-τ -quasi-well preparedmorphism.

Recall that all exponents are positive in (15.27) - (15.30), and thus in (15.33)and (15.34). Thus we can choose a possibly larger η(q1) so that ord(σk) is suffi-ciently large in (15.33) and (15.34) that R′ satisfies 5 of the conditions of Definition13.1 (as well as 1 – 4). Thus f is pre-τ -quasi-well prepared with respect to R′.Further, f is τ -prepared.

For qi ∈ U(Rqi,k), let Ω(Rqi,k) be an affine neighborhood of qi on the surfacewith local equation wqi,k = 0 at qi, such that

1. Ω(Rqi,k) is nonsingular and makes SNCs with DY .2. Ω(Rqi,k) ∩ U(R′) = qi

We now restrict R′ so that U(R′) = GY (f , τ) − Θ(f , Y ), f is pre-τ -quasi-wellprepared with relation R′ and R′ is algebraic.

By Theorem 15.3 there exists a pre-τ -quasi-well prepared diagram for R′

X ′ f ′→ Y ′

Φ′ ↓ ↓ Ψ′

X → Y

such that f ′ is τ -quasi-well prepared for the transformR′ of R′. By our construction,R′ has pre-algebraic structure. Further, R′ is algebraic. Thus the conclusions ofLemma 15.5 hold.

Remark 15.6. Suppose that f : X → Y is τ -quasi-well prepared and q ∈ U(Ri)is a 1-point. Let


, wi = wRi(q).

Then f is not toroidal above q if and only if q is contained in the nonfinite locus off .

Suppose that f is not toroidal above q. Then the germ of the nonfinite locusof f at q is a nonsingular (algebraic) curve, and u = wi = 0 are (formal) localequations of the nonfinite locus of f at q.


Proof. Suppose that f is not toroidal above q. u, v, wi are super parametersat p for all p ∈ f−1(q). From consideration of the local forms 5 and 6 of Definition10.1 of super parameters at a 1-point q, we see that u = wi = 0 are local equationsof (a formal branch of) the fundamental locus of f at q. Since the fundamentallocus of f is algebraic, we obtain the conclusions of the remark.

Theorem 15.7. Suppose that τ ≥ 1, f : X → Y is τ -quasi-well prepared withrelation R. Then there exists a τ -quasi-well prepared diagram

Xf→ Y

Φ ↓ ↓ Ψ

Xf→ Y

where R is the transform of R on X such that if q1 ∈ U(R) is on a component Eof DY such that Ψ(E) is not a point, then T (R) ∩ f−1(q1) = ∅.

Proof. Step 1. Let A0 be the set of 2-points q ∈ Y such that q ∈ U(Ri) for someRi associated to R and f−1(q)∩T (Ri) 6= ∅. A0 is a finite set since f is τ -prepared.

For q ∈ A0 ∩ U(Ri), set

(15.35) u = uRi(q), v = vRi(q)

, wi = wRi(q).

Let Ψ1 : Y1 → Y be the blowup of all q ∈ A0, and let

X1f1→ Y1

Φ1 ↓ ↓ Ψ1

Xf→ Y

be a τ -quasi-well prepared diagram of R and Ψ1. Such a diagram exits by Lemma14.7. Suppose that q ∈ A0 ∩ U(Ri) and q1 ∈ Ψ−1

1 (q) ∩ U(R1

i ) is a 1-point lying ona component E of DY1 and f−1

1 (q1) ∩ T (R1

i ) 6= ∅. Then q1 has regular parametersu1, v1, wi,1 with

u = u1, v = u1(v1 + α), wi = u1wi,1

with 0 6= α ∈ k, which implies that E has local equation u1 = 0, so that Ψ1(E) isa point.

Let A1 be the set of all 2-points q1 ∈ Y1 such that for some i, q1 ∈ U(R1

i ),f−11 (q1) ∩ T (R1

i ) 6= ∅ and q1 is on a component E of DY1 such that Ψ1(E) is nota point. We have A1 ⊂ Ψ−1

1 (A0). Let Ψ2 : Y2 → Y1 be the blowup of all q1 ∈ A1,and let (by Lemma 14.7)

X2f2→ Y2

Φ2 ↓ ↓ Ψ2

X1f1→ Y1

be a τ -quasi-well prepared diagram of R1 and Ψ2. Continue in this way to construct(for arbitrary n) a sequence of n blow ups of sets of 2-points Ψk+1 : Yk+1 → Yk for0 ≤ k ≤ n− 1 with τ -quasi-well prepared diagrams

Xk+1fk+1→ Yk+1

Φk+1 ↓ ↓ Ψk+1

Xkfk→ Yk

of Rk and Ψk+1. We have a resulting τ -quasi-well prepared diagram of R


(15.36)Xn

fn→ YnΦ ↓ ↓ Ψ

Xf→ Y.

Suppose that qn ∈ Yn is a 2-point such that qn is on a component E of DYn suchthat Ψ(E) is not a point, and qn ∈ U(R

n

i ), f−1n (qn) ∩ T (Rni ) 6= ∅ for some i. We

have permissible parameters

(15.37) u1 = uRni (qn), v1 = vRn

i (qn), wi,1 = wRni (qn)

at qn such that for Ψ(qn) = q and with notation of (15.35),

(15.38)u = u1

v = un1 v1wi = un1wi,1

or

u = u1vn1

v = v1wi = vn1wi,1.

Suppose that p ∈ T (Ri)∩ f−1(q) is a 1-point. First suppose that τ > 1. Thereare permissible parameters x, y, z at p, 0 6= α ∈ k, and a unit γ ∈ OX,p, such thatwe have

u = xa, v = xb(α+ y), wi = xcγ.

From the construction of (15.36) and the algorithm of Lemma 14.7, we havethat Φ is an isomorphism at points of Φ−1(p) ∩ f−1

n (qn). Thus for n ≥ maxab ,ba,

f−1n (qn) ∩ Φ−1(p) = ∅.

Suppose that τ = 1 (and p ∈ T (Ri) ∩ f−1(q) is a 1-point). Then there existpermissible parameters x, y, z at p, β ∈ k, c > 0 and 0 6= α ∈ k such that

u = xa, v = xb(α+ y), w = xc(β + z).

If β 6= 0, then Φ is an isomorphism at points of Φ−1(p) ∩ f−1n (qn). For n ≥

maxab ,ba, we have f−1

n (qn) ∩ Φ−1(p) = ∅.If β = 0, then Φ is a product of blow ups of sections over the curve x = z = 0 at

points of f−1n (qn), and in this case also, Φ−1(p) ∩ f−1

n (qn) = ∅ for n ≥ maxab ,ba.

Suppose that p ∈ T (Ri)∩ f−1(q) is a 2-point. First suppose that τ > 1. Thereare permissible parameters x, y, z at p and 0 6= α ∈ k such that we have one of theforms

(15.39) u = (xayb)k, v = (xayb)t(α+ z), wi = (xayb)lγ,

where 0 6= α ∈ k, gcd(a, b) = 1, γ ∈ OX,p is a unit,or we have

(15.40) u = xayb, v = xcyd, wi = xeyfγ

where ad− bc 6= 0, γ ∈ OX,p is a unit.Suppose that (15.39) holds. From the construction of (15.36) and the algorithm

of Lemma 14.7, we have that Φ is a sequence of blow ups of 2-curves at points ofΦ−1(p) ∩ f−1

n (qn), so that for n ≥ maxkt ,tk, f

−1n (qn) ∩ Φ−1(p) = ∅,


Suppose that (15.40) holds. If f−1n (qn) ∩Φ−1(p) ∩ T (Rni ) 6= ∅ for all n, we will

show that, after possibly interchanging u, v and x, y, (15.40) must be

(15.41) u = xa, v = xcyd, wi = xeyfγ,

with d, f > 0, and (15.38) holds.We see this as follows.Suppose that pn ∈ f−1

n (qn) ∩ T (Rni ) and Φ(pn) = p.Let ν be a valuation of k(X) whose center on Xn is pn. We identify ν with

an extension of ν to the quotient field of OXn,pnwhich dominates OXn,pn

. qn haspermissible parameters (15.37). After possibly interchanging u and v, we have arelation (15.38), so that ν(v) > nν(u), and ν(wi) > nν(u). We can reindex x, y, zso that 0 < ν(x) ≤ ν(y). Then

(c+ d− nb)ν(y) ≥ (d− nb)ν(y) + (c− na)ν(x) > 0,

and

(15.42) (e+ f − nb)ν(y) ≥ (f − nb)ν(y) + (e− na)ν(x) > 0.

Thus if b 6= 0, and n > c+ d, we have a contradiction.Taking n > c + d for all c, d in local forms (15.40) for 2-points p ∈ T (R),

we achieve that b = 0 in all local forms (15.40) which are the images of 2-pointspn ∈ T (Rn) which map to a point qn of Yn which is on a component E of DYn

suchthat Ψ(E) is not a point. d > 0 since ad− bc 6= 0.

We have f > 0 if n >> 0. In fact, if f = 0 in (15.40), we then have e > 0, andfor n > a

e , we have a contradiction to (15.42).Suppose that τ = 1 (and p ∈ T (Ri) ∩ f−1(q) is a 2-point). Then there exist

permissible parameters x, y, z at p and 0 6= α ∈ k such that we have one of theforms:

(15.43) u = (xayb)k, v = (xayb)t(α+ z), wi = xcyd

with 0 6= α ∈ k, ad− bc 6= 0, gcd(a, b) = 1, or we have

(15.44) u = xayb, v = xcyd, wi = xeyf (β + z)

with ad− bc 6= 0, β ∈ k.Suppose that (15.43) holds. From the construction of (15.36), we have that

Φ is a sequence of blow ups of 2-curves at points of Φ−1(p) ∩ f−1n (qn), so that for

n ≥ maxkt ,tk, f

−1n (qn) ∩ Φ−1(p) = ∅.

Suppose that (15.44) holds. If β 6= 0, the analysis of (15.40) shows that iff−1n (qn) ∩ Φ−1(p) ∩ T (Rni ) 6= ∅ for all n, then after possibly interchanging u, v and

interchanging x, y (15.44) has the form (15.41), with γ = β+z, d, f > 0 and (15.38)holds.

Suppose that β = 0 (in (15.44)), and f−1n (qn) ∩ Φ−1(p) ∩ T (Rni ) 6= ∅ for all n.

