ACC FOR MLD’S FOR SURFACES ACCORDING TO …ACC FOR MLD’S FOR SURFACES ACCORDING TO SHOKUROV 3...

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arXiv:2005.09626v2 [math.AG] 26 Jul 2020 ON BOUNDEDNESS OF DIVISORS COMPUTING MINIMAL LOG DISCREPANCIES FOR SURFACES JINGJUN HAN AND YUJIE LUO Dedicated to Vyacheslav Shokurov with our deepest gratitude on the occasion of his seventieth birthday Abstract. Let Γ be a finite set, and X x a fixed klt germ. For any lc germ (X x, B := i bi Bi ) such that bi Γ, Nakamura’s conjecture, which is equivalent to the ACC conjecture for minimal log discrepancies for fixed germs, predicts that there always exists a prime divisor E over X x, such that a(E,X,B) = mld(X x, B), and a(E,X, 0) is bounded from above. We extend Nakamura’s conjecture to the setting that X x is not necessarily fixed and Γ satisfies the DCC, and show it holds for surfaces. We also find some sufficient conditions for the boundedness of a(E,X, 0) for any such E. Contents 1. Introduction 1 2. Sketch of the proof 4 3. Preliminaries 8 3.1. Arithmetic of sets 8 3.2. Singularities of pairs 9 3.3. Dual graphs 9 3.4. Sequence of blow-ups 13 3.5. Extracting divisors computing MLDs 14 4. Proof of Theorem 1.3 17 5. Proof of Theorem 1.2 21 6. Proof of Theorem 1.4 23 7. An equivalent conjecture for MLDs on a fixed germ 28 Appendix A. ACC for MLDs for surfaces according to Shokurov 31 References 36 1. Introduction We work over the field of complex numbers C. The minimal log discrepancy (MLD for short) introduced by Shokurov is an important invariant in birational geometry. Shokurov conjectured that 2010 Mathematics Subject Classification. 14E30, 14J17. 1

Transcript of ACC FOR MLD’S FOR SURFACES ACCORDING TO …ACC FOR MLD’S FOR SURFACES ACCORDING TO SHOKUROV 3...

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ON BOUNDEDNESS OF DIVISORS COMPUTING

MINIMAL LOG DISCREPANCIES FOR SURFACES

JINGJUN HAN AND YUJIE LUO

Dedicated to Vyacheslav Shokurov with our deepest gratitude on the occasion of his

seventieth birthday

Abstract. Let Γ be a finite set, and X ∋ x a fixed klt germ. For anylc germ (X ∋ x,B :=

∑ibiBi) such that bi ∈ Γ, Nakamura’s conjecture,

which is equivalent to the ACC conjecture for minimal log discrepanciesfor fixed germs, predicts that there always exists a prime divisor E

over X ∋ x, such that a(E,X,B) = mld(X ∋ x,B), and a(E,X, 0) isbounded from above. We extend Nakamura’s conjecture to the settingthat X ∋ x is not necessarily fixed and Γ satisfies the DCC, and showit holds for surfaces. We also find some sufficient conditions for theboundedness of a(E,X, 0) for any such E.

Contents

1. Introduction 12. Sketch of the proof 43. Preliminaries 83.1. Arithmetic of sets 83.2. Singularities of pairs 93.3. Dual graphs 93.4. Sequence of blow-ups 133.5. Extracting divisors computing MLDs 144. Proof of Theorem 1.3 175. Proof of Theorem 1.2 216. Proof of Theorem 1.4 237. An equivalent conjecture for MLDs on a fixed germ 28Appendix A. ACC for MLDs for surfaces according to Shokurov 31References 36

1. Introduction

We work over the field of complex numbers C.The minimal log discrepancy (MLD for short) introduced by Shokurov is

an important invariant in birational geometry. Shokurov conjectured that

2010 Mathematics Subject Classification. 14E30, 14J17.

1

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2 J.Han, and Y.Luo

the set of MLDs should satisfy the ascending chain condition (ACC) [12,Problem 5], and proved that the conjecture on termination of flips in theminimal model program (MMP) follows from two conjectures on MLDs:the ACC conjecture for MLDs and the lower-semicontinuity conjecture forMLDs [15].

The ACC conjecture for MLDs is still widely open in dimension 3 ingeneral. We refer readers to [6] and references therein for a brief history andrelated progress. In order to study this conjecture for the case when X ∋ xis a fixed klt germ, Nakamura proposed Conjecture 1.1. It is proved byMustata-Nakamura [11, Theorem 1.5] and Kawakita [8, Theorem 4.6] thatConjecture 1.1 is equivalent to the ACC conjecture for MLDs in this case.

Conjecture 1.1 ([11, Conjecture 1.1]). Let Γ ⊆ [0, 1] be a finite set andX ∋ x a klt germ. Then there exists an integer N depending only on X ∋ xand Γ satisfying the following.

Let (X ∋ x,B) be an lc germ such that B ∈ Γ. Then there exists aprime divisor E over X ∋ x such that a(E,X,B) = mld(X ∋ x,B) anda(E,X, 0) ≤ N.

When dimX = 2, Conjecture 1.1 is proved by Mustata-Nakamura [11,Theorem 1.3], and Alexeev proved it when Γ satisfies the descending chaincondition (DCC) [1, Lemma 3.7] as one of key steps in his proof of the ACCfor MLDs for surfaces. See [4, Theorem B.1] for a proof of Alexeev’s resultin details. Kawakita proved Conjecture 1.1 for the case when dimX = 3, Xis smooth, Γ ⊆ Q, and (X ∋ x,B) is canonical [8, Theorem 1.3].

Naturally, one may ask whether Nakamura’s conjecture holds or not whenX ∋ x is not necessarily fixed and Γ satisfies the DCC. If the answer is yes,the boundedness conjecture of MLDs will be an immediate corollary. Themain goal of this paper is to give a positive answer to this question indimension 2.

Theorem 1.2. Let Γ ⊆ [0, 1] be a set which satisfies the DCC. Then thereexists an integer N depending only on Γ satisfying the following.

Let (X ∋ x,B) be an lc surface germ such that B ∈ Γ. Then there existsa prime divisor E over X ∋ x such that a(E,X,B) = mld(X ∋ x,B) anda(E,X, 0) ≤ N .

Theorem 1.2 implies that for any surface germ (X ∋ x,B) such thatB ∈ Γ, there exists a prime divisor E which could compute mld(X ∋ x,B)and is “weakly bounded”, in the sense that a(E,X, 0) is uniformly boundedfrom above, among all divisorial valuations.

A natural idea to prove Theorem 1.2 is to take the minimal resolution

f : X → X, and apply Conjecture 1.1 in dimension 2. However, the coef-ficients of B

Xmay not belong to a DCC set anymore, where K

X+ B

X:=

f∗(KX +B). In order to resolve this difficulty, we need to show Nakamura’sconjecture for the smooth surface germ while the set of coefficients Γ does

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On boundedness of divisors computing MLDs for surfaces 3

not necessarily satisfy the DCC. It would be interesting to ask if similarapproaches could be applied to solve some questions in birational geometryin high dimensions, that is we solve these questions in a more general settingof coefficients on a terminalization of X.

Theorem 1.3. Let γ ∈ (0, 1] be a real number. Then N0 := ⌊1 + 32γ2 + 1

γ⌋

satisfies the following.Let (X ∋ x,B :=

∑i biBi) be an lc surface germ, where X ∋ x is smooth,

and Bi are distinct prime divisors. Suppose that {∑

i nibi − 1 > 0 | ni ∈Z≥0} ⊆ [γ,+∞). Then there exists a prime divisor E over X such thata(E,X,B) = mld(X ∋ x,B), and a(E,X, 0) ≤ 2N0 .

We remark that for any DCC set Γ, there exists a positive real numberγ, such that {

∑i nibi − 1 > 0 | ni ∈ Z≥0, bi ∈ Γ} ⊆ [γ,+∞) (see Lemma

3.2). However, the converse is not true. For example, the set Γ := [12 +γ2 , 1]

satisfies our assumption, but it is not a DCC set. It is also worthwhileto remark that all previous works we mentioned before did not give anyeffective bound even when Γ is a finite set.

Theorem 1.3 indicates that there are some differences between Conjecture1.1 and the ACC conjecture for MLDs as the latter does not hold for suchkind of sets.

When we strengthen the assumption “{∑

i nibi − 1 > 0 | ni ∈ Z≥0} ⊆[γ,+∞)” to “{

∑i nibi − 1 ≥ 0 | ni ∈ Z≥0} ⊆ [γ,+∞)”, we may give an

explicit upper bound for a(E,X, 0) for all prime divisors E over X ∋ x,such that mld(X ∋ x,B) = a(E,X,B). We remark that Γ := [12 + γ

2 , 1)

satisfies the assumption in Theorem 1.4, while Γ := [12 + γ2 , 1] does not.

Theorem 1.4. Let γ ∈ (0, 1] be a real number. Then N0 := ⌊1 + 32γ2 + 1

γ⌋

satisfies the following.Let (X ∋ x,B :=

∑i biBi) be an lc surface germ, such that X ∋ x is

smooth, where Bi are distinct prime divisors. Suppose that {∑

i nibi − 1 ≥0 | ni ∈ Z≥0} ⊆ [γ,+∞). Then

(1) |S| ≤ N0, where S := {E | E is a prime divisor over X ∋ x, a(E,X,B) = mld(X ∋ x,B)}, and

(2) a(E,X, 0) ≤ 2N0 for any E ∈ S.

It is clear that (A2, B1+B2) satisfies the assumptions in Theorem 1.3 but|S| = +∞, whereB1 and B2 are defined by x = 0 and y = 0 respectively. Forklt germs (X,B), we always have |S| < +∞. In this case, Example 4.3 showsthat Theorem 1.3 does not hold for the set Γ := {1

2} ∪ {12 +

1k+1 | k ∈ Z≥0},

and Example 6.9 shows Theorem 1.4 may fail when bi =12 for any i, that is

the equality for “∑

i nibi−1 ≥ 0” is necessary. These examples also indicatethat the assumptions in Theorem 1.3 and Theorem 1.4 might be optimal.

We also give an effective bound for Theorem 1.3 (respectively Theorem1.4) when X ∋ x is a fixed lc surface germ (respectively X ∋ x is a fixed kltsurface germ), see Theorem 6.4 (respectively Theorem 6.5).

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4 J.Han, and Y.Luo

Another key ingredient in the proof of Theorem 1.2 is the ACC for MLDsfor surfaces.

Theorem 1.5 ([1, Theorem 3.6],[14]). Let Γ ⊆ [0, 1] be a set which satisfiesthe DCC. Then the set of minimal log discrepancies

Mld(2,Γ) := {mld(X ∋ x,B) | (X ∋ x,B) is lc,dimX = 2, B ∈ Γ}

satisfies the ACC.

Theorem 1.5 was proved by Alexeev [1] and Shokurov [14] independently.We refer readers to [4, Lemma 4.5, Theorem B.1, Theorem B.4] for a prooffollowing Alexeev’s arguments in details. Shokurov’s preprint was neverpublished. In this paper, we will give a simple proof following his idea.Yusuke Nakamura informed us that, together with Weichung Chen, andYoshinori Gongyo, they have another proof of Theorem 1.5 which is alsoinspired by Shokurov’s preprint [5, Theorem 1.5].

Structure of the paper. In Section 2, we give a sketch of the proofs ofTheorem 1.3, Theorem 1.2, and Theorem 1.5. In Section 3, we introducesome notation and tools which will be used in this paper, and prove certainresults. In Section 4, we prove Theorem 1.3. In Section 5, we prove Theorem1.2. In Section 6, we prove Theorem 1.4, Theorem 6.4 and Theorem 6.5.In Section 7, we introduce a conjecture on a fixed germ, and show it isequivalent to the ACC conjecture for MLDs. In Appendix A, we proveTheorem 1.5.

