Fitting ideals and multiple points of surfaceparameterizations

Laurent Buse

Galaad, INRIA Sophia AntipolisLaurent.Buse@inria.fr

June 4, 2013MEGA conference

Joint work with M. Chardin (University of Paris VI) and N. Botbol

(University of Buenos Aires).

The problem

Given a surface parameterization (over an alg. closed field K)

φ : P2K 99K P3

K

We would like to have a computational description of the sets

Multiple points locci

∆k(φ) = {P ∈ P3 : “ φ−1(P) has at least k points ”}

Motivation : determination of singular points of parameterizations forapplications in geometric modeling.

Approach : give a scheme structure to ∆k(φ) by means of Fitting ideals

Remark : Observe that ∆1(φ) corresponds to the image of φ.

The problem

Given a surface parameterization (over an alg. closed field K)

φ : P2K 99K P3

K

We would like to have a computational description of the sets

Multiple points locci

∆k(φ) = {P ∈ P3 : “ φ−1(P) has at least k points ”}

Motivation : determination of singular points of parameterizations forapplications in geometric modeling.

Approach : give a scheme structure to ∆k(φ) by means of Fitting ideals

Remark : Observe that ∆1(φ) corresponds to the image of φ.

The problem

Given a surface parameterization (over an alg. closed field K)

φ : P2K 99K P3

K

We would like to have a computational description of the sets

Multiple points locci

∆k(φ) = {P ∈ P3 : “ φ−1(P) has at least k points ”}

Motivation : determination of singular points of parameterizations forapplications in geometric modeling.

Approach : give a scheme structure to ∆k(φ) by means of Fitting ideals

Remark : Observe that ∆1(φ) corresponds to the image of φ.

The problem

Given a surface parameterization (over an alg. closed field K)

φ : P2K 99K P3

K

We would like to have a computational description of the sets

Multiple points locci

∆k(φ) = {P ∈ P3 : “ φ−1(P) has at least k points ”}

Motivation : determination of singular points of parameterizations forapplications in geometric modeling.

Approach : give a scheme structure to ∆k(φ) by means of Fitting ideals

Remark : Observe that ∆1(φ) corresponds to the image of φ.

Fitting image (after Teissier)

How to give a scheme structure to the image of the following map ?:

ψ : A1 → A2 : t 7→ (t2, t3).

Algebraically, ψ corresponds to h : K[x , y ]→ K[t].

I Classical image: subscheme of A2 defined by the ideal

ker(h) = {P(x , y) : P(t2, t3) = 0} = annK[x,y ](K[t]).

I Fitting image: subscheme of A2 defined by the initial Fitting ideal ofK[t] seen as a K[x , y ]-module (direct image):

K[x , y ]2

−y x2

x −y

−−−−−−−−−−−→ K[x , y ]2 → K[t]→ 0

Fitting image (after Teissier)

How to give a scheme structure to the image of the following map ?:

ψ : A1 → A2 : t 7→ (t2, t3).

Algebraically, ψ corresponds to h : K[x , y ]→ K[t].

I Classical image: subscheme of A2 defined by the ideal

ker(h) = {P(x , y) : P(t2, t3) = 0} = annK[x,y ](K[t]).

I Fitting image: subscheme of A2 defined by the initial Fitting ideal ofK[t] seen as a K[x , y ]-module (direct image):

K[x , y ]2

−y x2

x −y

−−−−−−−−−−−→ K[x , y ]2 → K[t]→ 0

Fitting image (after Teissier)

How to give a scheme structure to the image of the following map ?:

ψ : A1 → A2 : t 7→ (t2, t3).

Algebraically, ψ corresponds to h : K[x , y ]→ K[t].

I Classical image: subscheme of A2 defined by the ideal

ker(h) = {P(x , y) : P(t2, t3) = 0} = annK[x,y ](K[t]).