After possibly interchanging u and v, we may assume that (15.38) holds. Let ν bea valuation of k(X) whose center on Xn is pn. We identify ν with an extension ofν to the quotient field of OXn,pn which dominates OXn,pn . Then we see that

(15.45) ν(v) > nν(u) and ν(wi) > nν(u)

for all n.


Thus for n sufficiently large, after possibly interchanging x and y, (15.44) mustbe

(15.46) u = xa, v = xcyd, wi = xeyfz

with d > 0.We will now show that we can take n sufficiently large that f > 0 in (15.46).Suppose that f = 0 in (15.46). Then we have

(15.47) ν(y) > nν(x) and ν(z) > nν(x)

for all n.In the algorithm of Lemma 14.7, we see that Φ1 : X1 → X can be factored by

morphismsX1 = Zm → · · · → Z2 → Z1 → X

where Z1 → X is a sequence of blow ups of 2-curves and 3-points, and each Zi+1 →Zi is the blow up of a possible curve containing a 1-point, which is in the locuswhere IqOZi is not invertible.

By (15.47), we see that there exist permissible parameters x1, y1, z1 at thecenter p1 of ν on Z1 such that

x = x1, y = xg1y1, z = z1.

We haveu = xa1 , v = xc+gd1 yd1 , wi = xez1

with a ≤ c+ gd, since from the construction of Z1 → X1, we have that (u, v)OZ1,p1

is invertible.If e ≥ a, then IqOZ1,p1 is invertible, and X1 → Z1 is an isomorphism above p1.

Then at p1 ∈ X1, there are permissible parameters x1, y1, z1 such that

uR

1i (q1)

= xa1 , vR1i (q1)

= xc+gd−a1 yd1 , wR1i (q1)

= xe−a1 z1.

If e− a = 0, we have that f1 is toroidal at p1 (so that τf1(p1) = −∞).If e < a, we have that X1 → Z1 is not an isomorphism above p1. Without loss

of generality, we may assume that each Zi+1 → Zi is not an isomorphism at thecenter of ν.

We have that x1 = z1 = 0 are (formal) local equations at p1 of the curve blownup in Z2 → Z1.

Let p2 be the center of ν on Z2. By (15.47), we have that there are permissibleparameters x2, y2, z2 at p2 such that x1 = x2, y1 = y2, z1 = x2z2.

We haveu = xa2 , v = xc+gd2 yd2 , wi = xe+1

2 z2.

We see that in X1 = Zm, there are regular parameters x1, y1, z1 such that

uR

1i (q1)

= xa1 , vR1i (q1)

= xc+gd−a1 yd1 , wR1i (q1)

= z2.

thus f1 is toroidal at p1.Iterating this analysis for the morphisms Ψ2, . . . ,Ψn, we see that for n >> 0,

fn is toroidal at pn. In fact, if we take n ≥ ea in (15.36), we see that f−1

n (qn) ∩T (Rni ) ∩ Φ−1(p) = ∅.

We can thus take n sufficiently large so that f > 0 in all local forms (15.46),which are the images of 2-points pn ∈ T (Rn) which map to a point qn of Yn whichis on a component E of DYn such that Ψ(E) is not a point.


Suppose that p ∈ X is a 3-point such that p ∈ T (Ri) ∩ f−1(q). Then there arepermissible parameters x, y, z for u, v, wi at p such that

(15.48)u = xaybzc

v = xdyezf

wi = xgyhziγ

where γ is a unit series.We will show that we can choose n sufficiently large in the diagram (15.36),

so that if pn ∈ Xn is such that pn ∈ Φ−1(p) ∩ T (Rni ) and qn = fn(pn) is on acomponent E of DYn such that Ψ(E) is not a point, then (15.48) must have one ofthe following forms (after possibly interchanging u, v and x, y, z):

(15.49)u = xayb

v = xdyezf

wi = xgyhziγ

where b 6= 0, f 6= 0, i 6= 0 and (15.38) holds, or

(15.50)u = xa

v = xdyezf

wi = xgyhziγ

with e and f 6= 0, h or i 6= 0 and (15.38) holds.We will now prove this statement.Let ν be any valuation of k(X) which has center pn on Xn. We identify ν with

an extension of ν to the quotient field of OXn,pn which dominates OXn,pn .qn has permissible parameters (15.37).After possibly interchanging u and v, we have a relation (15.38), so that ν(v) >

nν(u). We can reindex x, y, z so that

0 < ν(x) ≤ ν(y) ≤ ν(z).

Then

(f + e+ d− nc)ν(z) ≥ (f − nc)ν(z) + (e− nb)ν(y) + (d− na)ν(x) > 0.

If c 6= 0, and n > f + e + d, we have a contradiction. Thus taking n > f + e + dfor all d, e, f in local forms (15.48) for 3-points p ∈ T (R), we achieve that c = 0 inall local forms (15.48) which are the images of 3-points pn ∈ T (Rn) which map toa point qn of Yn which is on a component E of DYn

such that Ψ(E) is not a point.If i = 0 (and c = 0) in (15.48) we have

(h+ g − nb)ν(y) ≥ (h− nb)ν(y) + (g − na)ν(x) > 0

so that if b 6= 0 and n > h+ g we have a contradiction. Thus, by taking n 0in (15.36), we see that if b 6= 0, then a form (15.49) must hold at p (since uv = 0 isa local equation of DX at p implies f 6= 0). If b = c = 0 in (15.48), then a similarcalculation shows that a form (15.50) must hold at p (for n 0).

We observe that in (15.49) we have

(15.51) (z) ∩ OY,q = (v, wi).

Suppose that (15.50) holds. If i 6= 0 then

(15.52) (z) ∩ OY,q = (v, wi).


If h 6= 0, then

(15.53) (y) ∩ OY,q = (v, wi).

Suppose that (15.41) or (15.46) hold. Then

(15.54) (y) ∩ OY,q = (v, wi).

We will show that in (15.49), v = wi = 0 is a formal branch of an algebraiccurve C in the nonfinite locus of f : X → Y . Let R = OY,q, S = OX,p.

Since p is a 3-point, there exist regular parameters x, y, z in OX,p and unitsλ1, λ2, λ3 ∈ OX,p such that x = xλ1, y = yλ2, z = zλ3.

z = 0 is a local equation for a component of DX . We have that v ∈ (z) ∩ Rand u 6∈ (z)∩R so that (z)∩R = (v) or (z)∩R = a where a is a height two primecontaining v. We have (zS) ∩ R = (v, wi). Suppose that (z) ∩ R = (v). We thenhave an induced morphism

R/(v)→ S/(z)which is an inclusion by the Zariski Subspace Theorem (Theorem 10.14 [Ab3]).This is impossible, so that a is a height 2 prime in R, and defines a curve C, whichis necessarily in the nonfinite locus of f since z = 0 is a local equation at p of acomponent of DX which dominates C. A similar argument shows that in (15.50),(15.41) and (15.46), v = wi = 0 is a formal branch of an algebraic curve C in thenonfinite locus of f .

Step 2. Let C be the reduced curve in Y whose components are the curves inthe nonfinite locus of f which are not 2-curves. Let C be the reduced curve in Ynwhich is the strict transform of C. The components of C are then in the nonfinitelocus of fn. By Theorem 15.2, we can perform a sequence of blow ups of prepared1-points, prepared 2-points, 2-curves and resolving curves Ψ′ : Y ′ → Yn so that wecan construct a τ -quasi-well prepared diagram of Ψ′ and Rn

(15.55)X ′ f ′→ Y ′

Φ′ ↓ ↓ Ψ′

Xn → Yn

where R′ is the transform of Rn on X ′, such that the strict transform C of C onY ′ is nonsingular, and makes SNCs with DY ′ . If q′ ∈ U(R′i) ∩ C for some i thenthe germ at q′ of C is contained in SR′

i(q′), and the (disjoint) components of C are

permissible centers for R′.Let Ψ(1) : Y (1) → Y ′ be the blow up of C. By Theorem 15.2, we have a

τ -quasi-well prepared diagram of Ψ(1) and R′

(15.56)X(1)

f(1)→ Y (1)Φ(1) ↓ ↓ Ψ(1)

X ′ f ′→ Y ′.

Let R(1) be the transform of R′ on X(1).Suppose that q ∈ U(R) ⊂ Y is a 2-point. Suppose that q ∈ (Ψ Ψ′)−1(q) and

(f ′)−1(q) ∩ T (R′i) 6= ∅. Then q ∈ A0.Let

u = uR′i(q)

, v = vR′i(q)

, wi = wR′i(q)

,


u = uR(q), v = vR(q), wi = wRi(q).

We have (since q ∈ A0 and ΨΨ′ is a nontrivial sequence of blow ups of points,followed by a sequence of blow ups of points and 2-curves at q) one of the followingforms (by (12.7) - (12.9):

(15.57) u = uavb, v = ucvd, wi = ulvmwi

with ad− bc = ±1 and l > 0 if c > 0, m > 0 if b > 0, or

(15.58) u = (uavb)tγ1(u, v), v = (uavb)kγ2(u, v), wi = uevf wiγ3(u, v)

where γ1, γ2, γ3 are unit series, a, b, t, k, e, f > 0, gcd(a, b) = 1, or

(15.59) u = uaγ1(u, v), v = ubγ2(u, v), wi = ucwiγ3(u, v)

where γ1, γ2, γ3 are unit series, a, b, c > 0.We have that (ΨΨ′)(E) = q for all components E of DY ′ containing q (which

implies q 6∈ C) unless q is a 2-point, and we have an expression

(15.60)u = uv = ucvwi = ulvmwi

with c, l > 0 and v = 0 is a local equation of E, or

u = uvb

v = vwi = ulvmwi

with b,m > 0 and u = 0 is a local equation of E.Let q∗ = Ψ′(q). q∗ is a 2-point and f−1(q∗) ∩ T (R′i) 6= ∅. After possibly

interchanging u and v we have that a form (15.38) holds at qn and q, and thus by(15.51) - (15.54) that v = wi = 0 are local equations of a formal component of Cat q. (15.60) thus holds at q, and since Ψ′ is a sequence of blow ups of points and2-curves at q, m = 0 in (15.60). We thus have an expression

(15.61) u = u, v = uev, wi = uf wi

for some e, f > 0. v = 0 is a local equation of the strict transform of DY at q, andv = wi = 0 are local equations of C at q since C is nonsingular.