Acknowledgments. We would like to thank Vyacheslav Shokurov for shar-ing with us his preprint [14]. We would like to thank Guodu Chen, JihaoLiu, Wenfei Liu, Yuchen Liu, and Yusuke Nakmura for their interests andhelpful comments. Part of this paper was written during the reading sem-inar on birational geometry at Johns Hopkins University, Spring 2019. Wewould like to thank Yang He and Xiangze Zeng for joining in the seminarand useful discussions.

2. Sketch of the proof

In this section, we give a brief account of some of the ideas in the proofsof Theorem 1.3, Theorem 1.2, and Theorem 1.5.

Sketch of the proof of Theorem 1.3. We may assume that B 6= 0, and byLemma 3.15, we may assume that mld(X ∋ x,B) ≤ 1. First, we extracta divisor E0 on some smooth model Y0 over X, such that a(E0,X,B) =mld(X ∋ x,B). Next, we may construct a sequence of MMPs, contractsome other exceptional divisors on Y0 which are “redundant”, and reach a“good” smooth model f : Y → X such that E0 is the only f -exceptional(−1)-curve, and the dual graph DG of f is a chain, see Lemma 3.18.

It suffices to find an upper bound of the number of vertices of DG. Let{Ei}−n1≤i≤n2

be the vertices of DG, n := n1 + n2 +1, wi := −(Ei ·Ei), and

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On boundedness of divisors computing MLDs for surfaces 5

ai := a(Ei,X,B) for −n1 ≤ i ≤ n2. We may assume that Ei is adjacent toEi+1. There are two cases.

Case 1. If n1 = 0, that is E0 is adjacent to only one vertex E1, then

a1 − a0 = f−1∗ B ·E0 − 1 ∈ {

i

nibi − 1 > 0 | ni ∈ Z≥0} ⊆ [γ,+∞).

Thus ai+1 − ai ≥ a1− a0 ≥ γ for any 0 ≤ i ≤ n2 − 1, and 1 ≥ an2≥ n2γ. So

n2 ≤1γ.

E0 E1 E2 En2

Case 2. E0 is adjacent to two vertices, that is E−1 and E1. If mld(X ∋x,B) has a positive lower bound, say mld(X ∋ x,B) ≥ γ

2 , then wi ≤ 4γ,

and we may show that DG belongs to a finite set depending only on γ, seeLemma 3.19.

The hard part is when mld(X ∋ x,B) < γ2 as we could not bound wi.

Recall that Y := Xn → X := X0 is the composition of a sequence ofblow-ups fi : Xi → Xi−1, such that fi is the blow-up of Xi−1 at a closedpoint xi−1 ∈ Xi−1 with the exceptional divisor Fi, and xi ∈ Fi for any1 ≤ i ≤ n− 1, where F0 = ∅, x0 := x. A key observation (see Lemma 4.1) isthat we may show

n ≤ n3 +min{W1(g),W2(g)},

where n3 is the largest integer such that xi ∈ Fi\Fi−1 for any 1 ≤ i ≤ n3−1,W1(g) :=

∑j<0wj and W2(g) :=

∑j>0wj .

E−n1E−1 E0 E1

Fn3−1

En2

F1

Since 0 < (a−1 − a0) + (a1 − a0) = f−1∗ B · E0 − a0, f

−1∗ B · E0 − a0 ≥

γ − γ2 = γ

2 by our assumption (see Lemma 3.5). Thus we may assume thata−1 − a0 = max{a−1 − a0, a1 − a0} ≥ γ

4 . Again, we may bound n1 as wellas wi for any i < 0. In particular, we may bound min{W1(g),W2(g)}.

Thus it is enough to bound n3. Let BXibe the strict transform of B

on Xi for 0 ≤ i ≤ n3, and let a′i := a(Fi,X,B) for 1 ≤ i ≤ n3, anda′0 := 1. Since xi ∈ Fi \ Fi−1, we may show a′i − a′i+1 = multxi

BXi− 1 ≥

min{a1 − a0, a−1 − a0} > 0 for any 0 ≤ i ≤ n3 − 2. Hence a′i − a′i+1 ≥ γ as{∑

i nibi − 1 > 0 | ni ∈ Z≥0} ⊆ [γ,+∞). Therefore,

0 ≤ a′n3−1 = a′0 +

n3−2∑

i=0

(a′i+1 − a′i) ≤ 1− (n3 − 1)γ,

and n3 ≤ 1 + 1γ.

Sketch of the proof of Theorem 1.2. We may assume that B 6= 0, and X ∋ x

is not a smooth germ. Let f : X → X ∋ x be the minimal resolution, such

that KX+B

X= f∗(KX +B) for some B

X=

∑i biBi ≥ 0.

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6 J.Han, and Y.Luo

If X ∋ x is a fixed germ, then bi belong to a DCC set (respectively afinite set if Γ is finite), and we are done by Theorem 1.3. However, the

coefficients bi do not even satisfy {∑

i nibi − 1 > 0 | ni ∈ Z≥0} ⊆ [γ,+∞)for any positive real number γ in general. For example, when X ∋ x areDu Val singularities. Thus Theorem 1.2 does not follow from Theorem 1.3directly.

We may assume that mld(X ∋ x,B) 6= pld(X ∋ x,B), where pld(X ∋x,B) := mini{1 − bi} is the partial log discrepancy of (X ∋ x,B). Again,we may construct a good resolution f : Y → X, such that E0 is the onlyf -exceptional (−1)-curve and a(E0,X,B) = mld(X ∋ x,B), and the dualgraph DG of f is a chain, see Lemma 3.18. We observed that the center of

all exceptional divisors of g : Y → X is a unique closed point x, which must

lie on a f -exceptional divisor E, such that a(E,X,B) = pld(X ∋ x,B), seeLemma 3.18(5).

By the ACC for MLDs for surfaces, or more precisely, the ACC for pld’s

for surfaces (c.f. Theorem 2.2), b := multEB

X= 1− pld(X ∋ x,B) belong

to a DCC set. Thus if x does not belong to any other f -exceptional divisor,

then we may replace (X ∋ x,B) by (X ∋ x, f−1∗ B+ bE), and we are done by

Theorem 1.3. We remark that even for this simple case, we need Theorem1.3 for any DCC set Γ rather than any finite set Γ.

Hence we may assume that x ∈ E′ ∩ E for some unique f -exceptional

divisor E′. Let b′ := multE′

BX

= 1− a(E′,X,B).We may show that there exist ǫ, γ ∈ (0, 1] depending only on Γ, such that

if a(E′,X,B)− a(E,X,B) < ǫ, or equivalently b− ǫ ≤ b′ ≤ b, then

{∑

i

nibi + nb+ n′b′ − 1 > 0 | ni, n, n′ ∈ Z≥0} ⊆ [γ,+∞),

see Lemma 3.4. Thus we may replace (X ∋ x,B) by (X ∋ x, f−1∗ B + bE +

b′E′), and we are done by Theorem 1.3. This is why we need to show Naka-mura’s conjecture for any set that satisfies {

∑i nibi − 1 > 0 | ni ∈ Z≥0} ⊆

[γ,+∞) rather than any DCC set. And when a(E′,X,B)− a(E,X,B) ≥ ǫ,we may show that mult

E′B

Xbelongs to a DCC set depending only on Γ.

Again, we are done by Theorem 1.3.

Sketch of the proof of Theorem 1.5. Let (X ∋ x,B :=∑

i biBi) be anlc surface germ. It is well known that if mld(X ∋ x,B) > 1, then X issmooth near x, and mld(X ∋ x,B) = 2 −

∑nibi satisfies the ACC, where

ni ∈ Z≥0 (c.f. Lemma 3.15). Thus it suffices to consider the case whenmld(X ∋ x,B) ≤ 1. We may assume that mld(X ∋ x,B) ≥ ǫ0 for somefixed ǫ0 ∈ (0, 1).

Before giving the main ingredients in the proof of Theorem 1.5 for thiscase in our paper, we would like to give a brief sketch of both Alexeev’sapproach and Shokurov’s idea.

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On boundedness of divisors computing MLDs for surfaces 7

Roughly speaking, Alexeev’s approach is from the bottom to the top. Asone of key steps in Alexeev’s proof [1, Theorem 3.8], he claimed that if thenumber of vertices of the dual graph of the minimal resolution of X ∋ x islarger enough, then pld(X ∋ x,B) = mld(X ∋ x,B), see [4, Lemma 4.5] fora precise statement. Hence it suffices to show:

(1) the ACC for pld’s, and(2) the ACC for MLDs for any fixed germ X ∋ x.

For (1), by the classification of surface singularities and results in linearalgebra, Alexeev showed that there is an explicit formula for computingpld(X ∋ x,B), and then the ACC for pld’s follows from it (see [1, Lemma3.3]). For (2), let E be an exceptional divisor over X ∋ x, such thata(E,X,B) = mld(X ∋ x,B), it is enough to show Conjecture 1.1 for smoothsurface germs when Γ is a DCC set.

Shokurov’s idea is that we should prove Theorem 1.5 from the top to thebottom, that is we should construct some resolution f : Y → X, such thata(E0,X,B) = mld(X ∋ x,B) for some f -exceptional divisor E0 on Y , andthen we should reduce Theorem 1.5 to the case when the dual graph of fbelongs to a finite set.

In this paper, we follow Shokurov’s idea and construct such a model byLemma 3.18. Again, E0 is the only f -exceptional (−1)-curve as well as theonly f -exceptional divisor such that a(E0,X,B) = mld(X ∋ x,B) providedthat mld(X ∋ x,B) 6= pld(X ∋ x,B). We may show that either the dualgraph of f : Y → X belongs to a finite set (see Theorem 2.1) or f is theminimal resolution of X ∋ x. As the final step, we will show the set ofMLDs satisfies the ACC in both cases. The latter case is the so called “theACC for pld’s for surfaces”.

Theorem 2.1 could be regarded as a counterpart of Alexeev’s claim whenpld(X ∋ x,B) 6= mld(X ∋ x,B).

Theorem 2.1. Let ǫ0 ∈ (0, 1) be a real number, and Γ ⊆ [0, 1] a set whichsatisfies the DCC. Then there exists a finite set of dual graphs {DGi}1≤i≤N ′

which only depends on ǫ0 and Γ satisfying the following.If (X ∋ x,B) is an lc surface germ, such that B ∈ Γ, mld(X ∋ x,B) 6=

pld(X ∋ x,B), and mld(X ∋ x,B) ∈ [ǫ0, 1], then there exists a projec-tive birational morphism f : Y → X, such that a(E,X,B) = mld(X ∋x,B) for some f -exceptional divisor E, and the dual graph of f belongs to{DGi}1≤i≤N ′ . Moreover, mld(X ∋ x,B) belongs to an ACC set which onlydepends on Γ.

We will give another proof of the ACC for pld’s for surfaces which doesnot rely on the explicit formula for pld’s.

Theorem 2.2. Let Γ ⊆ [0, 1] be a set which satisfies the DCC. Then

Pld(2,Γ) := {pld(X ∋ x,B) | (X ∋ x,B) is lc,dimX = 2, B ∈ Γ},

satisfies the ACC.

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8 J.Han, and Y.Luo

3. Preliminaries

We adopt the standard notation and definitions in [10], and will freelyuse them.

3.1. Arithmetic of sets.

Definition 3.1 (DCC and ACC sets). We say that Γ ⊆ R satisfies thedescending chain condition (DCC) or Γ is a DCC set if any decreasingsequence a1 ≥ a2 ≥ · · · in Γ stabilizes. We say that Γ satisfies the ascendingchain condition (ACC) or Γ is an ACC set if any increasing sequence in Γstabilizes.

Lemma 3.2. Let Γ ⊆ [0, 1] be a set which satisfies the DCC, and n a non-negative integer. There exists a positive real number γ which only dependson n and Γ, such that

{∑

i

nibi − n > 0 | bi ∈ Γ, ni ∈ Z≥0} ⊆ [γ,+∞).

Proof. The existence of γ follows from that the set {∑

i nibi−n | bi ∈ Γ, ni ∈Z≥0} satisfies the DCC. �

Definition 3.3. Let ǫ ∈ R, I ∈ R\{0}, and Γ ⊆ R a set of real numbers.We define Γǫ := ∪b∈Γ[b− ǫ, b], and 1

IΓ := { b

I| b ∈ Γ}.