I Fitting image: subscheme of A2 defined by the initial Fitting ideal ofK[t] seen as a K[x , y ]-module (direct image):

K[x , y ]2

−y x2

x −y

−−−−−−−−−−−→ K[x , y ]2 → K[t]→ 0

Fitting ideals

Let A be a ring and M a A-module of finite presentation:

Aq ψ−→ Ap → M → 0.

Definition

The k th Fitting ideal of M is the ideal Fk(M) generated by all the p − kminors of ψ. It does not depend on the choice of ψ.

Properties

I F0(M) ⊂ F1(M) ⊂ · · · ⊂ A.

I annA(M)p ⊂ F0(M) ⊂ annA(M)

I A→ B a ring map, then

Fk(M ⊗A B) = Fk(M).B

(Fitting ideals commute with base change)

Fitting ideals

Let A be a ring and M a A-module of finite presentation:

Aq ψ−→ Ap → M → 0.

Definition

The k th Fitting ideal of M is the ideal Fk(M) generated by all the p − kminors of ψ. It does not depend on the choice of ψ.

Properties

I F0(M) ⊂ F1(M) ⊂ · · · ⊂ A.

I annA(M)p ⊂ F0(M) ⊂ annA(M)

I A→ B a ring map, then

Fk(M ⊗A B) = Fk(M).B

(Fitting ideals commute with base change)

Some related works

I If X is a scheme and M a coherent sheaf on X , on can define thesheaves of ideals Fk(M).

I Given f : X → Y a finite map, one can consider Fk(f∗OX ).

Tessier’s idea

Define the image of a finite map as the Fitting ideal F0(f∗OX ) instead ofannOY

(f∗OX ), in order to get stability under base change.

In this settings, works by Teissier (1976), by Mond, Pellikaan (1989),and also by Piene (1978) and (Kleiman, Lipman, Ulrich 1992) inrelation with the double point formula.

The case of parameterized plane curves

The multiple points sets ∆k(φ) can be described in terms of the Fittingideals of the associated classical implicitizing Sylvester matrix.

Some related works

I If X is a scheme and M a coherent sheaf on X , on can define thesheaves of ideals Fk(M).

I Given f : X → Y a finite map, one can consider Fk(f∗OX ).

Tessier’s idea

Define the image of a finite map as the Fitting ideal F0(f∗OX ) instead ofannOY

(f∗OX ), in order to get stability under base change.

In this settings, works by Teissier (1976), by Mond, Pellikaan (1989),and also by Piene (1978) and (Kleiman, Lipman, Ulrich 1992) inrelation with the double point formula.

The case of parameterized plane curves

The multiple points sets ∆k(φ) can be described in terms of the Fittingideals of the associated classical implicitizing Sylvester matrix.

Surface parameterizationsSupose given a birational surface parameterization

φ : P2 99K P3

(s0 : s1 : s2) 7→ (f0 : f1 : f2 : f3)(s0, s1, s2)

I Notation: S := K[s0, s1, s2], I = (f0, f1, f2, f3) ⊂ Sd , d ≥ 1.I Base points: B = Proj(S/I ) ⊂ P2, assumed finite (w.l.o.g.).

Let Γ be the closure of the graph of P2 \ B φ−→ P3:

Γ

π1

��

π2

##

� � //P2 × P3

P2

φ//P3

I Fiber at a point P ∈ P3:

π−12 (P) = Γ×P3 Spec(κ(P)) ⊂ P2κ(P)

where κ(P) is the residue field.

Surface parameterizationsSupose given a birational surface parameterization

φ : P2 99K P3

(s0 : s1 : s2) 7→ (f0 : f1 : f2 : f3)(s0, s1, s2)

I Notation: S := K[s0, s1, s2], I = (f0, f1, f2, f3) ⊂ Sd , d ≥ 1.I Base points: B = Proj(S/I ) ⊂ P2, assumed finite (w.l.o.g.).