Suppose that q ∈ (Ψ Ψ′ Ψ(1))−1(q) and f(1)−1(q) ∩ T (Ri(1)) 6= ∅. Letq = Ψ(1)(q). then (f ′)−1(q) ∩ T (R′i) 6= ∅. If q 6∈ C (so that q = q) then we haveseen that (Ψ Ψ′)(E) = q for all components E of DY ′ containing q. Suppose thatq ∈ C. Then an expression (15.61) holds at q.

Ψ(1) is the blow up of v = wi = 0 above q. Since q ∈ U(Ri(1)), we must have

u = uRi(1)(q), v = vRi(1)(q)

, wi = vRi(1)(q)wRi(1)(q)

.

Substituting into (15.61), we have

(15.62) u = uRi(1)(q), v = ue

Ri(1)(q)vRi(1)(q)

, wi = ufRi(1)(q)

vRi(1)(q)wRi(1)(q)

with e, f ≥ 1. We have that q is a 2-point and q is a 2-point.


Suppose that q ∈ U(R) ⊂ Y is a 1-point. Then Ψ is an isomorphism over q.Suppose that q ∈ (Ψ Ψ′)−1(q) and (f ′)−1(q) ∩ T (R′i) 6= ∅. Let

u = uR′i(q)

, v = vR′i(q)

, wi = wR′i(q)

,


, wi = wRi(q).

We have that u = wi = 0 are local equations of C since T (Ri) ∩ f−1(q) 6= ∅, byRemark 15.6. Then Ψ′ is either an isomorphism at q, or factors at q as the blowup of q, followed by a sequence of blow ups of 2-points and 2-curves.

First suppose that Ψ′ is not an isomorphism at q. We have one of the followingforms (by (12.7) - (12.9):

(15.63) u = uavb, v = ucvd, wi = ulvmwi

with ad− bc = ±1 and l > 0 if c > 0, m > 0 if b > 0, or

(15.64) u = (uavb)tγ1(u, v), v = (uavb)kγ2(u, v), wi = uevf wiγ3(u, v)

where γ1, γ2, γ3 are unit series, a, b, t, k, e, f > 0, gcd(a, b) = 1, or

(15.65) u = uaγ1(u, v), v = ubγ2(u, v), wi = ucwiγ3(u, v)

where γ1, γ2, γ3 are unit series, a, b, c > 0.We have that Ψ′(E) = q for all components E of DY ′ containing q, (which

implies q 6∈ C) unless we have an expression

(15.66)u = uvb

v = vwi = ulvmwi

with b,m > 0.Since Ψ′ is an isomorphism over a generic point of every component of C, we

have that q is a 2-point and

(15.67) u = uvm, v = v, wi = vnwi

with m,n ≥ 1. u = wi = 0 are local equations of C at q.Suppose that q ∈ (Ψ Ψ′ Ψ(1))−1(q) and f(1)−1(q) ∩ T (Ri(1)) 6= ∅. Let

q = Ψ(1)(q). then (f ′)−1(q) ∩ T (R′i) 6= ∅. If q 6∈ C (so that q = q) then we haveseen that (Ψ Ψ′)(E) = q for all components E of DY ′ containing q. Suppose thatq ∈ C. Then an expression (15.67) holds at q. Then q is a 2-point with

u = uRi(1)(q), v = vRi(1)(q)

, wi = uRi(1)(q)wRi(1)(q)

,

and thus

(15.68) u = uRi(1)(q)vmRi(1)(q)

, v = vRi(1)(q), wi = uRi(1)(q)

vnRi(1)(q)

wRi(1)(q)

with m,n ≥ 1. We have that q is a 1-point and q is a 2-point.Now suppose that Ψ′ is an isomorphism over q (q ∈ U(R) is a 1-point), and

q ∈ Ψ(1)−1(q) is such that f(1)−1(q) ∩ T (Ri(1)) 6= ∅. then q is a 1-point, and (byRemark 15.6)

(15.69) u = uRi(1)(q), v = vRi(1)(q)

, wi = uRi(1)(q)wRi(1)(q)

.

We have that q is a 1-point and q is a 1-point.


Step 3. We now apply steps 1 and 2 of the proof to f(1) : X(1)→ Y (1) and R(1).We construct a τ -quasi-well prepared diagram

X(2)f(2)→ Y (2)

Φ(2) ↓ ↓ Ψ(2)

X(1)f(1)→ Y (1),

where R(2) is the transform of R(1) on X(2). Suppose that q2 ∈ U(Ri(2)) ⊂ Y (2)is on a component E2 of DY (2) such that Ψ Ψ′ Ψ(1) Ψ(2)(E2) is not a point ofY and there exists a point p2 ∈ f(2)−1(q2) ∩ T (Ri(2)) ⊂ X(2). Let q1 = Ψ(2)(q2).Ψ2(2)(E2) is necessarily not a point of Y (1).

Suppose that q1 is a 2-point. Then we have an expression (analogous to (15.62)):

(15.70)

uRi(1)(q1)= uRi(2)(q2)

vRi(1)(q1)= ue

Ri(2)(q2)vRi(2)(q2)

wRi(1)(q1)= uf

Ri(2)(q2)vRi(2)(q2)

wRi(2)(q2)

or

(15.71)

uRi(1)(q1)= uRi(2)(q2)

veRi(2)(q2)

vRi(1)(q1)= vRi(2)(q2)

wRi(1)(q1)= uRi(2)(q2)

vfRi(2)(q2)

wRi(2)(q2)

with e, f ≥ 1.Let q = (Ψ Ψ′ Ψ(1))(q1). Since q1 lies on a component E1 of DY (1) which

does not contract to a point of Y , and f(1)−1(q1) ∩ T (Ri(1)) 6= ∅, a form (15.62)or (15.68) holds for


, wi = wRi(q)

anduRi(1)(q1)

, vRi(1)(q1), wRi(1)(q1)

.

Suppose that (15.62) holds at q1. Substituting (15.70) and (15.71) into (15.62),we see that since q2 is on a component E2 of DY (2) which does not contract to q,then we have that (15.70) holds, and an expression

(15.72)

u = uRi(q) = uRi(2)(q2)

v = vRi(q) = ue2Ri(2)(q2)

vRi(2)(q2)

w = wRi(q) = uf2Ri(2)(q2)

v2Ri(2)(q2)

wRi(2)(q2)

with e2, f2 ≥ 2.Suppose that (15.68) holds at q1. Substituting (15.70) or (15.71) into (15.68),

we see that since q2 is on a component of DY (2) which does not contract to q, wehave that (15.71) holds we have and an expression

(15.73) u = uRi(2)(q2)ve2Ri(2)(q2)

, v = vRi(2)(q2), wi = u2

Ri(2)(q2)vf2Ri(2)(q2)

wRi(2)(q2)

with e2, f2 ≥ 2.Suppose that q1 is a 1-point. Then we have an expression (analogous to (15.68))

(15.74)

uRi(1)(q1)= uRi(2)(q2)

vmRi(2)(q2)

,

vRi(1)(q1)= vRi(2)(q2)

,

wRi(1)(q1)= uRi(2)(q2)

vnRi(2)(q2)

wRi(2)(q2)


with m,n ≥ 1, or (analogous to (15.69))(15.75)

uRi(1)(q1)= uRi(2)(q2)

, vRi(1)(q1)= vRi(2)(q2)

, wRi(1)(q1)= uRi(2)(q2)

wRi(2)(q2).

Since q1 lies on a component E1 of DY (1) which does not contract to a pointof Y , and f(1)−1(q1) ∩ T (Ri(1)) 6= ∅, a form (15.69) holds at q1 for


, wi = wRi(q)

anduR1(1)(q1)

, vRi(1)(q1), wRi(1)(q1)

.

Substituting (15.74) and (15.75) into (15.69), we see that since q1 is on a com-ponent E1 of DY (1) which does not contract to q, we have

(15.76) u = uRi(2)(q2)vmRi(2)(q2)

, v = vRi(2)(q2), wi = u2

Ri(2)(q2)vnRi(2)(q2)

wRi(2)(q2)

with m,n ≥ 0.Iterating steps 1 and 2, we construct a sequence of τ -quasi-well prepared dia-

grams

(15.77)

......

↓ ↓X(n)

f(n)→ Y (n)Φ(n) ↓ ↓ Ψ(n)

X(n− 1)f(n−1)→ Y (n− 1)

↓ ↓...

...↓ ↓

X(1)f(1)→ Y (1)

Φ(1) ↓ ↓ Ψ(1)

Xf→ Y.

We continue this algorithm as long as there exists qn ∈ U(Ri(n)) for some isuch that qn is on a component E of DY (n) which does not contract to a point ofY , and f(n)−1(qn) ∩ T (Ri(n)) 6= ∅.

Step 4. Suppose that the algorithm never terminates.Let ν be a 0-dimensional valuation of k(X). We will say that ν is resolved on

X(n) if the center of ν on X(n) is at a point pn of X(n) such that either pn 6∈T (R(n)) or pn ∈ T (R(n)) and all components E of DY (n) containing qn = f(n)(pn)contract to a point of Y .

By our construction, if ν is resolved on X(n), then ν is resolved on X(m) forall m ≥ n. Further, the set of ν in the Zariski-Riemann manifold Ω(X) of X ([Z])which are resolved on X is an open subset of Ω(X).

Suppose that ν is a 0-dimensional valuation of k(X) such that ν is not resolvedon X(n) for all n. Let pn be the center of ν on X(n), qn be the center of ν on Y (n).

For all n, we identify ν with an extension of ν to the quotient field of OXn,pn

which dominates OXn,pn .First suppose that the center of ν on Y is a 2-point.


There exists an i such that for all n, qn ∈ U(Ri(n)) and pn ∈ f(n)−1(qn) ∩T (Ri(n)). We have expressions (after possibly interchanging u and v)

(15.78)

u = uRi(q)= uRi(n)(qn)

v = vRi(q)= uen

Ri(n)(qn)vRi(n)(qn)

wi = wRi(q)= ufn

Ri(n)(qn)vnRi(n)(qn)

wRi(n)(qn)

with en, fn ≥ n for all n.From (15.78), we see that

(15.79) ν(v) > nν(u) for all n ∈ N.