Lemma 3.4. Let Γ ⊆ [0, 1] be a set which satisfies the DCC. Then thereexist positive real numbers ǫ, δ ≤ 1, such that

{∑

i

nib′i − 1 > 0 | b′i ∈ Γǫ ∩ [0, 1], ni ∈ Z≥0} ⊆ [δ,+∞).

Proof. We may assume that Γ \ {0} 6= ∅, otherwise we may take ǫ = δ = 1.Since Γ satisfies the DCC, by Lemma 3.2, there exists a real number

γ ∈ (0, 1] such that Γ \ {0} ⊆ (γ, 1], and {∑

i nibi − 1 > 0 | ni ∈ Z≥0, bi ∈Γ} ⊆ [γ,+∞). It suffices to prove that there exist 0 < ǫ, δ < γ

2 , such thatthe set {

∑i nib

′i − 1 ∈ (0, 1] | b′i ∈ Γǫ ∩ [0, 1], ni ∈ Z≥0} is bounded from

below by δ, or equivalently

{∑

i

nib′i − 1 > 0 | b′i ∈ Γǫ ∩ [0, 1], ni ∈ Z≥0,

i

ni ≤4

γ} ⊆ [δ,+∞).

We claim that ǫ = γ2

8 , δ = γ2 have the desired property. Let b′i ∈ Γǫ∩ [0, 1]

and ni ∈ Z≥0, such that∑

i nib′i−1 > 0 and

∑i ni ≤

4γ. We may find bi ∈ Γ,

such that 0 ≤ bi − b′i ≤ ǫ for any i. In particular,∑

i nibi − 1 > 0. By thechoice of γ,

∑i nibi − 1 ≥ γ. Thus

i

nib′i − 1 = (

i

nibi − 1)−∑

i

ni(bi − b′i) ≥ γ −4

γǫ =

γ

2,

and we are done. �

We will use the following lemma frequently without citing it in this paper.

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On boundedness of divisors computing MLDs for surfaces 9

Lemma 3.5. Let Γ ⊆ [0, 1] be a set, and γ ∈ (0, 1] be a real number. If{∑

i nibi − 1 > 0 | bi ∈ Γ, ni ∈ Z≥0} ⊆ [γ,+∞), then Γ \ {0} ⊆ [γ, 1].

Proof. Otherwise, we may find b ∈ Γ, such that 0 < b < γ. Then 0 <(⌊1

b⌋+ 1) · b− 1 ≤ b < γ, a contradiction. �

3.2. Singularities of pairs.

Definition 3.6. A pair (X,B) consists of a normal quasi-projective varietyX and an R-divisor B ≥ 0 such that KX + B is R-Cartier. A germ (X ∋x,B :=

∑i biBi) consists of a pair (X,B), and a closed point x ∈ X, such

that bi > 0, and Bi are distinct prime divisors on X with x ∈ ∩i SuppBi. Wecall it a surface germ if dimX = 2. (X ∋ x,B) is called an lc (respectivelyklt, canonical, plt, terminal) surface germ if (X ∋ x,B) is a surface germ,and (X,B) is lc (respectively klt, canonical, plt, terminal) near x.

Definition 3.7. Let (X ∋ x,B) be an lc germ. We say a prime divisor Eis over X ∋ x if E is over X and centerX E = x.

The minimal log discrepancy of (X ∋ x,B) is defined as

mld(X ∋ x,B) := min{a(E,X,B) | E is a prime divisor over X ∋ x}.

Let f : Y → X be a projective birational morphism, we denote it byf : Y → X ∋ x if centerX E = x for any f -exceptional divisor E.

Let f : X → X ∋ x be the minimal resolution of X ∋ x, and we may write

KX+B

X+∑

i(1−ai)Ei = f∗(KX +B), where BX

is the strict transform of

B, Ei are f -exceptional prime divisors and ai := a(Ei,X,B) for all i. Thepartial log discrepancy of (X ∋ x,B), pld(X ∋ x,B), is defined as follows.

pld(X ∋ x,B) :=

{mini{ai} if x ∈ X is a singular point,+∞ if x ∈ X is a smooth point.

3.3. Dual graphs.

Definition 3.8 (c.f. [10, Definition 4.6]). Let C = ∪iCi be a collection ofproper curves on a smooth surface U . We define the dual graph DG of C asfollows.

(1) The vertices of DG are the curves Cj .(2) Each vertex is labelled by the negative self intersection of the corre-

sponding curve on U , we call it the weight of the vertex (curve).(3) The vertices Ci, Cj are connected with Ci · Cj edges.

Let f : Y → X ∋ x be a projective birational morphism with exceptionaldivisors {Ei}1≤i≤m, such that Y is smooth. Then the dual graph DG off is defined as the dual graph of E = ∪1≤i≤mEi. In particular, DG is aconnected graph.

Definition 3.9. A cycle is a graph whose vertices and edges can be orderedv1, . . . , vm and e1, . . . , em (m ≥ 2), such that ei connects vi and vi+1 for1 ≤ i ≤ m, where vm+1 = v1.

Let DG be a dual graph with vertices {Ci}1≤i≤m. We call DG a tree if

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10 J.Han, and Y.Luo

(1) DG does not contain a subgraph which is a cycle, and(2) Ci · Cj ≤ 1 for all 1 ≤ i 6= j ≤ m.

Moreover, if C is a vertex of DG that is adjacent to more than three vertices,then we call C a fork of DG. If DG contains no fork, then we call it a chain.

Lemma 3.10. Let X ∋ x be a surface germ. Let Y, Y ′ be smooth surfaces,and let f : Y → X ∋ x and f ′ : Y ′ → X ∋ x be two projective birationalmorphisms, such that f ′ factors through f .

Y ′ g//

f ′

##●●

Y

f

��

X ∋ x

If the dual graph of f is a tree whose vertices are all smooth rational curves,then the dual graph of f ′ is a tree whose vertices are all smooth rationalcurves.

Proof. Let g : Y ′ → Y be the projective birational morphism such thatf ◦ g = f ′. Since g is a composition of blow-ups at smooth closed points,by inducion on the number of blow-ups, we may assume that g is a singleblow-up of Y at a smooth closed point y ∈ Y .

Let E′ be the g-exceptional divisor on Y ′, {Ei}1≤i≤m the set of distinctexceptional curves of f on Y , and {E′

i}1≤i≤m their strict transforms on Y ′.By assumption E′

i · E′j ≤ g∗Ei · E

′j = Ei · Ej ≤ 1 for 1 ≤ i 6= j ≤ m. Since

Ei is smooth, 0 = g∗Ei · E′ ≥ (E′

i + E′) · E′. It follows that E′ · E′i ≤ 1 for

1 ≤ i ≤ m.If the dual graph of f ′ contains a cycle, then E′ must be a vertex of this

cycle. Let E′, E′i1, . . . , E′

ikbe the vertices of this cycle, 1 ≤ k ≤ m. Then the

vertex-induced subgraph by Ei1 , . . . , Eik of the dual graph of f is a cycle, acontradiction. �

The following lemma maybe well-known to experts (c.f. [4, Lemma 4.2]).For the readers convenience, we include the proof here.

Lemma 3.11. Let γ, ǫ0 ∈ (0, 1] be two real numbers. Let (X ∋ x,B) bean lc surface germ, Y a smooth surface, and f : Y → X ∋ x a projectivebirational morphism with the dual graph DG. Let {Ek}1≤k≤m be the set ofvertices of DG, and wk := −Ek · Ek, ak := a(Ek,X,B) for each k. Supposethat ak ≤ 1 for any 1 ≤ k ≤ m, then we have the following:

(1) wk ≤ 2ak

if ak > 0, and in particular, wk ≤ 2ǫ0

for 1 ≤ k ≤ m if

mld(X ∋ x,B) ≥ ǫ0.(2) If wk ≥ 2 for some k, then for any Ek1 , Ek2 which are adjacent to

Ek, we have 2ak ≤ ak1 + ak2 . Moreover, if the equality holds, thenf−1∗ B ·Ek = 0, and either wk = 2 or ak = ak1 = ak2 = 0.

(3) If Ek0 is a fork, then for any Ek1 , Ek2 , Ek3 which are adjacent toEk0 with wki ≥ 2 for 0 ≤ i ≤ 2, ak3 ≥ ak0. Moreover, if the equalityholds, then wki = 2 and f−1

∗ B · Eki = 0 for 0 ≤ i ≤ 2.

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On boundedness of divisors computing MLDs for surfaces 11

(4) Let Ek0 , Ek1 , Ek2 be three vertices, such that Ek1 , Ek2 are adjacent toEk0 . Assume that ak1 ≥ ak2, ak1 ≥ ǫ0, and wk0 ≥ 3, then ak1−ak0 ≥ǫ03 .

(5) If Ek0 is a fork, and there exist three vertices Ek1 , Ek2 , Ek3 which areadjacent to Ek0 with wki ≥ 2 for 0 ≤ i ≤ 3, then a(E,X,B) ≥ ak0for any vertex E of DG.

(6) Let {Eki}0≤i≤m′ be a set of distinct vertices such that Eki is adjacentto Eki+1

for 0 ≤ i ≤ m′−1, where m′ ≥ 2. If ak0 = akm′= mld(X ∋

x,B) > 0 and wki ≥ 2 for 1 ≤ i ≤ m′ − 1, then ak0 = ak1 = · · · =akm′

and wki = 2 for 1 ≤ i ≤ m′ − 1.

Proof. For (1), we may write

KY + f−1∗ B +

1≤i≤m

(1− ai)Ei = f∗(KX +B).

For each 1 ≤ k ≤ m, we have

0 = (KY + f−1∗ B +

1≤i≤m

(1− ai)Ei) ·Ek,

or equivalently,

akwk = 2− 2pa(Ek)−∑

i 6=k

(1− ai)Ei ·Ek − f−1∗ B ·Ek.(3.1)

So akwk ≤ 2, and wk ≤ 2ak.

For (2), by (3.1),

2ak ≤ akwk ≤ ak1 + ak2 − f−1∗ B · Ek ≤ ak1 + ak2 .

If 2ak = ak1 + ak2 , then f−1∗ B · Ek = 0, and either wk = 2 or ak = ak1 =

ak2 = 0.

For (3), let k = ki in (3.1) for i = 1, 2,

akiwki ≤ 1 + ak0 − (∑

j 6=k0,ki

(1− aj)Ej ·Eki + f−1∗ B ·Eki) ≤ 1 + ak0 ,

or aki ≤1+ak0wki

. Thus let k = k0 in (3.1), we have

ak3 ≥ak0wk0 + 1− ak1 − ak2 + f−1∗ B ·Ek0

≥ak0(wk0 −1

wk1

−1

wk2

) + (1−1

wk1

−1

wk2

) ≥ ak0 .

If the equality holds, then wki = 2 and f−1∗ B ·Ei = 0 for 0 ≤ i ≤ 2.

For (4), by (3.1), we have ak0wk0 ≤ ak1 +ak2 −d, where d := f−1∗ B ·Ek0 +∑

j 6=k1,k2,k0(1− aj)Ej ·Ek0 . Hence

ak1 − ak0 ≥(wk0 − 1)ak1 − ak2 + d

wk0

≥(wk0 − 2)ak1

wk0

≥ǫ03.

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12 J.Han, and Y.Luo

For (5), we may assume that E 6= Ek0 . There exist m′+1 distinct vertices

{Fi}0≤i≤m′ of DG, such that

• F0 = Ek0 , Fm′ = E, and• Fi is adjacent to Fi+1 for 0 ≤ i ≤ m′ − 1.

Denote a′i := a(Fi,X,B) for 0 ≤ i ≤ m′. By (3), we have a′1 ≥ a′0, and by(2), a′i+1 − a′i ≥ a′i − a′i−1 for 1 ≤ i ≤ m′ − 1. Thus a′m − a′0 ≥ 0.

For (6), by (2) ak0 ≤ ak1 ≤ . . . ≤ akm′−1

≤ akm′. Thus ak0 = . . . = akm′

.