Let Γ be the closure of the graph of P2 \ B φ−→ P3:

Γ

π1

��

π2

##

� � //P2 × P3

P2

φ//P3

I Fiber at a point P ∈ P3:

π−12 (P) = Γ×P3 Spec(κ(P)) ⊂ P2κ(P)

where κ(P) is the residue field.

Blow-up algebras

Algebraically, Γ = Proj(ReesS(I )), the blow up of φ along B.

Property

If B is a local complete intersection, then

Γ = Proj(ReesS(I )) = Proj(SymS(I ))

Compared to the Rees algebra, the symmetric algebra is a more friendlyto computations:

SymS(I ) = S [x0, x1, x2, x3]/( 3∑i=0

gixi :3∑

i=0

gi fi = 0

)

⇒ from now on, B is assumed to be a local complete intersection.

Blow-up algebras

Algebraically, Γ = Proj(ReesS(I )), the blow up of φ along B.

Property

If B is a local complete intersection, then

Γ = Proj(ReesS(I )) = Proj(SymS(I ))

Compared to the Rees algebra, the symmetric algebra is a more friendlyto computations:

SymS(I ) = S [x0, x1, x2, x3]/( 3∑i=0

gixi :3∑

i=0

gi fi = 0

)

⇒ from now on, B is assumed to be a local complete intersection.

Fitting ideals associated to φ

Notation : R := K[x0, x1, x2, x3], so that P3 = Proj(R)

Projecting Γ on P3 amounts to take graded parts of SymS(I )

Definition

For all integer ν ≥ 0, we have

(Z1)ν ⊂ ⊕3i=0R[s0, s1, s2]ν

M(φ)ν−−−−→ R[s0, s1, s2]ν → SymS(I )ν → 0

(g0, g1, g2, g3) :3∑

i=0

gi fi = 0 7→3∑

i=0

gixi

I get a family of matrices M(φ)ν indexed by ν ∈ NI define the associated Fitting ideals:

∀k ∈ N : Fk(SymS(I )ν) ⊂ R.

Example: the sphere

A parameterization of the sphere:

P2 φ−→ P3

(s0 : s1 : s2) 7→(s20 + s21 + s22 : s20 − s21 − s22 : 2s0s1 : 2s0s2

)

Matrix representation of the sphere

M(φ)1 =

0 −x3 −x2 −x0 + x1−x3 0 x0 + x1 x2x2 x0 + x1 0 x3

I F0(SymS(I )1) =ideal generated by the 3-minors of M(φ)1I Fk(SymS(I )1) =ideal generated by the (3− k)-minors of M(φ)1

I F0(SymS(I )ν) =ideal generated by the(ν+22

)-minors of M(φ)ν

I etc.

Support of these Fitting ideals

For any P ∈ P3 we have

Fk(SymS(I )ν)⊗R κ(P) = Fk(SymS(I )ν ⊗R κ(P))

= Fk ((SymS(I )⊗R κ(P))ν)

Therefore, we have

Fk(SymS(I )ν)⊗R κ(P) =

{0 if 0 ≤ k < NP

ν

κ(P) if NPν < k

where NPν := HFSymS (I )⊗κ(P)(ν) (Hilbert Function of the fiber et P).

Corollary

For all ν � 0, the Fitting ideal Fk(SymS(I )ν) is supported on ∆k+1(π2)

Support of these Fitting ideals

For any P ∈ P3 we have

Fk(SymS(I )ν)⊗R κ(P) = Fk(SymS(I )ν ⊗R κ(P))

= Fk ((SymS(I )⊗R κ(P))ν)

Therefore, we have

Fk(SymS(I )ν)⊗R κ(P) =

{0 if 0 ≤ k < NP

ν

κ(P) if NPν < k

where NPν := HFSymS (I )⊗κ(P)(ν) (Hilbert Function of the fiber et P).