Thus ν is a composite valuation, and there exists a prime ideal P of the valuationring V of ν such that v ∈ P , u 6∈ P . Let ν1 be a valuation whose valuation ring isVP . We have ν1(u) = 0, ν1(v) > 0. From (15.78) we see that

(15.80) ν1(wi) > nν1(v) > 0

for all n ∈ N.At p = p0 ∈ X, we have a form (15.41), (15.46), (15.49) or (15.50). In (15.41)

we have ν1(x) = 0 and d > 0, a contradiction to (15.80). In (15.49) we haveν1(y) = 0. uv = 0 is a local equation of DX at p. Thus either a > 0 or d > 0.If a > 0 then ν1(x) = 0 and ν1(z) > 0, a contradiction to (15.80) since f 6= 0. Ifd > 0, we again have a contradiction to (15.80). In (15.50) we have a contradictionto (15.80), since ν1(x) = 0 and e, f > 0.

Suppose that (15.46) holds. In this case a more detailed analysis is required. Byour construction and with the notation of steps 1 and 2, there exists a factorization

X(1)f(1)→ Y (1)

Φ(1) ↓ ↓ Ψ(1)

X ′ f ′→ Y ′

Φ′ ↓ ↓ Ψ′

Xnfn→ Yn

Φ ↓ ↓ Ψ

Xf→ Y.

Recall that p1 is the center of ν on X(1), p is the center of ν on X, q1 is thecenter of ν on Y (1), and q is the center of ν on Y .

Let p′ be the center of ν on X ′, q′ be the center of ν on Y ′.At q′, Y ′ → Y is a sequence of blow ups of prepared 2-points of type 1, and

2-curves, and at p′,X ′ → Y ′

↓ ↓X → Y

is obtained by iterating the constructions of Remark 14.3 and Lemma 14.7.Let

u′ = uR′i(q

′), v′ = vR′

i(q′), w

′i = wR′

i(q′).

By equations (15.79), (15.80) and (15.46) we see that

(15.81) ν(y) > nν(x) and ν1(x) = 0, ν1(z) > nν1(y)

for all n ∈ N.


We see (by a variant of the analysis of (15.46) in Step 1, using (15.81)), thatthere exist permissible parameters x′, y′, z′ at p′ such that

u′ = (x′)a, v′ = (x′)c′(y′)d, w′i = (x′)e

′(y′)fz′

where a, d, f are the constants of (15.46) and c′, e′ ∈ N. Ψ(1) is the blow up ofthe curve C which has local equations v′ = w′i = 0 at q′. The construction ofX(1) → X ′ at p1 (from Remark 14.12) is analogous to the analysis of Lemma14.11. There exists a factorization

X(1) = Wm → · · · →W2 →W1 → X ′

where W1 → X ′ is a sequence of blow ups of 2-curves and 3-points, and eachWi+1 → Wi is a curve containing a 1-point in the locus where ICOWi is notinvertible.

Let pi be the center of ν on Wi. By the analysis of the proof of Lemma 14.11,and (15.81), we see that there exist permissible parameters x1, y1, z1 at p1 such that

u′ = xa1 , v′ = xc11 y

d1 , w

′i = xe11 y

f1 z1

with (c1, d) ≤ (e1, f), or (c1, d) > (e1, f).If (c1, d) ≤ (e1, f), then X(1)→W1 is an isomorphism at p1, and we have

u′ = uRi(1)(q1), v′ = vRi(1)(q1)

, w′i = wRi(1)(q1)vRi(1)(q1)

.

We have

(15.82) uRi(1)(q1)= xa1 , vRi(1)(q1)

= xc11 yd1 , wRi(1)(q1)

= xe1−c11 yf−d1 z1.

Suppose that (c1, d) > (e1, f). We may assume without loss of generality thateach Wi+1 →Wi is not an isomorphism at the center of ν.

W2 →W1 is the blow up of a curve which either has local equations

(15.83) x1 = z1 = 0

(and c1 > e1), or

(15.84) y1 = z1 = 0

(and d > f).By (15.81), there exist permissible parameters x2, y2, z2 at p2 such that

x1 = x2, y1 = y2, z1 = x2z2

if (15.83) holds,x1 = x2, y1 = y2, z1 = y2z2

if (15.84) holds.We then have that

u′ = xa2 , v′ = xc12 y

d2 , w

′i = xe1+1

2 yf2 z1

oru′ = xa2 , v

′ = xc12 yd2 , w

′i = xe12 y

f+12 z1.

By iteration of this analysis for local equations of Wi+1 → Wi, we see that atp1 = pm, we have permissible parameters xm, ym, zm such that

u′ = xam, v′ = xc1m y

dm, w

′i = xc1m y

dmzm.

We have


(15.85) uRi(1)(q1)= xam, vRi(1)(q1)

= xc1m ydm, wRi(1)(q1)

= zm.

We see that ν is resolved on X(1) if (15.85) holds, since (15.85) is a toroidalform, so we must have that (15.82) holds at p1. Observe that we must have areduction f1 = f − d < f in (15.82) from (15.46).

Iterating this analysis, we see that we must reach the case (15.85) after a finitenumber of iterations of step 2. This is a contradiction to the assumption that ν isnever resolved on X(n).

Now suppose that the center of ν on Y is a 1-point. Then there exists an i suchthat for all n, qn ∈ U(Ri(n)) and pn ∈ f(n)−1(qn) ∩ T (Ri(n)), and either thereexists n0 such that qn is a 1-point for n < n0 and qn is a 2-point for n ≥ n0 or qnis a 1-point for all n.

q = q0 ∈ Y is a 1-point and p = p0 ∈ f−1(q) ∩ T (Ri). Let


, wi = wRi(q).

For all n, let

u(n) = uRi(n)(qn), v(n) = vRi(n)(qn), w(n) = wRi(n)(qn).

If τ > 1, there exist permissible parameters x, y, z at p such that one of thefollowing forms hold: p a 1-point

(15.86) u = xa, v = y, wi = xb(γ(x, y) + xc−bz)

where γ is a unit series or p a 2-point

(15.87) u = (xayb)k, v = z, wi = (xayb)t(γ(xayb, z) + xcyd)

where γ is a unit series.If τ = 1, then there exist permissible parameters x, y, z at p such that either p

is a 1-point and

(15.88) u = xa, v = y, wi = xb(β + z)

with β ∈ k, b > 0, or p is a 2-point and

(15.89) u = (xayb)k, v = z, wi = xcyd

with ad− bc 6= 0.Suppose that there exists n0 such that qn is a 2-point for all n ≥ n0 (and qn is

a 1-point for n < n0). By iterating (15.69), we see that

(15.90) u = u(n0 − 1), v = v(n0 − 1), wi = u(n0 − 1)n0−1w(n0 − 1).

After possibly interchanging u(n0−1) and v(n0−1), we have that u(n0−1), v(n0−1), w(n0 − 1) and u(n0), v(n0), w(n0) are related by an expression (15.68), andafter possibly interchanging u(n0) and v(n0), we have that u(n0), v(n0), w(n0) andu(n0 + 1), v(n0 + 1), w(n0 + 1) are related by an expression (15.62). From then on,u(n− 1), v(n− 1), w(n− 1) are related to u(n), v(n), w(n) by an expression (15.62).

Since we assume some component of DY (n) containing qn does not contract toq, we have an expression:

u = u(n)v(n)en ,v = v(n),wi = u(n)fnv(n)gnw(n),

and en, fn, gn ≥ n− n0 + 1.


We have ν(u) > nν(v) > 0 for all n ∈ N. Thus ν is a composite valuation, andthere exists a prime ideal P of the valuation ring V of ν such that u ∈ P , v 6∈ P .Let ν1 be a valuation whose valuation ring is VP . We have ν1(v) = 0, ν1(u) > 0.We further have

(15.91) ν1(wi) > nν1(u) > 0

for all n ∈ N.Suppose that τ > 1 (and qn is a 2-point for n ≥ n0). At p = p0 ∈ X, we have

a form (15.86) or (15.87). In (15.86) we have

ν1(wi) =b

aν1(u)

and in (15.87) we have

ν1(wi) =t

kν1(u),

a contradiction (to (15.91).Suppose that τ = 1. If β 6= 0 in (15.88), then the analysis is the same as for

the τ > 1 case, so we may assume that β = 0 if (15.88) holds.If (15.88) holds with β = 0, or if (15.89) holds, we finish the analysis in a similar

way to the proof when q is a 2-point given above. We must end up with f(n) beingtoroidal at pn, which is impossible.

The final case is when qn is a 1-point for all n. From (15.69) we see that

u = u(n), v = v(n), wi = u(n)nw(n)

so that

(15.92) ν(wi) > nν(u) > 0

for all n ∈ N.Suppose that τ > 1 (and qn is a 1-point for all n). At p = p0 ∈ X we have a

form (15.86) or (15.87), which implies

ν(wi) =b

aν(u)

or

ν(wi) =t

kν(u),

a contradiction to (15.92).When τ = 1, the proof follows in a similar way to the proof when q is a 2-point

and τ = 1.We have shown that for all 0-dimensional valuations ν of k(X), there exists n

such that ν is resolved on X(n).By compactness of the Zariski-Riemann manifold [Z] there exists N such that

all ν ∈ Ω(X) are resolved on X(N), a contradiction to our assumption that (15.77)is of infinite length. The diagram

X(N) → Y (N)↓ ↓X → Y

thus satisfies the conclusions of Theorem 15.7.


Theorem 15.8. Suppose that τ ≥ 1, f : X → Y is τ -quasi-well prepared withrelation R. Then there exists a commutative diagram

X1f1→ Y1

Φ ↓ ↓ Ψ

Xf→ Y

such that Φ and Ψ are products of blow ups of possible centers and f1 is τ -quasi-wellprepared with relation R1 and pre-algebraic structure. Further, R has an algebraicstructure. (R1 will in general not be the transform of R.)

Proof. By Theorem 15.7 there exists a τ -quasi-well prepared diagram (15.26)as in the hypothesis of Lemma 15.5. Then Lemma 15.5 implies that the conclusionsof Theorem 15.8 hold.

Lemma 15.9. Suppose that τ ≥ 1, f : X → Y is τ -quasi-well prepared withrelation R and pre-algebraic structure (or τ -well prepared with relation R), q ∈U(R) is a 2-point and p ∈ f−1(q)∩T (Ri) for some i. Suppose that E is a componentof DY containing q. Let C = E · SRi

(q).Let Ψn : Yn → Y be obtained by blowing up q, then blowing up the point q1

which is the intersection of the exceptional divisor over q and the strict transformof C on Y1, and iterating this procedure n times, blowing up the intersection pointof the last exceptional divisor with the strict transform of C. Let

Xnfn→ Yn

Φn ↓ ↓ Ψn

Xf→ Y

be a τ -quasi-well prepared (or τ -well prepared) diagram of R and Ψn obtained fromLemma 14.7 (so that Φn is an isomorphism above f−1(Y − Σ(Y ))).