By (2) again, wki = 2 for 1 ≤ i ≤ m′ − 1. �

Lemma 3.12. Let (X ∋ x,B) be an lc surface germ. Let Y be a smoothsurface and f : Y → X ∋ x a birational morphism with the dual graph DG.If DG contains a (−1)-curve E0, then

(1) E0 can not be adjacent to two (−2)-curves in DG,(2) if either mld(X ∋ x,B) 6= pld(X ∋ x,B) or mld(X ∋ x,B) > 0,

then E0 is not a fork in DG, and(3) if E,E0, . . . , Em are distinct vertices of DG such that E is adjacent

to E0, Ei is adjacent to Ei+1 for 0 ≤ i ≤ m − 1, and −Ei · Ei = 2for 1 ≤ i ≤ m, then m+ 1 < −E ·E = w.

w 1 2 2

Proof. For (1), if E0 is adjacent to two (−2)-curves Ek1 and Ek2 in DG, thenwe may contract E0 and get a smooth model f ′ : Y ′ → X ∋ x over X, whosedual graph contains two adjacent (-1)-curves, this contradicts the negativitylemma.

By [10, Theorem 4.7] and the assumptions in (2), the dual graph of theminimal resoltion of X ∋ x is a tree. If E0 is a fork, we may contract E0 andget a smooth model f ′ : Y ′ → X ∋ x, whose dual graph contains a cycle,this contradicts Lemma 3.10.

For (3), we will construct a sequence of contractions of (−1)-curve X0 :=X → X1 → . . . Xm → Xm+1 inductively. Let EXk

be the strict transformof E on Xk, and wXk

:= −EXk· EXk

. For simplicity, we will always denotethe strict transform of Ek on Xj by Ek for all k, j. Let f1 : X0 → X1 be thecontraction of E0 on X0, then wX1

= w − 1, and E1 · E1 = −1 on X1. Letf2 : X1 → X2 be the contraction of E1 on X1, then wX2

= wX1− 1 = w− 2,

and E2 · E2 = −1 on X2. Repeating this procedure, we have fk : Xk−1 →Xk the contraction of Ek−1, and wXk

= w − k, Ek · Ek = −1 on Xk for1 ≤ k ≤ m+ 1. By the negativity lemma, wXm+1

= w − (m + 1) > 0, andwe are done. �

Lemma 3.13. Let γ ∈ (0, 1] be a real number. Let (X ∋ x,B :=∑

i biBi) bean lc surface germ, where Bi are distinct prime divisors. Let Y be a smoothsurface and f : Y → X ∋ x a birational morphism with the dual graph DG.Let {Ek}0≤k≤m be a vertex-induced sub-chain of DG, such that Ek is adjacentto Ek+1 for 0 ≤ k ≤ m − 1, and let wk := −Ek · Ek, ak := a(Ek,X,B) for

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On boundedness of divisors computing MLDs for surfaces 13

all k. Suppose that w0 = 1, and E0 is adjacent to only one vertex E1 of DG,then

(1) if {∑

i nibi−1 > 0 | ni ∈ Z≥0} ⊆ [γ,+∞), and a0 < a1, then m ≤ 1γ,

(2) if∑

i nibi − 1 6= 0 for all ni ∈ Z≥0, and a0 ≤ a1, then a0 < a1, and

(3) if {∑

i nibi−1 ≥ 0 | ni ∈ Z≥0} ⊆ [γ,+∞), and a0 ≤ a1, then m ≤ 1γ.

1 w1 w2 wm

Proof. We may write KY + f−1∗ B +

∑i(1− ai)Ei = f∗(KX +B), then

−2 + f−1∗ B ·E0 + w0a0 +

i 6=0

(1− ai)Ei · E0 = 0.(3.2)

Since E0 is adjacent to only one vertex E1 of DG, by (3.2) we have

a1 − a0 = f−1∗ B ·E0 − 1.

For (1), since 1 ≥ a1 > a0, it follows that f−1∗ B ·E0 − 1 ∈ {

∑i nibi − 1 >

0 | ni ∈ Z≥0} ⊆ [γ,+∞), Thus a1 − a0 ≥ γ. By Lemma 3.11(2), we haveai+1−ai ≥ a1−a0 ≥ γ for any 0 ≤ i ≤ m−1, and 1 ≥ am ≥ γm. So m ≤ 1

γ.

For (2), since f−1∗ B ·E0−1 =

∑i nibi−1 6= 0 for some ni ∈ Z≥0, a0 < a1.

(3) follows immediately from (1) and (2). �

3.4. Sequence of blow-ups.

Definition 3.14. Let X ∋ x be a smooth surface germ. We say Xn →Xn−1 → · · · → X1 → X0 := X is a sequence of blow-ups with the data(fi, Fi, xi ∈ Xi) if

• fi : Xi → Xi−1 is the blow-up of Xi−1 at a closed point xi−1 ∈ Xi−1

with the exceptional divisor Fi for any 1 ≤ i ≤ n, where x0 := x,and

• xi ∈ Fi for any 1 ≤ i ≤ n− 1.

In particular, Fn is the only exceptional (−1)-curve over X.For convenience, we will always denote the strict transform of Fi on Xj

by Fi for any n ≥ j ≥ i.

The following lemma is well known. For the reader’s convenience, we givea proof in details.

Lemma 3.15. Let (X ∋ x,B) be an lc surface germ such that mld(X ∋x,B) > 1, then mld(X ∋ x,B) = 2 − multxB, and there is exactly oneprime divisor E over X ∋ x such that a(E,X,B) = mld(X ∋ x,B).

Proof. By [10, Theorem 4.5], X is smooth near x. Let E be any primedivisor over X ∋ x, such that a(E,X,B) = mld(X ∋ x,B). By [10, Lemma2.45], there exists a sequence of blow-ups

Xn → Xn−1 → · · · → X1 → X0 := X(3.3)

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14 J.Han, and Y.Luo

with the data (fi, Ei, xi ∈ Xi), such that En = E, and a(Ei,X,B) ≥a(En,X,B) = mld(X ∋ x,B) for any 1 ≤ i ≤ n− 1.

Let gi : Xi → X and hi : Xi → X1 be the natural birational morphisminduced by (3.3), and BXi

the strict transform of B on Xi.Since f∗

1B = BX1+(multxB)E1, 0 = f∗

1B ·E1 = (BX1+(multxB)E1)·E1,

which implies multxB = BX1· E1. By the projection formula,

multxB = BX1·E1 = BXn−1

· h∗n−1E1 ≥ BXn−1·En−1 ≥ multxn−1

BXn−1.

Since a(Ei,X,B) > 1 for any i ≥ 1, we have

a(En,X,B) ≥ a(En,Xn−1, BXn−1) = 2−multxn−1

BXn−1,

and a(En,X,B) = 2−multxn−1BXn−1

if and only if n = 1. It follows that

a(En,X,B) ≥ 2−multxn−1BXn−1

≥ 2−multxB = a(E1,X,B).

Thus a(En,X,B) = 2 − multxn−1BXn−1

= a(E1,X,B). This implies thatn = 1, mld(X ∋ x,B) = a(E1,X,B) = 2 − multxB, and E1 is the onlyprime divisor over X ∋ x, such that a(E1,X,B) = mld(X ∋ x,B). �

Lemma 3.16. Let X ∋ x be a smooth surface germ, and Xl0 → · · · →X1 → X0 := X a sequence of blow-ups with the data (fi, Ei, xi ∈ Xi), thena(En,X, 0) ≤ Fib(l0 + 1) + 1, where Fib(i) is the i-th term of the Fibonaccisequence. In particular, a(En,X, 0) ≤ 2l0 .

Proof. We prove the lemma by induction on i for 1 ≤ i ≤ l0. When l0 = 1,a(E1,X, 0) = 2 = Fib(2) + 1. Suppose that a(Ei,X, 0) ≤ Fib(i+ 1) + 1 forall 1 ≤ i ≤ k. For i = k + 1, there exists at most one j 6= k, such thatxk ∈ Ej . By induction,

a(Ek+1,X, 0) ≤ a(Ek,X, 0) + max1≤j≤k−1

{a(Ej ,X, 0)} − 1

≤ Fib(k + 1) + Fib(k) + 1

= Fib(k + 2) + 1.

The induction is finished, and we are done. �

3.5. Extracting divisors computing MLDs.

Lemma 3.17. Let (X ∋ x,B) be an lc surface germ. Let h : W → (X,B)be a log resolution, and S = {Ej} a finite set of valuations of h-exceptionalprime divisors over X ∋ x such that a(Ej ,X,B) ≤ 1 for all j. Then thereexist a smooth surface Y and a projective birational morphism f : Y → X ∋x with the following properties.

(1) KY +BY = f∗(KX +B) for some R-divisor BY ≥ 0 on Y ,(2) each valuation in S corresponds to some f -exceptional divisor on Y ,

and(3) each f -exceptional (−1)-curve corresponds to some valuation in S.

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On boundedness of divisors computing MLDs for surfaces 15

Proof. We may write

KW +BW = h∗(KX +B) + FW ,

where BW ≥ 0 and FW ≥ 0 are R-divisors with no common components. Weconstruct a sequence of (KW +BW )-MMP over X as follows. Each time wewill contract a (−1)-curve whose support is contained in FW . Suppose thatKW +BW is not nef over X, then FW 6= 0. By the negativity lemma, thereexists a h-exceptional irreducible curve C ⊆ SuppFW , such that FW · C =(KW+BW )·C < 0. Since BW ·C ≥ 0, KW ·C < 0. Thus C is a h-exceptional(−1)-curve. We may contract C, and get a smooth surface Y0 := W → Y1

over X. We may continue this process, and finally reach a smooth modelYk on which KYk

+ BYkis nef over X, where BYk

is the strict transform ofBW on Yk. By the negativity lemma, FW is contracted in the MMP, thusKYk

+ BYk= h∗k(KX + B), where hk : Yk → X. Since a(Ej ,X,B) ≤ 1, Ej

is not contracted in the MMP for any Ej ∈ S.

We now construct a sequence of smooth models over X, Yk → Yk+1 → · · · ,by contracting a curve C ′ satisfying the following conditions in each step.

• C ′ is an exceptional (−1)-curve over X, and• C ′ /∈ S.

Since each time the Picard number of the variety will drop by one, afterfinitely many steps, we will reach a smooth model Y over X, such thatf : Y → X and (Y,BY ) satisfy (1)–(3), where BY is the strict transform ofBYk

on Y . �

We will need Lemma 3.18 to prove our main results. It maybe well knownto experts. Lemma 3.18(1)–(4) could be proved by constructing a sequenceof blow-ups (c.f. [4, Lemma 4.3]). We give another proof here.

We remark that Lemma 3.18(5) will only be applied to prove Theorem1.2.

Lemma 3.18. Let (X ∋ x,B) be an lc surface germ such that 1 ≥ mld(X ∋x,B) 6= pld(X ∋ x,B). There exist a smooth surface Y and a projectivebirational morphism f : Y → X with the dual graph DG, such that

(1) KY +BY = f∗(KX +B) for some R-divisor BY ≥ 0 on Y ,(2) there is only one f -exceptional divisor E0 such that a(E0,X,B) =

mld(X ∋ x,B),(3) E0 is the only (−1)-curve of DG, and(4) DG is a chain.

Moreover, if X ∋ x is not smooth, let f : X → X ∋ x be the minimal

resolution of X ∋ x, and let g : Y → X be the morphism such that f ◦g = f ,then

(5) there exist a f -exceptional prime divisor E on X and a closed point

x ∈ E, such that a(E,X,B) = pld(X ∋ x,B), and centerXE = x

for all g-exceptional divisors E.

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16 J.Han, and Y.Luo

Proof. By Lemma 3.17, we can find a smooth surface Y0 and a birationalmorphism h : Y0 → X ∋ x, such that a(E′

0,X,B) = mld(X ∋ x,B) for someh-exceptional divisor E′

0, and KY0+ BY0

= h∗(KX + B) for some BY0≥ 0

on Y0.We now construct a sequence of smooth models over X, Y0 → Y1 → · · · ,

by contracting a curve C ′ satisfying the following conditions in each step.