Corollary

For all ν � 0, the Fitting ideal Fk(SymS(I )ν) is supported on ∆k+1(π2)

Support of these Fitting ideals

For any P ∈ P3 we have

Fk(SymS(I )ν)⊗R κ(P) = Fk(SymS(I )ν ⊗R κ(P))

= Fk ((SymS(I )⊗R κ(P))ν)

Therefore, we have

Fk(SymS(I )ν)⊗R κ(P) =

{0 if 0 ≤ k < NP

ν

κ(P) if NPν < k

where NPν := HFSymS (I )⊗κ(P)(ν) (Hilbert Function of the fiber et P).

Corollary

For all ν � 0, the Fitting ideal Fk(SymS(I )ν) is supported on ∆k+1(π2)

Our main result

Theorem

Setν0 := 2d− 2− indeg(I : (s0, s1, s2)∞).

Then, for all ν ≥ ν0 and all k ∈ N, the Fitting ideal Fk(SymS(I )ν) issupported on ∆k+1(π2).

I For k = 0, we get a Fitting image for the parameterization φ.

I The proof relies on bounding regularity of the fibers in terms of theregularity of the symmetric algebra.

I The more difficult case is the case of fibers of positive dimension.

Computational aspects – drop of rank of M(φ)ν

Theorem

Let P be a point in P3. Then, for all ν ≥ ν0I finite fibers: if dimπ−12 (P) ≤ 0,

corank(M(φ)ν(P)) = deg(π−12 (P)).

I 1-dim fibers: if dimπ−12 (P) = 1,

corank(M(φ)ν(P)) = deg(π−12 (P))ν + c

I Determination of the dimension and degree of the fibers bycomparing the ranks of M(φ)ν0 and M(φ)ν0+1.

I For all integer 0 ≤ n ≤ ν0

corank(M(φ)2d−2(P)) = n⇔ π−12 (P) is finite of degree n.

Computational aspects – drop of rank of M(φ)ν

Theorem

Let P be a point in P3. Then, for all ν ≥ ν0I finite fibers: if dimπ−12 (P) ≤ 0,

corank(M(φ)ν(P)) = deg(π−12 (P)).

I 1-dim fibers: if dimπ−12 (P) = 1,

corank(M(φ)ν(P)) = deg(π−12 (P))ν + c

I Determination of the dimension and degree of the fibers bycomparing the ranks of M(φ)ν0 and M(φ)ν0+1.

I For all integer 0 ≤ n ≤ ν0

corank(M(φ)2d−2(P)) = n⇔ π−12 (P) is finite of degree n.

Come back to the sphere

P2 φ−→ P3

(s0 : s1 : s2) 7→(s20 + s21 + s22 : s20 − s21 − s22 : 2s0s1 : 2s0s2

)

M(φ)1 =

0 −x3 −x2 −x0 + x1−x3 0 x0 + x1 x2x2 x0 + x1 0 x3

I F0(M(φ)1) defines the sphere plus an embedded pointQ = (1 : −1 : 0 : 0) with multiplicity 2.

I F1(M(φ)1) defines Q with multiplicity 1

I F2(M(φ)1) defines the empty set

The fiber at Q is a line

because corank(M(φ)1(Q)) = 2 and corank(M(φ)2(Q)) = 3.

Come back to the sphere

P2 φ−→ P3

(s0 : s1 : s2) 7→(s20 + s21 + s22 : s20 − s21 − s22 : 2s0s1 : 2s0s2

)

M(φ)1 =

0 −x3 −x2 −x0 + x1−x3 0 x0 + x1 x2x2 x0 + x1 0 x3

I F0(M(φ)1) defines the sphere plus an embedded pointQ = (1 : −1 : 0 : 0) with multiplicity 2.

I F1(M(φ)1) defines Q with multiplicity 1

I F2(M(φ)1) defines the empty set

The fiber at Q is a line

because corank(M(φ)1(Q)) = 2 and corank(M(φ)2(Q)) = 3.