Suppose that for all n > 0 there exists a point pn ∈ Φ−1n (p) ∩ T (Rni ) such that

fn(pn) = qn ∈ Ψ−1n (q) ∩ Cn, where Cn is the strict transform of C on Yn. Then C

is a component of the nonfinite locus of f .

Proof. The proof follows from Step 1 of the proof of Theorem 15.7. Let


, wi = wRi(q).

After possibly interchanging u and v, v = wi = 0 are local equations of C at q.By our construction of Ψn, we have that

u1 = uRni (qn), v1 = vRn

i (qn), wi,1 = wRni (qn)

are defined byu = u1, v = un1 v1, wi = un1wi,1.

We must have an expression (15.41), (15.46), (15.49) or (15.50). It follows, asin the analysis of step 1 of Theorem 15.7 that C is in the nonfinite locus of f .

Theorem 15.10. Suppose that τ ≥ 1 and f : X → Y is τ -quasi-well preparedwith relation R and pre-algebraic structure. Further suppose that R has an algebraic


structure. Then there exists a commutative diagram

X1f1→ Y1

Φ ↓ ↓ Ψ

Xf→ Y

such that Φ,Ψ are products of blow ups of possible centers and f1 is τ -well preparedwith relation R1. (In general, R1 is not the transform of R).

Proof. 1 and 2 of Definition 13.5 of a τ -well prepared relation R hold byassumption.

For i 6= j, Let Hij be the set of points q ∈ U(Ri)∩U(Rj) such that neither anexpression (13.7) nor (13.8) of 3 of Definition 13.5 holds between wRi(q)

and wRj(q).

We see by 4 of Definition 13.1 (and (12.7) - (12.9)) that Hij is a finite set.Let Ψ1 : Y1 → Y be the blow up of the union of the sets

q ∈ Hij | f−1(Hij) ∩ [T (Ri) ∪ T (Rj)] 6= ∅.By Lemmas 14.7 and 14.8, there exists a τ -quasi-well prepared diagram

X1f1→ Y1

Φ1 ↓ ↓ Ψ1

Xf→ Y.

We define finite sets H1ij in the same way for the transform R1 of R, and iterate to

construct a τ -quasi-well prepared diagram

(15.93)

Xnfn→ Yn

Φn ↓ ↓ Ψn

Xn−1fn−1→ Yn−1

↓ ↓...

...↓ ↓

X1f1→ Y1

Φ1 ↓ ↓ Ψ1

Xf→ Y,

continuing as long as f−1n (Hn

ij) ∩ [T (Rni ) ∪ T (Rnj )] 6= ∅ for some i 6= j.Suppose that (15.93) doesn’t terminate after a finite number of blow ups n.

Then there exist i 6= j and a valuation ν of the function field k(X) of X suchthat the center pn of ν on Xn and the center qn of ν on Yn satisfy qn ∈ Hn

ij ,pn ∈ f−1

n (Hnij) ∩ T (Rni ) for all n.

For all n, we may identify ν with an extension of ν to the quotient field ofOXn,pn

which dominates OXn,pn.

Letun = uRn

i (qn), vn = vRni (qn), wni = wRn

i (qn), wnj = wRnj (qn).

We have(15.94)

un = un+1, vn = un+1(vn+1 + αn+1), wni = un+1wn+1,i, wnj = un+1wn+1,j


with αn+1 ∈ k, or

(15.95) un = un+1vn+1, vn = vn+1, wni = vn+1wn+1,i, wnj = vn+1wn+1,j

for all n.The relation

wnj = wni + λnij(un, vn)

of 4 of Definition 13.1 transforms as

λn+1ij (un+1, vn+1) =

λnij(un+1, un+1(vn+1 + αn+1))un+1

if (15.94) holds, and transforms as

λn+1ij (un+1, vn+1) =

λnij(un+1vn+1, vn+1)vn+1

if (15.95) holds.Let Fn be the germ at qn of the divisor λnij = 0. By embedded resolution of

plane curve singularities, there exists n0 such that DXn+ Fn is a SNC divisor at

qn for all n ≥ n0. If qn is a 2-point for some n ≥ n0, we have that qn 6∈ Hnij , a

contradiction, so we have that qn is a 1-point for all n ≥ n0. Thus a form (15.94)holds for all n ≥ n0. We have

wn0i = un−n0n0

wni.

Thus

(15.96) ν(wn0i) > nν(un0) > 0

for all positive n.Suppose that τ > 1. Since qn0 ∈ U(Rn0

i ) is a 1-point, we have permissibleparameters x, y, z at pn0 such that

(15.97) un0 = xa, vn0 = y, wn0i = xcγ

where γ ∈ OXn0 ,pn0is a unit series, or

(15.98) un0 = (xayb)k, vn0 = z, wn0,i = (xayb)lγ

where γ ∈ OXn0 ,pn0is a unit series. In either case, we have a contradiction to

(15.96).If τ = 1, then we have one of the following forms at pn0 .

un0 = xa, vn0 = y, wn0i = xc(z + β)

with β ∈ k, orun0 = (xayb)k, vn0 = z, wn0,i = xcyd

with a, b > 0 and ad− bc 6= 0.We obtain a contradiction to (15.96), as in the τ = 1 case, unless we have the

formun0 = xa, vn0 = y, wn0i = xcz.

Now by an argument similar to the case τ = 1 of Theorem 15.7, we obtain

τfn(pn) = −∞

for n >> 0, a contradiction.


Thus (15.93) terminates in a finite number of steps n. Let (Rn)′ be the restric-tion of Rn, defined by

U((Rn)′i) = U(Rni )− ∪jHij ,

where Hij is defined by Rn, Let

Ω((Rn)′i) = Ω(Rni )− ∪jHij .

We have that fn : Xn → Yn with relation (Rn)′ satisfies 1, 2 and 3 of Definition13.5 of a τ -well prepared morphism.

There exists a sequence of blow ups of 2-curves Ψ1 : Y1 → Yn such that 4 (aswell as 3) of Definition 13.5 hold for the transforms R1

i of the (Rn)′i on Y1 atall 2-points of U((R

n)′i), by Lemma 5.14 [C6]). By Lemma 14.1, there exists a

τ -quasi-well prepared diagram

X1f1→ Y1

Φ1 ↓ ↓ Ψ1

Xnfn→ Yn

of (Rn)′i and Ψ1 where Φ1 is a product of blow ups of 2-curves. Let R1 be thetransform of (Rn)′ on X1. f1 is τ -well prepared with relation R1.

Theorem 15.11. Suppose that τ ≥ 1 and f : X → Y is τ -well prepared withrelation R. Then there exists a commutative diagram

X1f1→ Y1

Φ ↓ ↓ Ψ

Xf→ Y

such that Φ,Ψ are products of blow ups of possible centers and f1 is τ -very-wellprepared with relation R1. (In general, R1 is not the transform of R).

Proof. Let Ω = GY (f, τ)−Θ(f, Y ). Ω is a finite set by Remark 11.12.Let R′ be the restriction of R to Ω. R′ has an algebraic structure determined

by the algebraic structure of R.By Theorem 15.3, there exists a τ -well prepared diagram

X2f2→ Y2

Φ2 ↓ ↓ Ψ2

Xf→ Y

for R where Φ2 and Ψ2 are products of blow ups of possible centers such that f2 isτ -well prepared for the transform R2 of R′, and there exists an open subset V ⊂ Ysuch that Y − V is a finite set of points and f2 is toroidal over Ψ−1

2 (V ). Further,V ∩GY (f, τ) = Θ(f, Y ).

Let R2i be a primitive relation associated to R2. Then U(R′i) = qi for some

qi ∈ Ω. Ω(R′i) is a neighborhood of qi on a surface in Y , and Ω(R2i ) → Ω(R′i)

is a projective birational map. Suppose that E is a component of DY2 such thatγ = E · Ω(R2

i ) dominates a curve (containing qi) of Ω(R′i). Then a general point ηof γ is a 1-point over which f2 is toroidal, since Ψ2(η) ∈ V . Hence f2 is finite overa general point of γ, and γ is not in the nonfinite locus of f2. Further, by Remark15.6, if q ∈ γ ∩ U(R2

i ), and f−12 (q) ∩ T (R2

i ) 6= ∅, then q is a 2-point.By Lemma 15.9, there exists a τ -well prepared diagram (for R2)


X3f3→ Y3

Φ3 ↓ ↓ Ψ3

X2f2→ Y2

where Ψ3 is a product of blow ups of prepared 2-points (of type 1 in Definition13.7) such that if R3

i is a primitive relation associated to the transform R3 of R2,E is a component of DY3 , and γ = E ·Ω(R3

i ) is not exceptional for Ω(R3i )→ Ω(R′i),

then γ ∩ f3(T (R3i )) = ∅. We further have that f3(T (R3)) ∩ γ = ∅. We see this as

follows. Suppose that p ∈ f3(T (R3)) ∩ γ. Ψ3(E) is a component of DY2 . Thus p ison the strict transform γ′ of Ψ3(E) · Ω(R2

j ) on Y3, which is not exceptional for Ψ2.Since γ′ = E · Ω(R3

j ), we have a contradiction.Let

W3 =

γ = SR3

i(q) · E such that E is a component of DY3 ,

R3i is associated to R3 and q ∈ f3(T (R3)) ∩ U(R

3

i )

.

We have that all γ ∈W3 contract to a point on Y1. Let

Z3 =q ∈ U(R3)− f3(T (R3)) such that there exist γi, γj ∈W3

such that γi 6= γj and q ∈ γi ∩ γj .

Suppose that q ∈ Z3. Then there exist γi = SR3

i(pi) · E1 ∈ W3 and γj =

SR3j(pj) · E2 ∈W3 such that q ∈ γi ∩ γj and γi 6= γj .γi and γj are exceptional, so they contract to the common point qi = qj ∈

U(R′). thus γi and γj are contained in Ω(R3i ) and Ω(R3

j ) respectively.The points of Z3 are prepared 2-points for R3 (of type 1 of Definition 13.7) by

Lemmas 14.1, 14.7. 14.8. Let Ψ4 : Y4 → Y3 be the blow up of Z3. By Lemma 14.7,there exists a τ -well prepared diagram

X4f4→ Y4

Φ4 ↓ ↓ Ψ4

X3f3→ Y3

of R3 and Ψ4. Let R4 be the transform of R3 on X4.Define

W4 =

γ = S

R4i(q) · E such that E is a component of DY4 ,

R4i is associated to R4 and q ∈ Ψ−1

4 (f3(T (R3))) ∩ U(R4

i )

,

Z4 =q ∈ U(R4)−Ψ−1

4 (f3(T (R3))) such that there exist γi, γj ∈W4

such that γi 6= γj and q ∈ γi ∩ γj .