• C ′ is an exceptional (−1)-curve over X, and• there exists C ′′ 6= C ′ over X, such that a(C ′′,X,B) = mld(X ∋x,B).

Since each time the Picard number of the variety will drop by one, afterfinitely many steps, we will reach a smooth model Y over X, such thatf : Y → X and (Y,BY ) satisfy (1), where BY is the strict transform of BY0

on Y . Since mld(X ∋ x,B) 6= pld(X ∋ x,B), by the construction of Y ,there exists a curve E0 on Y satisfying (2)–(3).

For (4), by [10, Theorem 4.7], the dual graph of the minimal resolution

f : X → X ∋ x is a tree whose vertices are smooth rational curves. Since

Y is smooth, f factors through f . By Lemma 3.10, the dual graph DG off is a tree whose vertices are smooth rational curves. It suffices to showthat there is no fork in DG. By Lemma 3.12(2), E0 is not a fork. Supposethat DG contains a fork E′ 6= E0, by (3) and (5) of Lemma 3.11, we havea(E′,X,B) ≤ a(E0,X,B), this contradicts (2). Thus DG is a chain.

For (5), since there exists only one f -exceptional (−1)-curve, there is at

most one closed point x ∈ X, such that centerXE = x for all g-exceptional

divisors E. Thus the dual graph of g, which is denoted by DG ′, is a vertex-induced connected sub-chain of DG by all g-exceptional divisors. Sincemld(X ∋ x,B) 6= pld(X ∋ x,B), we have DG′ ( DG.

E−n1E

−n′

1−1 E−n′

1E0

center at x ∈ XDG′

En′

2En′

2+1 En2

Figure 1. The dual graph of f

We may index the vertices of DG as {Ei}−n1≤i≤n2for n1, n2 ∈ Z≥0, such

that Ei is adjacent to Ei+1, and ai := a(Ei,X,B) for all possible i. We mayassume that the set of vertices of DG′ is {Ej}−n′

1≤j≤n′

2, where 0 ≤ n′

1 ≤ n1

and 0 ≤ n′2 ≤ n2 (see Figure 1). If n1 > n′

1, then by Lemma 3.11(2),ak − a−n′

1−1 ≥ min{0, a−1 − a0} ≥ 0 for all −n1 ≤ k < −n′

1. If n2 > n′2,

then again by Lemma 3.11(2), ak′ − an′

2+1 ≥ min{0, a1 − a0} ≥ 0 for all

n′2 < k′ ≤ n2. Set a−n1−1 = 1, E−n1−1 = E−n1

if n1 = n′1, and set an2+1 =

1, En2+1 = En2if n2 = n′

2. Then min{an2+1, a−n1−1} = pld(X ∋ x,B), and

x = g(E−n′

1−1) ∩ g(En2+1) ∈ E, where a(E,X,B) = pld(X ∋ x,B). �

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On boundedness of divisors computing MLDs for surfaces 17

The following lemma gives an upper bound for number of vertices of cer-tain kind of DG constructed in Lemma 3.18, with the additional assumptionthat mld(X ∋ x,B) and the coefficients of B are bounded from below by apositive real number.

Lemma 3.19. Let ǫ0 ∈ (0, 1] be a real number. Then N ′0 := ⌊ 8

ǫ0⌋ satisfies

the following properties.Let (X ∋ x,B :=

∑biBi) be an lc surface germ, such that mld(X ∋

x,B) ≥ ǫ0, and bi ≥ γ for any i, where Bi are distinct prime divisors. LetY be a smooth surface, and f : Y → X ∋ x a birational morphism with thedual graph DG, such that

• KY +BY = f∗(KX +B) for some R-divisor BY ≥ 0 on Y ,• DG is a chain with only one (−1)-curve E0,• a(E0,X,B) = mld(X ∋ x,B), and• E0 is adjacent to two vertices of DG.

Then the number of vertices of DG is bounded from above by N ′0.

Proof. Let {Ei}−n1≤i≤n2be the vertices of DG, such that Ei is adjacent to

Ei+1 for −n1 ≤ i ≤ n2 − 1, and wi := −(Ei · Ei), ai := a(Ei,X,B) for all i.We may assume that E0 is adjacent to two vertices E−1, E1 of DG.

w−n1

≤ 3

ǫ0

w−1 1 2

(−2)-curves

2 wn′+1

≤ 3

ǫ0

wn2

By Lemma 3.12(1), we may assume that w−1 > 2. By (2) and (4) ofLemma 3.11, ai−1 − ai ≥

ǫ03 for any −n1 +1 ≤ i ≤ −1, and a−1 ≥

ǫ03 . Since

a−n1≤ 1, n1 ≤ 3

ǫ0. Similarly, n2 − n′ ≤ 3

ǫ0, where n′ is the largest non-

negative integer such that wi = 2 for any 1 ≤ i ≤ n′. By Lemma 3.11(1),w−1 ≤ 2

ǫ0, and by Lemma 3.12(3), n′ ≤ 2

ǫ0− 1. Hence n1 + n2 + 1, the

number of vertices of DG, is bounded from above by 8ǫ0. �

4. Proof of Theorem 1.3

Lemma 4.1 is crucial in the proof of Theorem 1.3. Before providing theproof, we introduce some notations first.Notation (⋆). Let X ∋ x be a smooth surface germ, and let g : Xn →Xn−1 → · · · → X1 → X0 := X be a sequence of blow-ups with the data(fi, Fi, xi ∈ Xi). Let DG be the dual graph of g, and assume that DG is achain.

Let n3 ≥ 2 be the largest integer, such that xi ∈ Fi \Fi−1 for any 1 ≤ i ≤n3 − 1, where we set F0 := ∅. Let {Ej}−n1≤j≤n2

be the vertices of DG, suchthat E0 := Fn is the only g-exceptional (−1)-curve on Xn, En2

:= F1, andEi is adjacent to Ei+1 for any −n1 ≤ i ≤ n2 − 1(see Figure 2).

We define ni(g) := ni for 1 ≤ i ≤ 3, n(g) = n, wj(g) := −Ej ·Ej for all j,and W1(g) :=

∑j<0wj(g) and W2(g) :=

∑j>0wj(g).

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18 J.Han, and Y.Luo

E−n1

Fn3

E−1 E0 E1

Fn3−1

En2

F1

Figure 2. The dual graph of g

Lemma 4.1. With Notation (⋆). Then

(W1(g) − n1(g)) + n3(g)− 1 = W2(g) − n2(g).(4.1)

In particular, n(g) = n1(g) + n2(g) + 1 ≤ n3(g) + min{W1(g),W2(g)}.

Proof. For simplicity, let n := n(g), ni := ni(g) for 1 ≤ i ≤ 3, wj := wj(g) =−Ej · Ej for all j, and Wj := Wj(g) for j = 1, 2.

We prove (4.1) by induction on the non-negative integer n− n3.

1

Fn3

2

Fn3−1

2

F2

2

F1

Figure 3. The dual graph for the case n = n3.

2

Fn3

1

Fn3−1

3 2 2

F1

Figure 4. The dual graph for the case n = n3 + 1.

If n = n3, then n1 = W1 = 0, n2 = n3 − 1, and W2 = 2n3 − 2, thus (4.1)holds (see Figure 3). If n = n3 + 1, then xn3

∈ Fn3∩ Fn3−1. In this case,

n1 = 1, W1 = 2, n2 = n3−1, and W2 = 2n3−1, thus (4.1) holds (see Figure4).

In general, suppose (4.1) holds for any sequence of blow-ups g as in Nota-

tion (⋆) with positive integers n, n3 satisfying 1 ≤ n− n3 ≤ k. For the casewhen n− n3 = k + 1, we may contract the (−1)-curve on Xn, and considerg′ : Xn−1 → · · · → X0 := X, a subsequence of blow-ups of g with the data(fi, Fi, xi ∈ Xi) for 0 ≤ i ≤ n − 1. Denote n′

i := ni(g′) for any 1 ≤ i ≤ 3,

and W ′j := Wj(g

′) for any 1 ≤ j ≤ 2. By Lemma 3.12(1), either w−1 = 2 or

w1=2. In the former case, W ′1 = W1−2,W ′

2 = W2−1, n′1 = n1−1, n′

2 = n2,and n′

3 = n3. In the latter case, W ′1 = W1 − 1,W ′

2 = W2 − 2, n′1 = n1, n

′2 =

n2 − 1, and n′3 = n3. In both cases, by induction,

W ′2 − n′

2 − (W ′1 − n′

1) = (W2 − n2)− (W1 − n1) = n3 − 1.

Hence we finish the induction, and (4.1) is proved.Since wj ≥ 2 for j 6= 0, we have W1 =

∑−n1≤j≤−1wj ≥ 2n1 and W2 =∑

1≤j≤n2wj ≥ 2n2. By (4.1),

n1 + n2 + 1 ≤ n1 +W2 − n2 + 1 = W1 + n3,

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On boundedness of divisors computing MLDs for surfaces 19

andn1 + n2 + 1 ≤ W1 − n1 + n2 + n3 − 1 = W2,

which imply that n = n1 + n2 + 1 ≤ n3 +min{W1,W2}. �

We will need Lemma 4.2 to prove both Theorem 1.3 and Theorem 1.4.

Lemma 4.2. Let γ ∈ (0, 1] be a real number. Let N0 := ⌊1 + 32γ2 + 1

γ⌋, then

we have the following.Let (X ∋ x,B :=

∑i biBi) be an lc surface germ, such that X ∋ x is

smooth, and Bi are distinct prime divisors. Suppose that {∑

i nibi − 1 > 0 |ni ∈ Z≥0} ⊆ [γ,+∞). Let Y be a smooth surface and f : Y → X ∋ x be abirational morphism with the dual graph DG, such that

• KY +BY = f∗(KX +B) for some BY ≥ 0 on Y ,• DG is a chain that contains only one (−1)-curve E0,• E0 is adjacent to two vertices of DG, and• either E0 is the only vertex of DG such that a(E0,X,B) = mld(X ∋x,B), or a(E0,X,B) = mld(X ∋ x,B) > 0 and

∑i nibi 6= 1 for all

ni ∈ Z≥0.

Then the number of vertices of DG is bounded from above by N0.

Proof. By Lemma 3.5, bi ≥ γ for all i.If mld(X ∋ x,B) ≥ γ

2 , then by Lemma 3.19 with ǫ0 = γ2 , the number of

vertices of DG is bounded from above by 16γ.

Thus we may assume that 0 ≤ mld(X ∋ x,B) ≤ γ2 . We may index

the vertices of DG as {Ej}−n1≤j≤n2for some positive integer n1, n2, where

Ej is adjacent to Ej+1 for −n1 ≤ j ≤ n2 − 1. Let wj := −Ej · Ej andaj := a(Ej ,X ∋ x,B) for all j.

For all −n1 ≤ k ≤ n2, we have

(KY + f−1∗ B +

j

(1− aj)Ej) ·Ek = f∗(KX +B) ·Ek = 0.(4.2)

Let k = 0, (4.2) becomes 0 = −2+ f−1∗ B ·E0 + (1− a−1) + (1− a1) +w0a0,

thus(a1 − a0) + (a−1 − a0) = f−1

∗ B ·E0 − a0.

By the last assumption in the lemma, either (a−1−a0)+ (a1 −a0) > 0 ora0 > 0, thus f−1

∗ B ·E0 > 0 in both cases. Hence f−1∗ B · E0 − a0 ≥ γ − γ

2 =γ2 . Possibly switching Ej (j < 0) with Ej (j > 0), we may assume thata−1 − a0 ≥

γ4 .

By Lemma 3.11(2), a−j − a−j+1 ≥ a−1 − a0 ≥ γ4 for 1 ≤ j ≤ n1, thus

n1 ·γ4 ≤ a−n1

≤ 1, and n1 ≤ 4γ. Since aj ≥ γ

4 for all −n1 ≤ j ≤ −1, by

Lemma 3.11(1), wj ≤8γfor all −n1 ≤ j ≤ −1. Thus

∑−n1

j=−1wj ≤ n1 ·8γ≤ 32

γ2 .