We necessarily have that the curves in W4 are strict transforms of curves in

W3. We can iterate, blowing up Z4, and constructing a τ -well prepared diagram,and repeating until we eventually construct a τ -well prepared diagram of R4

X5f5→ Y5

Φ5 ↓ ↓ Ψ5

X4f4→ Y4

such that Ψ5 is a sequence of blow ups of prepared 2-points (of type 1 of Defi-nition 13.7) and if γ1 = S

R5i(qi) · Ei, γ2 = S

R5j(qj) · Ej , for qi ∈ U(R

5

i ) ∩ (Ψ4


Ψ5)−1(f3(T (R3))) (where R5 is the transform of R4) and E1, E2 components ofDY5 , are such that γ1 6= γ2, then γ1 ∩ γ2 ⊂ U(R5) ∩ (Ψ4 Ψ5)−1(f3(T (R3)).

We now construct pre-relations R∗i on Y5 with associated primitive relations

R∗i for f5.Let T (R∗i ) = T (R5

i ) and let

(15.99) U(R∗i ) = U(R5i ) ∩ (Ψ4 Ψ5)−1(f3(T (R3))).

For q′ ∈ U(R∗i ), define R

∗i (q

′) = R5

i (q′). For p ∈ T (R∗i ) define R∗i (p) =

R5

i (f5(p)). Let R∗ be the relation for f5 defined by the R∗i . Let Ω(R∗i ) = Ω(R

5

i ).For all R

∗i , let

Vi(Y5) =γ = Eα · SR∗

i(q) such that q ∈ U(R

∗i ), Eα is a component of DY5

.

Recall that these curves are all exceptional for Ψ2 Ψ3 Ψ4 Ψ5.By our construction, Lemmas 14.5, 14.1, 14.7, 14.9 and 14.8, and Remark 14.2

and 2 of Remark 14.12, every curve γ ∈ Vi(Y5) is prepared for R5 of type 6. By(15.99) and our construction of f4 from f2, we now conclude that every curveγ ∈ Vi(Y5) is prepared for R∗ of type 6. 1 and 2 of Definition 13.9 thus hold for f5and R∗. 3 of Definition 13.9 holds for f5 and R∗ since for all R

∗i , Vi(Y5) consists of

exceptional curves of Ω(R∗i ) contracting to a nonsingular point qi ∈ Ω(R

′i). Thus

f5 is τ -very-well prepared with relation R∗.

Theorem 15.12. Suppose that f : X → Y is prepared, and τ = τf (X) ≥ 1.Then there exists a commutative diagram

X1f1→ Y1

Φ1 ↓ ↓ Ψ1

Xf1→ Y

such that Φ1 and Ψ1 are products of blow ups of possible centers and f1 is τ -very-well prepared with a relation R1.

Proof. By Theorem 14.4, there exists a commutative diagram

X1f1→ Y1

Φ ↓ ↓ Ψ

Xf→ Y

such that Φ and Ψ are products of 2-curves, and f1 is τ -prepared. Now by Theorems15.4, 15.8, 15.10 and 15.11, there exists a commutative diagram

X2f2→ Y2

↓ ↓X1

f1→ Y1

where the vertical arrows are products of blow ups of possible centers such that f2is τ -very-well prepared.

CHAPTER 16

Toroidalization

Suppose that f : X → Y is a proper, dominant morphism of nonsingular 3-foldswith toroidal structures DY and DX = f−1(DY ), such that DX contains the locuswhere f is not smooth.

Theorem 16.1. Suppose that τ ≥ 1 and f : X → Y is τ -very-well preparedwith relation R. Then there exists a τ -very-well prepared diagram

X1f1→ Y1

↓ ↓X

f→ Y

such that the transform R1 of R is resolved (T (R1) = ∅). In particular, f1 isprepared and τf1(X1) < τ .

Proof. Fix a pre-relation Rt associated to R on Y , with associated primitiverelation Rt. By induction on the number of pre-relations associated to R, it sufficesto resolve Rt by a τ -very-well prepared diagram (of R).

Recall (Definition 13.9)

Vt(Y ) =E · S such that E is a component of DY ,S = SRt

(q) for some q ∈ U(Rt)

.

Ft =∑γ∈Vt(Y ) γ is a SNC divisor on Ω(Rt) whose intersection graph is a forest.

If γ1 = E1 · SRt(q1) ∈ Vt(Y ) and q ∈ γ1, we will say that γ1 is good at q if

whenever q ∈ U(Ri) for some i, then SRi(q) contains the germ of γ1 at q (so that

γ1 = E1 · SRi(q) ⊂ Ω(Ri)). Otherwise, say that γ1 is bad at q. Say that γ1 is good

if γ1 is good at q for all q ∈ γ1.Let Y0 = Y , X0 = X, f0 = f . We will show that there exists a sequence of

τ -very-well prepared diagrams of the transform of R,

(16.1)Xi+1

fi+1→ Yi+1

Φi+1 ↓ Ψi+1 ↓Xi

fi→ Yi

for 0 ≤ i ≤ m− 1 such that the transform Rmt of Rt on Xm is resolved.Suppose that γ1 ∈ Vt(Y ) and q ∈ γ1 is a bad point. By Remark 13.10, we

have that q ∈ U(Rt). By (13.8) of Definition 13.5, we have that q is a 2-point.Suppose that E1, E2 are the two components of DY containing q, ordered so thatγ1 = E1 · SRt

(q). Let γ2 = E2 · SRt(q).

We will show that q is a good point of γ2.

211

212 16. TOROIDALIZATION

γ1 not good at q implies there exists j 6= t such that q ∈ U(Rj) and the germof γ1 at q is not contained in SRj

(q). Let

u = uRt(q), v = vRt

(q), wt = wRt(q).

After possibly interchanging u and v we have that u = wt = 0 are local equationsof γ1, v = wt = 0 are local equations of γ2 at q. Let wj = wRj

(q). In the equation

wj = wt + uatjvbtjφtj

of (13.7) of Definition 13.5 we thus have atj = 0.If q is not a good point for γ2 then there exists k 6= t such that q ∈ U(Rk) and

the germ of γ2 at q is not contained in SRk(q). Let wk = wRk

(q). In the equation

wk = wt + uatkvbtkφtk

of (13.7) we thus have btk = 0. But we must have

(0, btj) ≤ (atk, 0) or (atk, 0) ≤ (0, btk)

by 4 of Definition 13.5, which is impossible. Thus q is a good point for γ2.Suppose that all γ ∈ Vt(Y ) are bad. Pick γ1 ∈ Vt(Y ). Since γ1 is bad there

exists γ2 ∈ Vt(Y ) − γ1 such that γ2 is good at q1 = γ1 ∩ γ2 (as shown above).γ1∩γ2 is a single point since Vt(Y ) is a forest. Since γ2 is bad and Vt(Y ) is a forest,there exists γ3 ∈ Vt(Y ) which intersects γ2 at a single point q2 and is disjoint fromγ1 such that γ3 is good at q2. Since Vt(Y ) is a finite set, and the intersection graphof Vt(Y ) is a forest, we must eventually find a curve which is good, a contradiction.

Let γ ∈ Vt(Y ) be a good curve, so that it is prepared for R of type 6, and is a*-permissible center (Lemma 14.11) and let Ψ′

1 : Y ′1 → Y be the blow up of γ.

By Lemma 14.11 we can construct a τ -very-well prepared diagram of the formof (13.11) of Definition 13.12

(16.2)

X1f1→ Y1

↓ ↓↓ Y ′

1

↓ ↓ Ψ′1

X → Y.

where Y1 → Y ′1 is a sequence of blow ups of 2-points which are prepared for the

transform of R of type 2 of Definition 13.7. Observe that if γ1 ∈ Vt(Y ) is a goodcurve, with γ1 6= γ, then the strict transform of γ1 in Y1 is a good curve in Vt(Y1).

We now iterate this process. We order the curves in Vt(Y ), and choose γ =E · SRt(q) ∈ Vt(Y ) in the construction of the diagram (16.2) so that it is theminimum good curve in Vt(Y ).

We inductively define a sequence of τ -very well prepared diagrams (16.1) byblowing up the good curve in Vt(Yi) with smallest order, and then constructing avery well prepared diagram (16.1) of the form of (16.2). Then we define the totalordering on Vt(Yi+1) so that the ordering of strict transforms in Yi+1 of elements ofVt(Yi) is preserved, and these strict transforms have smaller order than the elementof Vt(Yi+1) which is not a strict transform of an element of Vt(Yi). We repeat, aslong as Rit is not resolved (T (Rit) 6= ∅).

16. TOROIDALIZATION 213

Suppose that the algorithm does not converge in the construction of fm : Xm →Ym such that the transform Rmt of Rt is resolved. Then there exists a diagram

(16.3)

......

↓ ↓Xn


Xn−1fn−1→ Yn−1

↓ ↓...

...Φ1 ↓ ↓ Ψ1

X0 = Xf0=f→ Y0 = Y

constructed by infinitely many iterations of the algorithm such that T (Rnt ) 6= ∅ forall n.

Suppose that qn ∈ U(Rn

t ) is an infinite sequence of points such that Ψn(qn) =qn−1 for all n and Ψn is not an isomorphism for infinitely many n. qn is either a2-point or a 1-point for all n.

First suppose that qn is a 2-point for all n.By construction, the restriction of Ψn to SRn

t(qn) is an isomorphism onto

SR

n−1t

(qn−1) for all n. Thus the restriction

Ψn = Ψ1 · · · Ψn : SRnt(qn)→ SRt

(q)

is an isomorphism, where q = q0 = Ψ1 · · · Ψn(qn). Without loss of generality,we may assume that no Ψn is an isomorphism (on Yn) at qn. We have permissibleparameters ui = u

Rit(qi), vi = v

Rit(qi), wt,i = w

Rit(qi) at qi for all i such that either

(16.4) ui = ui+1, vi = vi+1, wt,i = ui+1wt,i+1

or

(16.5) ui = ui+1, vi = vi+1, wt,i = vi+1wt,i+1.

Suppose there exists k 6= t such that qn ∈ U(Rn

k ) for all n.Let wk,i = w

Rik(qi) for i ≥ 0.

The relationswk,i − wt,i = uatk

i vbtki φt,k

of (13.7) of Definition 13.5 transform to

wk,i+1 − wt,i+1 = uatk−1i+1 vbtk

i+1φt,k

under (16.4), and transform to

wk,i+1 − wt,i+1 = uatki+1v

btk−1i+1 φt,k

under (16.5). But we see that after a finite number of iterations qn 6∈ U(Rn

k ), unlessatk = btk = −∞. Thus there exists n0, such that whenever n ≥ n0, qn 6∈ U(R

n

k ) ifk 6= t and akt, bkt 6= −∞ .

Now suppose that qn is a 1-point for all n. We have permissible parameters

ui = uR

it(qi), vi = v

Rit(qi), wt,i = w

Rit(qi)


at qi for all i such that

ui = ui+1, vi = vi+1, wt,i = ui+1wt,i+1

for all i.Suppose that there exists k 6= t such that qn ∈ U(R

n

k ) for all n. Let

wk,i = wR

ik(qi)

for i ≥ 0. The relationwk,i − wt,i = u

ct,k

i φt,k

of (13.8) of Definition 13.5 transforms to

wk,i+1 − wt,i+1 = uct,k−1i+1 φt,k.

Thus after a finite number of iterations, qn 6∈ U(Rn

k ) unless ct,k = −∞. Thus thereexists n0 such that when n ≥ n0, qn 6∈ U(R

n

k ) if k 6= t and ct,k 6= −∞.By our ordering, we have that there exists an n0 such that if n ≥ n0, γ ∈ Vt(Yn)

is good and if k is such that γ ∩ U(Rn

k ) 6= ∅ then atk, btk = −∞ (or ctk = −∞), sothat the Zariski closures of Ω(R

n

k ) and Ω(Rn

t ) are the same. Thus all elements ofVt(Yn) are good for n ≥ n0, since otherwise, there would be a bad curve γ1 ∈ V (Yn)which intersects a good curve γ2 at a point q′ at which γ1 is not good. But thenwe must have that there exists k 6= t such that the Zariski closure of Ω(R

n

k ) is notequal to the Zariski closure of Ω(R

n

t ), and q′ ∈ U(Rn

k ), so that γ2 ∩ U(Rn

k ) 6= ∅, acontradiction.

Our birational morphism of Ω(Rn

t ) to Ω(Rt) is an isomorphism in a neighbor-hood of U(R

n

t ). Thus we have a natural identification of Vt(Yn) and Vt(Y ), and wesee that for n ≥ n0, the Ψn cyclically blow up the different curves of Vt(Y ).

There are points pn ∈ T (Rnt ) ⊂ Xn such that Φn(pn) = pn−1, and fn(pn) =qn ∈ U(R

n

t ) for all n. Without loss of generality, we may assume that no Ψn is anisomorphism at qn.

We have that all qn are 1-points or all qn are 2-points.First suppose that all qn are 2-points.With the above notation at qn = fn(pn), we have that (16.4) and (16.5) must

alternate in the diagram (16.3) for n ≥ n0, by our ordering of Vt(Yn). Let p = p0 ∈X = X0, q = q0 = f(p).

We have

(16.6) u = un, v = vn, wt = uann vbn

n wt,n

whereu = uRt

(q), v = vRt(q), wt = wRt

(q)and an, bn are positive integers which both go to infinity as n goes to infinity.

There exists (by Theorem 4 of Section 4, Chapter VI [ZS]) a valuation ν of k(X)which dominates the (non-Noetherian) local ring ∪n≥0OXn,pn , and thus dominatesthe local rings OXn,pn for all n. Without loss of generality, we may identify ν withan extension of ν to the quotient field of OXn,pn which dominates OXn,pn for all n.

Let x, y, z be permissible parameters for u, v, wt at p. Suppose that p is a3-point. Write (in OX,p)

(16.7)u = xaybzc

v = xdyezf

wt = xgyhziγ

16. TOROIDALIZATION 215

where xyz = 0 is a local equation of DX at p and γ is a unit series.We may permute x, y, z so that 0 < ν(x) ≤ ν(y) ≤ ν(z). We have (from (16.6))

ν(wt)− nν(u)− nν(v) > 0

for all n ∈ N. Thus

0 < (g−na−nd)ν(x)+(h−ne−nb)ν(y)+(i−nf−nc)ν(z) ≤ ((g+h+i)−nf−nc)ν(z)

for all n. Thus f = c = 0, but this is impossible, since uv = 0 is a local equation ofDX at p.

There is a similar but simpler algorithm if p is a 1-point or a 2-point and τ > 1,since wt = 0 is supported on DX at p.

Suppose that τ = 1 (and all qn are 2-points). We have that wt = 0 is a divisorsupported on DX at p, so that the argument for τ > 1 works in this case also, orwe have one of the following two special forms:

p a 1-point

(16.8) u = xa, v = xb(α+ y), wt = xcz

or p a 2-point

(16.9) u = xayb, v = xcyd, wt = xeyfz

with ad− bc 6= 0.In the diagram (16.2) we have that q1 ∈ U(R1

i ) implies Y1 → Y ′1 is an isomor-

phism near q1. X1 → X factors above a suitable neighborhood of q1 as a diagram

X1 = WmΛm→ Wm−1 → · · · →W1

Λ1→ X

where Λ1 : W1 → X is a sequence of blow ups of 2-curves and 3-points, and eachΛj+1 is the blow up of a possible curve Σj containing a 1-point and the center pjof ν on Wj , such that IγOW is not invertible.

Without loss of generality, we may assume that u = wt = 0 are local equationsof γ at q.

We have a form (16.8) or (16.9) at the center p1 of ν on W1, where (a, b) ≤ (e, f)or (a, b) > (e, f) if (16.9) holds.

Suppose that a ≤ c in (16.8), or (a, b) ≤ (e, f) in (16.9). Then IγOW1,p1 isinvertible. Thus Wm = W1, and above a suitable neighborhood of q1, X1 → X is asequence of blow ups of 2-curves. Further, we have an expression of the form (16.8)or (16.9) at p1.

Suppose that a > c in (16.8). Then Λ2 is the blow up of a curve with localequations x = z = 0 at p1. Since ν(z) > nν(x) for all n ∈ N, at p2 we have regularparameters x2, y2, z2 defined by

x = x2, y = y2, z = x2z2

and we thus haveu = xa2 , v = xb2(α+ y2), wt = xc+1

2 z2.

Iterating, we see that pm has permissible parameters xm, ym, zm such that

u = xam, v = xbm(α+ ym), wt = xamzm.

The permissible parameters u1, v1, wt,1 at q1 are defined by

u = u1, v = v1, wt = u1wt1.


Thusu1 = xam, v1 = xbm(α+ ym), wt1 = zm

and we have τf1(p1) = −∞, a contradiction.We have a similar analysis if (a, b) > (e, f) in (16.9), leading to the conclusions

that τf1(p1) = −∞, a contradiction.We thus see that for all n in (16.3), Xn → Xn−1 factors as a sequence of blow

ups of 2-curves at qn.Suppose that there exists n0 such that pn0 is a 1-point, and thus pn is a 1-point

for all n ≥ n0. Then we see that Xn+1 → Xn is an isomorphism at pn for alln ≥ n0. At pn0 there are regular parameters x, y, z such that

(16.10) un0 = xa, vn0 = xb(α+ y), wtn0 = xcz,

and qn has regular parameters

un0 = un, vn0 = vn, wtn0 = un−n0n wtn.

Substituting into (16.10), we have a contradiction as soon as

n >c

a+ n0.

Now suppose that pn is a 2-point for all n. Then a form (16.9) holds at p1, andat pn, we have regular parameters xn, yn, zn defined by

(16.11) x = xrn11n y

rn12n , y = x

rn21n y

rn22n , z = zn

such that rn11rn22 − rn12rn21 = ±1.

Now from (16.6), (16.9) and (16.11), we see that

ν(xeyf )− nν(xayb)− nν(xcyd) > 0

for all n ∈ N, a contradiction, since ad− bc 6= 0.The argument is simpler in the case when qn is a 1-point for all n. (16.6)

becomes

(16.12) u = un, v = vn, wt = uann wt,n

where an goes to infinity as n goes to infinity.Suppose that τ > 1. If p ∈ f−1(q), u = 0 is a local equation of DX at p, and

wt = 0 is supported on DX at p. Thus (16.12) leads to a contradiction.If τ = 1, there is a similar argument to the above case of qn a 2-point for all n

and τ = 1.Thus the algorithm converges in a τ -very-well prepared morphism fm : Xm →

Ym such that T (Rmt ) = ∅, and after iterating for each primitive relation associatedto R, we obtain the construction of f1 : X1 → Y1, as in the conclusions of thetheorem, such that f1 is prepared and τf1(X1) < τ .

CHAPTER 17

Proofs of the main results

Proof of Theorem 1.2First suppose that X and Y are projective over k. By resolution of singu-

larities and resolution of indeterminacy [H] (cf. Section 6.8 [C6]), there exists acommutative diagram

X1f1→ Y1

Φ1 ↓ ↓ Ψ1

Xf→ Y

where Φ1, Ψ1 are products of possible blow ups of points and nonsingular curvessupported above DX and DY , such that X1 and Y1 are nonsingular and projective.Further, DY1 = Ψ−1

1 (DY ) and DX1 = Φ−11 (DX) = f−1

1 (DY1) are SNC divisors, andDX1 contains the locus where f1 is not smooth.

If Y is a curve, the proof of Theorem 1.2 now follows from embedded resolutionof singularities (c.f. Introduction to [C3]). If Y is a surface, the proof of Theorem1.2 follows from Theorem 19.11 [C3].

Assume that Y is a 3-fold. By Theorem 1.3, we can construct a commutativediagram

X2f2→ Y2

Φ2 ↓ ↓ Ψ2

X1f1→ Y1

such that Φ2 and Ψ2 are products of possible blow ups of points and nonsingularcurves, such that f2 is prepared for DY2 = Ψ−1

2 (DY1) and DX2 = Φ−12 (DX1).

Now by descending induction on τ = τf2(X2) and Theorems 15.12 and 16.1,there exists a commutative diagram

X3f3→ Y3

Φ3 ↓ ↓ Ψ3

X2f2→ Y2

such that Φ2 and Ψ3 are products of blow ups of possible centers, f3 is prepared,and τf3(X3) = −∞.