Note that X ∋ x is smooth and DG has only one (−1)-curve, thus f : Y → Xis a sequence of blow-ups as in Definition 3.14. Moreover, DG is a chain,thus by Lemma 4.1, 1 + n1 + n2 ≤ n3 +

32γ2 , where n3 = n3(f) is defined as

in Notation (⋆).

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20 J.Han, and Y.Luo

It suffices to show that n3 is bounded, we may assume that n3 > 2. Bythe definition of n3, there exists a sequence of blow-ups Xn3

→ . . . X1 →X0 := X with the data (fi, Fi, xi ∈ Xi), such that xi ∈ Fi \ Fi−1 for any1 ≤ i ≤ n3 − 1. Here F0 := ∅.

Let BXibe the strict transform of B on Xi for 0 ≤ i ≤ n3, and let

a′i := a(Fi,X,B) for 1 ≤ i ≤ n3, and a′0 := 1. Since xi ∈ Fi \ Fi−1,a′i − a′i+1 = multxi

BXi− 1 for any n3 − 1 ≥ i ≥ 0. By Lemma 3.11(2),

a′i − a′i+1 ≥ min{a1 − a0, a−1 − a0} ≥ 0 for 1 ≤ i ≤ n3 − 2 (see Figure 2).Thus by the last assumption in the lemma, either min{a1−a0, a−1−a0} > 0,or multxi

BXi− 1 > 0, in both cases we have a′i − a′i+1 = multxi

BXi− 1 > 0.

Hence a′i−a′i+1 = multxiBXi

−1 ≥ γ for any 1 ≤ i ≤ n3−2 as {∑

i nibi−1 >0 | ni ∈ Z≥0} ⊆ [γ,+∞). Therefore,

0 ≤ a′n3−1 = a′0 +

n3−2∑

i=0

(a′i+1 − a′i) ≤ 1− (n3 − 1)γ,

and n3 ≤ 1 + 1γ.

To sum up, the number of vertices of DG is bounded from above by⌊1 + 32

γ2 + 1γ⌋. �

Now we are ready to prove Theorem 1.3.

Proof of Theorem 1.3. By Lemma 3.15, we may assume that mld(X ∋ x,B) ≤1.

Let f : Y → X ∋ x be the birational morphism constructed in Lemma3.18 with the dual graph DG. We claim that the number of vertices of DGis bounded from above by N0 := ⌊1 + 32

γ2 + 1γ⌋.

Assume the claim holds, then by Lemma 3.16, a(E,X, 0) ≤ 2N0 for someexceptional divisor E such that a(E,X,B) = mld(X ∋ X, b), we are done.It suffices to show the claim.

If the f -exceptional (−1)-curve is adjacent to only one vertex of DG, thenby Lemma 3.13(1), the number of vertices of DG is bounded from above by1 + 1

γ.

If the f -exceptional (−1)-curve is adjacent to two vertices of DG, thenby Lemma 4.2, the number of vertices of DG is bounded from above by⌊1 + 32

γ2 + 1γ⌋. Thus we finish the proof. �

To end this section, we provide an example which shows that Theorem1.3 does not hold if we do not assume that {

∑i nibi − 1 > 0 | ni ∈ Z≥0} is

bounded from below by a positive real number.

Example 4.3. Let {(A2 ∋ 0, Bk)}k be a sequence of klt surface germs, suchthat Bk := 1

2Bk,1+(12 +1

k+1)Bk,2, and Bk,1 (respectively Bk,2) is defined by

the equation x = 0 (respectively x− yk = 0) at 0 ∈ A2.For each k, we may construct a sequence of blow-upsXk → Xk−1 → · · · →

X1 → X0 := X with the data (fi, Fi, xi ∈ Xi), such that xi−1 ∈ Fi−1 is the

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On boundedness of divisors computing MLDs for surfaces 21

intersection of the strict transforms of Bk,1 and Bk,2 on Fi−1 for 1 ≤ i ≤ k.Let gk : Xk → X be the natural morphism induced by {fi}1≤i≤k, we haveKXk

+ BXk= g∗k(KX + B) for some snc divisor BXk

≥ 0 on Xk, and the

coefficients of BXkare no more than k

k+1 . Thus we will need at least kblow-ups as constructed above to extract an exceptional divisor Fk suchthat a(Fk,A

2, Bk) = mld(A2 ∋ 0, Bk) =1

k+1 , and a(Fk,A2, 0) ≥ k.

5. Proof of Theorem 1.2

Proof of Theorem 1.2. We may assume that Γ \ {0} 6= ∅.Let (X ∋ x,B) be an lc surface germ with B ∈ Γ. By Lemma 3.15, we

may assume that mld(X ∋ x,B) ≤ 1.If mld(X ∋ x,B) = pld(X ∋ x,B), then a(E,X, 0) ≤ 1 for some prime

divisor E over X ∋ x such that a(E,X,B) = mld(X ∋ x,B). So we mayassume that mld(X ∋ x,B) 6= pld(X ∋ x,B).

By Lemma 3.18, there exists a birational morphism f : Y → X ∋ x which

satisfies Lemma 3.18(1)–(5). Let f : X → X be the minimal resolution of

X ∋ x, g : Y → X ∋ x the birational morphism such that f ◦ g = f , where

x ∈ X is chosen as in Lemma 3.18(5), and there exists a f -exceptional prime

divisor E over X ∋ x such that a(E,X,B) = pld(X ∋ x,B) and x ∈ E.

And the is at most one other vertex E′ of DG such that x ∈ E′.Let DG be the dual graph of f , and {Fi}−n1≤i≤n2

the vertices of DG, suchthat n1, n2 ∈ Z≥0, Fi is adjacent to Fi+1, wi := −Fi · Fi, ai := a(Fi,X,B)

for all i, and F0 := E, F1 := E′ (see Figure 5). We may write KX

+

BX

= f∗(KX + B), where BX

:= f−1∗ B +

∑i(1 − ai)Fi, and we define

B := f−1∗ B +

∑x∈Fi

(1− ai)Fi.

F−n1F−1 F0

xF1 Fn2

F0x

F1 Fn2

Figure 5. Cases when x ∈ F0 ∩ F1 and when x /∈ Fi for i 6= 0.

If x /∈ Fi for all i 6= 0, then we consider the surface germ (X ∋ x, B =

f−1∗ B+(1−a0)F0), where B ∈ Γ′ := Γ∪{1−a | a ∈ Pld(2,Γ)}. By Lemma2.2, Γ′ satisfies the DCC. Thus by Theorem 1.3, we may find a positive

integer N1 which only depends on Γ, and a prime divisor E over X ∋ x,

such that a(E, X, B) = a(E,X,B) = mld(X ∋ x,B), and a(E,X, 0) ≤

a(E, X, 0) ≤ N1.So we may assume that x = F0 ∩ F1. By Lemma 3.4, there exist positive

real numbers ǫ, δ ≤ 1 depending only on Γ, such that {∑

i nibi− 1 > 0 | bi ∈Γ′ǫ ∩ [0, 1], ni ∈ Z≥0} ⊆ [δ,+∞). Recall that Γ′

ǫ = ∪b′∈Γ′ [b′ − ǫ, b′].

If a1 − a0 ≤ ǫ, then we consider the surface germ (X ∋ x, B = f−1∗ B +

(1 − a0)F0 + (1 − a1)F1), where B ∈ Γ′ǫ ∩ [0, 1]. By Theorem 1.3, there

exist a positive integer N2 which only depends on Γ, and a prime divisor

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22 J.Han, and Y.Luo

E over X ∋ x, such that a(E, X, B) = a(E,X,B) = mld(X ∋ x,B) and

a(E,X, 0) ≤ a(E, X, 0) ≤ N2.If a1 − a0 ≥ ǫ, then we claim that there exists a DCC set Γ′′ depending

only on Γ, such that 1− a1 ∈ Γ′′.

F−n1F−1 F0

xF1

finite graphFn2

Figure 6. Cases when a1 − a0 ≥ ǫ.

Assume the claim holds, then we consider the surface germ (X ∋ x, B =

f−1∗ B + (1 − a0)F0 + (1 − a1)F1), where B ∈ Γ′′ ∪ Γ′. By Theorem 1.3, wemay find a positive integer N3 which only depends on Γ, and a prime divisor

E over X ∋ x, such that a(E, X, B) = a(E,X,B) = mld(X ∋ x,B) and

a(E,X, 0) ≤ a(E, X, 0) ≤ N3. Let N := max{N1, N2, N3}, and we are done.

It suffices to show the claim. By Lemma 3.11(1), wi ≤2ǫfor any 0 < i ≤

n2. Since 1 ≥ an2= a0 +

∑n2−1i=0 (ai+1 − ai) ≥ n2ǫ, n2 ≤

1ǫ. We may write

KX+ f−1

∗ B +∑

−n1≤i≤n2

(1− ai)Fi = f∗(KX +B),

For each 1 ≤ j ≤ n2, we have

(KX+ f−1

∗ B +∑

−n1≤i≤n2

(1− ai)Fi) · Fj = 0,

which implies∑

−n1≤i≤n2(ai−1)Fi ·Fj = −Fj

2−2+f−1∗ B·Fj , or equivalently,

F1 · F1 · · · Fn2· F1

.... . .

...F1 · Fn2

· · · Fn2· Fn2

a1 − 1...

an2− 1

=

w1 − 2 + f−1

∗ B · F1 + (1− a0)...

wn2− 2 + f−1

∗ B · Fn2

.

By assumption, wj − 2 + f−1∗ B · Fj belongs to a DCC set, and by Lemma

2.2, 1− a0 belongs to the DCC set {1− a | a ∈ Pld(2,Γ)}.By [10, Lemma 3.40], (Fi · Fj)1≤i,j≤n2

is a negative definite matrix. Let(sij)n2×n2

be the inverse matrix of (Fi · Fj)1≤i,j≤n2. By [10, Lemma 3.41],

sij < 0 for any 1 ≤ i, j ≤ n2, and this shows that

1− a1 = −s11(w1 − 2 + f−1∗ B · F1 + (1− a0))−

n2∑

j=2

s1j(wj − 2 + f−1∗ B · Fj)

belongs to a DCC set. �

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On boundedness of divisors computing MLDs for surfaces 23

6. Proof of Theorem 1.4

In this section, we will first show Theorem 1.4, then we will generalizeboth Theorem 1.3 and Theorem 1.4, see Theorem 6.4 and Theorem 6.5.

Lemma 6.1. Let (X ∋ x,B :=∑

i biBi) be an lc surface germ, where Bi

are distinct prime divisors. Let h : W → X ∋ x be a log resolution of(X ∋ x,B), and let S = {Ej} be a finite set of valuations of h-exceptionalprime divisors such that a(Ej ,X,B) = mld(X ∋ x,B) for all j. Supposethat

∑i nibi 6= 1 for any ni ∈ Z≥0, mld(X ∋ x,B) ∈ (0, 1], and Ej is

exceptional over X for some j, where X → X is the minimal resolution ofX ∋ x. Then there exist a smooth surface Y and a birational morphismf : Y → X ∋ x with the dual graph DG, such that

(1) KY +BY = f∗(KX +B) for some R-divisor BY ≥ 0 on Y ,(2) each valuation in S corresponds to some vertex of DG,(3) DG contains only one (−1)-curve E0, and it corresponds to a valu-

ation in S, and(4) DG is a chain.

Proof. By Lemma 3.17, there exist a smooth surface Y and a birationalmorphism f : Y → X ∋ x with the dual graph of DG which satisfy (1)–(2),and each f -exceptional (−1)-curve corresponds to some valuation in S.

For (3), by the assumption on S, DG contains at least one (−1)-curve E0.If there exist two (−1)-curves E′

0 6= E0, then a(E0,X,B) = a(E′0,X,B) =

mld(X ∋ x,B) > 0 by construction, and there exists a set of distinct vertices{Ck}0≤k≤n of DG such that n ≥ 2, Cn := E′

0, C0 is a (−1)-curve, −Ck ·Ck ≥−2 for 1 ≤ k ≤ n − 1, and Ck is adjacent to Ck+1 for 0 ≤ k ≤ n − 1.Since a(C0,X,B) = a(Cn,X,B) = mld(X ∋ x,B) > 0, by Lemma 3.11(6),−Ck · Ck = 2 for 1 ≤ k ≤ n− 1. Let E :=

∑nk=0Ck, then E · E = 0, which

contradicts the negativity lemma.