Thus f3 is toroidal, and the conclusions of the theorem follow.Now suppose thatX and Y are quasi-projective varieties. There exist projective

k-varieties X and Y such that X is an open subset of X, and Y is an open subsetof Y . After possibly modifying X and Y by blowing up X −X and Y −Y , we mayassume that F1 = X −X and F2 = Y − Y are closed subsets of pure codimension1 in X, Y respectively.

Let DX be the Zariski closure of DX in X, DY be the Zariski closure of DY inY .

217

218 17. PROOFS OF THE MAIN RESULTS

Let DX = DX + F1, DY = DY + F2. By resolution of indeterminancy, afterpossibly modifying X by blowing up subvarieties of X supported above DX , wehave that the rational map f : X → Y which extends f : X → Y is a morphism.

The hypotheses of Theorem 1.2 are satisfied for f : X → Y , so by the first partof this proof, there exists a commutative diagram

X1f1→ Y 1

Φ ↓ ↓ Ψ

Xf→ Y

satisfying the conclusions of Theorem 1.2.Let X1 = Φ

−1(X), Y1 = Ψ

−1(Y ), f1 = f1 | X1, Φ = Φ | D1, Ψ = Ψ | Y1. Then

X1f1→ Y1

Φ ↓ ↓ Ψ

Xf→ Y

satisfies the conclusions of Theorem 1.2.

The proof of Theorem 1.1 is a simplification of the proof of Theorem 1.2.

CHAPTER 18

List of technical terms

(See also Chapter 3, Notation)

admissible center: Definition 12.2Fg: Chapter 3, Notationfundamental locus: Chapter 3, NotationGX(f, τ): Definition 9.4GY (f, τ): Definition 9.4good at p for f : Definition 11.1monomial form: Definition 4.4NFf : Chapter 3, Notationnonfinite locus: Chapter 3, Notationperfect for f : Definition 11.9.permissible center: Definition 13.11*-permissible center: Definition 13.12permissible parameters: before and after Definition 4.1possible center: Chapter 3, Notationprepared point or curve of type 1-5: Definition 13.7prepared curve of type 6: Definition 13.8prepared morphism

Prepared morphism from 3-fold to surface: before Remark 4.5prepared morphism of 3-folds: Definition 4.6τ -prepared morphism: Definition 9.4pre-τ -quasi-well prepared: Definition 13.1τ -quasi-well prepared: Definition 13.1τ -well prepared: Definition 13.5τ -very-well prepared: Definition 13.9

relationquasi-pre-relation: Definition 12.1pre-relation: Definition 12.3algebraic pre-relation: Definition 12.4primitive relation: Definition 12.5relation: Definition 12.5algebraic relation: Definition 12.5

resolved quasi-pre-relation: after Definition 12.1resolved relation: after Definition 12.5resolving curve: Definition 13.6Σ(Y ): Chapter 3, Notationsuper parameters: Definition 10.1τf (p): Definition 9.1

219

220 18. LIST OF TECHNICAL TERMS

τf (X): after Definition 9.1τ -quasi-well prepared diagram: after Definitions 13.11 and 13.13Θ(f, Y ): Definition 11.11toroidal forms for u, v, w: after Definition 4.3toroidal forms for u, v: Definition 4.1toroidal ideal: Chapter 3, Notationtorodial morphism: Definition 4.3transform of a pre-relation: after Definition 12.2transform of a relation: after Definition 12.6uniformizing parameters: Chapter 3, Notationweakly good at p for f : Definition 11.2Zariski-Riemann manifold: Chapter 3, Notation

Bibliography

[Ab1] Abhyankar, S., On the valuations centered in a local domain, Amer. J. Math. 78 (1956),321 – 348.

[Ab2] Abhyankar, S., Algebraic Geometry for Scientists and Engineers, Amer. Math. Soc., 1990.[Ab3] Abhyankar, S., Resolution of Singularities of Embedded Surfaces, Academic Press, New

York, 1966.[Ab4] Abhyankar, S., On the ramification of algebraic functions, I., Amer. J. Math. 77 (1955).[AK] Abramovich D., Karu K., Weak semistable reduction in characteristic 0, Invent. Math. 139

(2000), 241 – 273.[AKMW] Abramovich, D., Karu, K., Matsuki, K. and Wlodarczyk, J., Torification and factor-

ization of birational maps, JAMS 15 (2002), 531 – 572.[AMR] Abramovich, D., Matsuki, K., Rashid, S., A note on the factorization theorem of toric

birational maps after Morelli and its toroidal extension, Tohoku Math J. 51 (1999), 489 –537, Correction: Tohoku Math J. 52 (2000), 629 – 631.

[AkK] Akbulut, S. and King, H., Topology of algebraic sets, MSRI publications 25, Springer-Verlag, Berlin.

[BaM] Bartlet, D., Maire, H.M., Asymptotique des integrales-fibres, Ann. Inst. Fourier 43, 1267 -1299 (1993).

[BrM] Bierstone, E. and Millman, P., Canonical desingularization in characteristic zero by blow-ing up the maximal strata of a local invariant, Inv. Math 128 (1997), 207 – 302.

[BEV] Bravo, A., Encinas, S., Villamayor, O.,A simplified proof of desingularization and appli-cations, to appear in Revista Matematica Iberamericana.

[Ch] Christensen, C., Strong domination/weak factorization of three dimensional regular localrings, Journal of the Indian Math. Soc., 45 (1981), 21 – 47.

[Cr] Crauder, B., Two reduction theorems for threefold birational morphisms, Math Ann. 260(1984), 13 – 26.

[C1] Cutkosky, S.D., Local factorization of birational maps, Advances in Mathematics 132 (1997),167 – 315.

[C2] Cutkosky, S.D., Local monomialization and factorization of morphisms, Asterisque 260,1999.

[C3] Cutkosky, S.D., Monomialization of Morphisms from 3-folds to surfaces, Lecture Notes inMathematics 1786, Springer-Verlag, Berlin, Heidelberg, New York, 2002.

[C4] Cutkosky, S.D., Local monomialization of trancendental extensions, Annales de L’InstitutFourier 55 (2005), 1517 – 1586.

[C5] Cutkosky, S.D., Toroidalization of birational morphisms of 3-folds, AG/0407258.[C6] Cutkosky, S.D. Resolution of Singularities, American Mathematical Society, 2004.[C7] Cutkosky, S.D., Strong Toroidalization of birational morphisms of 3-folds, preprint,

AG/0412497.[C8] Cutkosky, S.D., Strong Toroidalization of dominant morphisms of 3-folds, preprint,

AG/0601037.[CK] Cutkosky, S.D., Kascheyeva, O., Monomialization of strongly prepared morphisms from

nonsingular n-folds to surfaces J. Algebra (2004), 275-320.[CP] Cutkosky, S.D. and Piltant, O., Monomial resolutions of morphisms of algebraic surfaces,

Comm. in Alg. 28 (2000), 5935 – 5959.[CS] Cutkosky, S.D. and Srinivasan, H. Factorizations of birational extensions of local rings,

preprint, AC/0601393.[D1] Danilov, V., Birational geometry of toric 3-folds, Math USSR Izv. 21 (83), 269 – 280.

221

222 BIBLIOGRAPHY

[D2] V. Danilov. Decomposition of some birational morphisms, Izv. Akad. Nauk SSSR 44 (1980),465-477.

[EH] Encinas, S., Hauser, H., Strong resolution of singularities in characteristic zero, CommentMath. Helv. 77 (2002), 821 – 845.

[E] Ewald, E., Blow ups of smooth toric 3-varieties, Abh. math. Sem. Univ. Hamburg 57 (1987).[F] Fulton, W. Introduction to Toric Varieties, Annals of Math. Studies, Princeton Univ. Press,

Princeton, 1993.[Go] R. Goward. A simple algorithm for the principalization of monomial ideals, Transactions of

the AMS 357 (2005), 4805-4812.[G] Grothendieck, A., Revetements etales et groupe fondamental (SGA1), Lecture notes in math-

ematics 176, Springer-Verlag, Berlin, Heidelberg, New York, 1970.[H] Hironaka, H., Resolution of singularities of an algebraic variety over a field of characteristic

zero, Annals of Math, 79 (1964), 109 – 326.[K] Karu, K., Local strong factorization of birational maps, J. Alg. Geom 14 (2005), 165 – 175.[KKMS] Kempf, G., Knudsen, F., Mumford, D., Saint-Donat, B., toroidal embeddings I, LNM

339, Springer Verlag (1973).[L] Lichtin, B., On a question of Igusa, II, uniform asymptotic bounds for Fourier transforms in

seceral variables, Compositio Math. 141 (2005), 192 -2006.[Li] Lipman, J., Appendix to Chapter II in Zariski, O., Algebraic Surfaces, second supplemented

edition, Springer-Verlag, New York, Heidelberg, Berlin, 1971.[Mo] Morelli, R., The birational geometry of toric varieties, J. Algebraic Geometry 5 (1996), 751

– 782.[N] Nagata, M., Local Rings, John Wiley and Sons, Inc. (Interscience Publishers), New York,

1962.[O] Oda, T., Torus embeddings and applications, TIFR, Bombay, 1978.[S] Sally, J., Regular overrings of regular local rings, Trans. Amer. Math. Soc. 171 (1972) 291 –

300.[Sh] Shannon, D.L., Monoidal transforms, Amer. J. Math 45 (1973), 284 – 320.[W1] Wlodarcyzk, J., Decomposition of birational toric maps in blowups and blowdowns, Trans.

Amer. Math. Soc. 349 (1997), 373-411.[Z] Zariski, O., The compactness of the Riemann manifold of an abstract field of algebraic func-

tions, Bull. Amer. Math. Soc., 45 (1044), 683 – 691.[Z1] Zariski, O., Introduction to the problem of minimal models in the theory of algebraic surfaces,

Publications of the Math. Soc. of Japan, 1958.[Z2] Zariski, O., Algebraic Surfaces, second supplemented edition, Springer-Verlag, New York,

Heidelberg, Berlin, 1971.[ZS] Zariski, O. and Samuel P., Commutative Algebra Volume II, Van Nostrand, Princeton, 1960.

Toroidalization of Dominant Morphisms of 3-Folds Steven Dale

Documents

Transcript of Toroidalization of Dominant Morphisms of 3-Folds Steven Dale