For (4), suppose that DG contains a fork F . Since E0 is a (−1)-curve,by Lemma 3.12(2), E0 6= F . By (2) and (5) of Lemma 3.11, we havea(E0,X,B) ≥ a(F,X,B) ≥ mld(X ∋ x,B), thus a(E0,X,B) = a(F,X,B) =mld(X ∋ x,B) > 0. There exists a set of distinct vertices {C ′

i}0≤i≤m, suchthat C ′

0 := E0, C′m := F , and C ′

i is adjacent to C ′i+1 for 0 ≤ i ≤ m− 1. We

may denote w′i := −C ′

i ·C′i and a′i := a(C ′

i,X,B) for 0 ≤ i ≤ m. By Lemma3.11(6), we have a′0 = · · · = a′m, and w′

k = 2 for 1 ≤ k ≤ m− 1. Since C ′m is

a fork and a′m−1 = a′m, by Lemma 3.11(3), w′m = 2.

If C ′0 is adjacent to only one vertex of DG, which is C ′

1 by our construction,then since

∑i nibi 6= 1 for any ni ∈ Z≥0, a

′0 6= a′1 by Lemma 3.13(2), a

contradiction.Thus we may assume that C ′

0 is adjacent to a vertex C ′−1 of DG other

than C ′1 (see Figure 7). We may contract C ′

k for 0 ≤ k ≤ m−1 step by step,and will end up with a fork which is a (−1)-curve, this contradicts Lemma3.12(2). �

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24 J.Han, and Y.Luo

C′

−1 C′

0

1

C′

m−1

2

C′

1

2 2

C′

m

Figure 7. C ′0 is adjacent to C ′

−1 and C ′1

Lemma 6.2. Let (X ∋ x,B :=∑

i biBi) be an lc surface germ, where Bi

are distinct prime divisors. Suppose that X ∋ x is a klt germ, mld(X ∋x,B) = 0, and

∑i nibi 6= 1 for any ni ∈ Z≥0. Then there is only one prime

divisor E over X ∋ x such that a(E,X,B) = 0.

Proof. By [2, Lemma 2.7], we can find a plt blow up g : Y → X ∋ x, suchthat

• KY +BY = g∗(KX +B) for some R-divisor BY ≥ 0 on Y ,• there is only one g-exceptional prime divisor E,• SuppE ⊆ ⌊BY ⌋, and• (Y,E) is plt.

By the adjunction formula ([13, §3], [9, §16]), KE + BE := (KY + BY )|E ,where

BE =∑

i

mi − 1 +∑

j ni,jbj

mipi

for some distinct closed points pi on E and some mi ∈ Z>0 and ni,j ∈ Z≥0

such that∑

j ni,jbj ≤ 1 for all i. By assumption,∑

j ni,jbj 6= 1, thus∑j ni,jbj < 1 for all i, which implies ⌊BE⌋ = 0, thus (E,BE) is klt. By the

inversion of adjunction ([13, §3], [9, §17.6-7]), (Y,BY ) is plt. Thus E is theonly prime divisor over X ∋ x such that a(E,X,B) = 0. �

Proof of Theorem 1.4. By Lemma 3.15, we may assume that mld(X ∋ x,B) ≤1.

If mld(X ∋ x,B) = 0, since X ∋ x is smooth, then by Lemma 6.2, thereexists a unique prime divisor E over X ∋ x such that a(E,X,B) = 0. ByTheorem 1.3, a(E,X, 0) ≤ 2N0 . Thus we may assume that mld(X ∋ x,B) >0.

We may apply Lemma 6.1 to S, there exist a smooth surface Y and abirational morphism f : Y → X ∋ x with the dual graph DG which satisfyLemma 6.1(1)–(4). Let E0 be the unique (−1)-curve in DG. It suffices togive an upper bound for the number of vertices of DG.

If E0 is adjacent to only one vertex of DG, then by Lemma 3.13(2)–(3),the number of vertices of DG is bounded from above by 1 + 1

γ, and |S| = 1.

If E0 is adjacent to two vertices of DG, then by Lemma 4.2, the numberof vertices of DG is bounded from above by 1+ 32

γ2 +1γ. Thus |S| ≤ N0, and

by Lemma 3.16, a(E,X, 0) ≤ 2N0 for any E ∈ S. �

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On boundedness of divisors computing MLDs for surfaces 25

Now we are going to introduce and prove Theorem 6.4 and Theorem 6.5.First, we need to introduce the following definition.

Definition 6.3. Let X ∋ x be a klt surface germ. Let f : X → X ∋ x

be the minimal resolution, and {Ei}1≤i≤n the set of f -exceptional primedivisors. The determinant of X ∋ x is defined by

det(X ∋ x) :=

{|det(Ei ·Ej)1≤i,j≤n| if x ∈ X is a singular point,1 if x ∈ X is a smooth point.

Theorem 6.4. Let γ ∈ (0, 1] be a real number, and I a positive integer.

Then N0 := ⌊1 + 32I2

γ2 + Iγ⌋ satisfies the following.

Let (X ∋ x,B :=∑

i biBi) be an lc surface germ, where Bi are distinctprime divisors. Suppose that det(X ∋ x)|I, and {

∑i nibi − t > 0 | ni ∈

Z≥0, t ∈ Z ∩ [1, I]} ⊆ [γ,+∞). Then there exists a prime divisor E overX ∋ x such that a(E,X,B) = mld(X ∋ x,B), and a(E,X, 0) ≤ 2N0 .

Theorem 6.5. Let γ ∈ (0, 1] be a real number, and X ∋ x a klt surfacegerm. Let N be the number of vertices of the dual graph of the minimal

resolution of X ∋ x, and I := det(X ∋ x). Then N0 := ⌊1 + 32I2

γ2 + Iγ⌋+N

satisfies the following.Let (X ∋ x,B :=

∑i biBi) be an lc surface germ, where Bi are distinct

prime divisors. Let S := {E | E is a prime divisor over X ∋ x, a(E,X,B) =mld(X ∋ x,B)}. Suppose that {

∑i nibi − t ≥ 0 | ni ∈ Z≥0, t ∈ Z ∩ [1, I]} ⊆

[γ,+∞). Then |S| ≤ N0, and a(E,X, 0) ≤ 2N0 for any E ∈ S.

Remark 6.6. Theorem 6.5 does not hold if X ∋ x is not klt (see Example6.8), or if we only bound det(X ∋ x) as in Theorem 6.4 (see Example 6.10).

Theorem 6.4 follows from Theorem 1.3, and Theorem 6.5 follows fromTheorem 1.4 and Theorem 6.4. We will need the following lemma to provethe above theorems.

Lemma 6.7. Let (X ∋ x,B :=∑

i biBi) be an lc surface germ, where Bi

are distinct prime divisors. Let f : X → X ∋ x be the minimal resolution,

and we may write KX

+ BX

= f∗(KX + B) for some R-divisor BX

:=∑j bjBj ≥ 0, where Bj are distinct prime divisors on X. Let I ∈ Z>0 such

that I|det(X ∋ x), then

(1) {∑

j n′j bj − 1 > 0 | n′

j ∈ Z≥0} ⊆ 1I{∑

i nibi − t |∑

i nibi − t > 0, ni ∈

Z≥0, t ≤ I}, and

(2) {∑

j n′j bj − 1 ≥ 0 | n′

j ∈ Z≥0} ⊆ 1I{∑

i nibi − t |∑

i nibi − t ≥ 0, ni ∈

Z≥0,∑

i ni > 0, t ≤ I}.

Proof. Let DG be the dual graph of f . Let {Ei}1≤i≤m be the set of f -exceptional divisors, and wi := −Ei ·Ei, ai := a(Ei,X ∋ x,B) for all i.

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26 J.Han, and Y.Luo

For each 1 ≤ j ≤ m, (KY + f−1∗ B +

∑mi=1(1 − ai)Ei) · Ej = 0, which

implies that∑m

i=1(ai − 1)Ei · Ej = −Ej2 − 2 + f−1

∗ B ·Ej , or equivalently,

E1 ·E1 · · · Em ·E1...

. . ....

E1 · Em · · · Em · Em

a1 − 1...

am − 1

=

w1 − 2 + f−1∗ B ·E1

...

wm − 2 + f−1∗ B ·Em

.

By [10, Lemma 3.40], (Ei · Ej)1≤i,j≤m is a negative definite matrix. Let(sij)m×m be the inverse matrix of (Ei ·Ej)1≤i,j≤m. By [10, Lemma 3.41] andthe assumption on I, we have Isij ∈ Z<0 for 1 ≤ i, j ≤ m. Thus for all i,

1− ai =1

I

m∑

j=1

(−Isij) · (wj − 2 + f−1∗ B · Ej).(6.1)

Since wj ≥ 2 for all j, (1) and (2) follows immediately from (6.1) and the

equation BX

=∑

j bjBj = f−1∗ B +

∑mi=1(1− ai)Ei. �

Proof of Theorem 6.4. By Lemma 3.15, we may assume that mld(X ∋ x,B) ≤1.

If mld(X ∋ x,B) = pld(X ∋ x,B), then a(E,X, 0) ≤ 1 for some primedivisor E over X ∋ x such that a(E,X,B) = mld(X ∋ x,B). So we mayassume that mld(X ∋ x,B) 6= pld(X ∋ x,B).

By Lemma 3.18, there exists a birational morphism f : Y → X ∋ x which

satisfies Lemma 3.18(1)–(4). Let f : X → X be the minimal resolution of

X ∋ x, and g : Y → f the natural morphism induced by f . We may write

KX

+ BX

= f∗(KX + B) for some R-divisor BX

:=∑

j bjBj ≥ 0 on X,

where Bj are distinct prime divisors.

By Lemma 6.7(1), we have {∑

j n′j bj−1 > 0 | n′

j ∈ Z≥0} ⊆ 1I{∑

i nibi−t >

0 | ni ∈ Z≥0, t ≤ I} ⊆ [γI,∞) by the assumption. Since the dual graph DG

of f contains only one (−1)-curve, there exists a closed point x ∈ X suchthat center

XE = x for any g-exceptional divisor E. Apply Theorem 1.3 to

the surface germ (X ∋ x, BX), we are done. �

Proof of Theorem 6.5. By Lemma 3.15, we may assume that mld(X ∋ x,B) ≤1.

We may assume that B 6= 0, otherwise we may take f : X → X ∋x the minimal resolution, and K

X+ B

X= f∗KX , where B

Xis an snc

divisor. Since X ∋ x is klt, all elements in S are on X , thus |S| ≤ N , anda(E,X, 0) ≤ 1 for all E ∈ S. For the same reason, we may assume that not

all elements of S are on the minimal resolution X of X ∋ x.If mld(X ∋ x,B) = 0, then by Lemma 6.2, there exists a unique excep-

tional divisor E over X ∋ x such that a(E,X,B) = 0. By Theorem 6.4,

a(E,X, 0) ≤ 2⌊1+ 32I2

γ2+ I

γ⌋. Thus we may assume that mld(X ∋ x,B) > 0

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On boundedness of divisors computing MLDs for surfaces 27

from now on. Since∑

i nibi 6= 1 for any ni ∈ Z≥0, we have ⌊B⌋ = 0, and(X,B) is klt near x ∈ X, hence by [10, Proposition 2.36], S is a finite set.

Now, we may apply Lemma 6.1 to the set S, there exists a birational

morphism f : Y → X ∋ x that satisfies Lemma 6.1(1)–(4). Let f : X → X

be the minimal resolution of X ∋ x, and g : Y → X the natural morphism

induced by f . We may write KX+ B

X= f∗(KX + B) for some R-divisor

BX

:=∑

j bjBj ≥ 0 on X , where Bj are distinct prime divisors on X.

By Lemma 6.7(2), we have {∑

j n′j bj−1 ≥ 0 | n′

j ∈ Z≥0} ⊆ 1I{∑

i nibi−t ≥

0 | ni ∈ Z≥0,∑

i ni > 0, t ≤ I} ⊆ [γI,∞). Since the dual graph DG of f

contains only one (−1)-curve, there exists a closed point x ∈ X such thatcenter

XE = x for any g-exceptional divisor E. Apply Theorem 1.4 to the

surface germ (X ∋ x, BX), we are done. �

To end this section, we provide some examples which show that Theorem1.4 and Theorem 6.5 do not hold if we try to relax their assumptions.

Example 6.8 below shows that Theorem 1.4 does not hold when X ∋ x isnot klt.

Example 6.8. Let X ∋ x be an lc surface germ, such that mld(X ∋ x) = 0and the dual graph DG of the minimal resolution of X ∋ x is a cycle ofsmooth rational curves. Then both S = {E | E is a prime divisor over X ∋x, a(E,X) = mld(X ∋ x)}, and supE∈S{a(E,X, 0)} are not bounded fromabove.

Theorem 1.4 does not hold if∑

i nibi = 1 for some ni ∈ Z≥0, even whenthe elements in {(X ∋ x,Bk)}k are all klt surface germs. We remark thatin thise case Sk := {E | E is a prime divisor over X ∋ x,mld(X ∋ x,Bk) =a(E,X,Bk)} < +∞ for all k.

Example 6.9. Let (X ∋ x) := (A2 ∋ 0), Dk the Cartier divisor which isdefined by the equation x−yk = 0, and Bk := 1

2Dk+12Dk+1 for any positive

integer k. Then (X ∋ x,Bk) is canonical near x for any k as multxBk ≤ 1.For each k, we may construct a sequence of blow-upsXk → Xk−1 → · · · →

X1 → X0 := X with the data (fi, Fi, xi ∈ Xi), such that xi−1 ∈ Fi−1 is theintersection of the strict transforms of Dk and Dk+1 on Fi−1 for 1 ≤ i ≤ k.Then a(Fi,X,Bk) = 1 = mld(X ∋ x,Bk) for any k and 1 ≤ i ≤ k. Thusboth |Sk| and supE∈Sk

{a(E,X, 0)} are not bounded from above as |Sk| = k,and a(Fk,X, 0) ≥ k for each k.

The following example shows that Theorem 6.5 does not hold if we do notfix the germ, even when we bound the determinant of the surface germs.

Example 6.10. Let {(Xk ∋ xk)}k≥1 be a sequence of surface germs, suchthat each Xk ∋ xk is a Du Val singularity of type Dk+3 ([10, Theorem 4.22]).

Now we fix a positive integer k ≥ 1. Let fk : Xk → Xk ∋ xk be the

minimal resolution, DGk the dual graph for fk, and {Ei,k}−2≤i≤k the vertices

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28 J.Han, and Y.Luo

of DGk as in Figure 8. Let Bk be a smooth curve on Xk, such that Bk∪Ek,k

are snc for all k, and Bk ·Ek,k = pk ∈ Z>0, Bk ·Ej,k = 0 for −2 ≤ j < k on Xk.

Let Bk := 13pk

fk∗Bk. We may consider the following sequence of klt surface

germs {(Xk ∋ xk, Bk)}k, and let ai,k := a(Ei,k,Xk, Bk) for −2 ≤ i ≤ k.

We have aj,k = 23 for j 6= −1,−2, and a−1,k = a−2,k = 5

6 , thusKX+B

Xk=

f∗(KXk+Bk), where B

Xk:= 1

3pkBk +

16

∑i=−1,−2Ei,k +

13

∑j 6=−1,−2Ej,k is

an snc divisor. Thus (Xk, BXk) is terminal, and

a0,k = · · · ak,k = mld(Xk ∋ xk, Bk) = pld(Xk ∋ xk, Bk).

We have det(Xk ∋ xk) = 4 for all k ≥ 4, but |Sk| = k+ 1 for k ≥ 4 is notbounded from above.

Bk

Ek,k E1,k

E−2,k

E0,k E−1,k

Figure 8. The dual graph of fk

7. An equivalent conjecture for MLDs on a fixed germ

Conjecture 7.2 could be regarded as an inversion of stability type con-jecture for divisors which compute MLDs, see [3, Main Proposition 2.1],[4, Theorem 5.10] for other inversion of stability type results.

Definition 7.1. Let X be a normal variety, and B :=∑

biBi an R-divisoron X, where Bi are distinct prime divisors. We define ||B|| := maxi{|bi|}.Let E be a prime divisor over X, and Y → X a birational model, such thatE is on Y . We define multE Bi to be the multiplicity of the strict transformof Bi on Y along E for each i, and multE B :=

∑i bimultE Bi.

Conjecture 7.2. Let Γ ⊆ [0, 1] be a finite set, X a normal quasi-projectivevariety, and x ∈ X a closed point. Then there exists a positive real numberτ depending only on Γ and x ∈ X satisfying the following.

Assume that (X ∋ x,B) and (X ∋ x,B′) are two lc germs such that

(1) B′ ≤ B, ||B −B′|| < τ,B ∈ Γ, and(2) a(E,X,B′) = mld(X ∋ x,B′) for some prime divisor E over X ∋ x.

Then a(E,X,B) = mld(X ∋ x,B).

Proposition 7.3. For any fixed Q-Gorenstein germ X ∋ x, Conjecture 7.2is equivalent to Conjecture 1.1, thus Conjecture 7.2 is equivalent to the ACCconjecture for MLDs.

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On boundedness of divisors computing MLDs for surfaces 29

Proof. Suppose that Conjecture 7.2 holds. Let t := min{1, τ} > 0. Then||(1 − t)B − B|| < τ . Let E be a prime divisor over X ∋ x, such thata(E,X, (1−t)B) = mld(X ∋ x, (1−t)B). Then mld(X ∋ x, 0) ≥ a(E,X, (1−t)B) = a(E,X,B) + tmultE B ≥ tmultE B, and

multE B ≤1

tmld(X ∋ x, 0).

By assumption, mld(X ∋ x, 0) ≥ mld(X ∋ x,B) = a(E,X,B) = a(E,X, 0)−multE B. Hence

a(E,X, 0) ≤ (1 +1

t)mld(X ∋ x, 0),

and Conjecture 1.1 holds.

Suppose that Conjecture 1.1 holds, then there exists a positive real num-ber N which only depends on Γ and X ∋ x, such that a(E0,X, 0) ≤ N forsome E0 satisfying a(E0,X,B) = mld(X ∋ x,B). In particular, multE0

B =−a(E0,X,B) + a(E0,X, 0) ≤ N .

By [7, Theorem 1.1], there exists a positive real number δ which onlydepends on Γ and X ∋ x, such that a(E,X,B) ≥ mld(X ∋ x,B) + δ forany (X ∋ x,B) and prime divisor E over X ∋ x such that B ∈ Γ anda(E,X,B) > mld(X ∋ x,B).

We may assume that Γ \ {0} 6= ∅. Let t := δ2N and τ := t ·min{Γ \ {0}}.

We claim that τ has the required properties.For any R-divisor B′, such that ||B′−B|| < τ , we have B−B′ < tB. Let

E be a prime divisor over X ∋ x, such that a(E,X,B′) = mld(X ∋ x,B′).Suppose that a(E,X,B) > mld(X ∋ x,B). Then a(E,X,B) ≥ mld(X ∋x,B) + δ = a(E0,X,B) + δ. Thus

a(E0,X, 0) −multE0B′ = a(E0,X,B′)

≥a(E,X,B′) ≥ a(E,X,B)

≥a(E0,X,B) + δ = a(E0,X, 0) −multE0B + δ.

Hence

δ ≤ multE0(B −B′) ≤ tmultE0

B ≤ tN =δ

2,

a contradiction. �

Proposition 7.4. For any fixed Q-Gorenstein germ X ∋ x, Conjecture1.1 implies Conjecture 7.2 holds for any DCC set Γ when B has only onecomponent.

Proof. Otherwise, there exist two sequence of germs {(X ∋ x, tiBi)}i and{(X ∋ x, t′iBi)}i, and prime divisors Ei over X ∋ x, such that

• for each i, Bi is a prime divisor on X, ti ∈ Γ is increasing, andt′i < ti,

• t := limi→∞ ti = limi→∞ t′i,• a(Ei,X, t′iBi) = mld(X ∋ x, t′iBi), and• a(Ei,X, tiBi) 6= mld(X ∋ x, tiBi).

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30 J.Han, and Y.Luo

By Proposition 7.3, there exists a positive real number τ which only dependson t and X ∋ x, such that if |t − t′i| < τ , then a(Ei,X, tBi) = mld(X ∋x, tBi). Since a(Ei,X, t∗Bi) is a linear function respect to the variable t∗,we have a(Ei,X, tiBi) = mld(X ∋ x, tiBi), a contradiction. �

To end this section, we provide some examples which show that Theorem7.2 does not hold for surface germs (X ∋ x,B) in general if either Γ is aDCC set (see Example 7.5), or we do not fix X ∋ x (see Example 7.6).

Before we give the following two examples, we fix some notations first. Let

X ∋ x be an lc surface germ, f : X → X ∋ x the minimal resolution, and

DG the dual graph of f . Assume that DG consists of two smooth rational

curves E1 and E2, and E1 · E2 = 1. Let B1 and B2 be smooth curves on

X, such that B1 ∪ E1 ∪ B2 ∪ E2 is an snc divisor, B1 · E2 = B2 · E1 = 0

(see Figure 9). Let p := B1 · E1 ∈ Z>0, q := B2 · E2 ∈ Z>0, Bi := f∗Bi for

i = 1, 2, and B := b1pB1 +

b2qB2 (respectively B′ :=

b′1

pB1 +

b′2

qB2). Let ai :=

a(Ei,X ∋ x,B) (respectively a′i := a(Ei,X ∋ x,B′)), and wi := −Ei ·Ei fori = 1, 2.

B1

E1 E2

B2

Figure 9. The dual graph of f

Example 7.5. For any positive real number τ , we may choose k ∈ Z>0 suchthat 1

k< τ . Let

• Γ = {12 − 1

m| m ∈ Z>0},

• wi = 2 for all 1 ≤ i ≤ 2,• b1 =

12 , b2 =

12 − 1

k, and

• b′1 =12 −

1k, b′2 =

12 −

1k+1 .

Since KX+B

X= f∗(KX +B) (respectively K

X+B′

X= f∗(KX +B′)), and

BX

(respectively B′X) is an snc Q-divisor with all coefficients≤ 1

2 , (X,BX)

(respectively (X,B′X)) is canonical, and mld(X ∋ x,B) = pld(X ∋ x,B)

(respectively mld(X ∋ x,B′) = pld(X ∋ x,B′)).Now a1 = 1

2 = mld(X ∋ x,B), while a′1 = 12 + 1

k> a′2 = 1

2 + 1k+1 =

mld(X ∋ x,B′). We get the desired counterexample for Conjecture 7.2

when Γ is a DCC set. We remark that we can not even find a f -exceptionaldivisor Ei such that ai = mld(X ∋ x,B) and a′i = mld(X ∋ x,B′) in thisexample.

Example 7.6. For any positive real number τ , we may choose k ∈ Z>0 suchthat 1

k< τ . For any fixed positive integer n ≥ 2, let

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On boundedness of divisors computing MLDs for surfaces 31

• Γ = {12},

• w1 = 2 + 2k,w2 = 1 + 2k,• b1 = b2 =

12 , and

• b′1 =12 −

1k, b2′ =

12 − 1

2k+1 .

Since KX

+ BX

= f∗(KX + B) (respectively KX

+ B′X

= f∗(KX + B′)),

and BX

(respectively B′X) is an snc Q-divisor with all coefficients≤ 1

2 , thus

(X,BX) (respectively (X,B′

X)) is canonical, and mld(X ∋ x,B) = pld(X ∋

x,B) (respectively mld(X ∋ x,B′) = pld(X ∋ x,B′)).

Now a1 = 12(2k+1) = mld(X ∋ x,B), while a′1 =

1

2+ 1

k

2k+1 > a′2 =1

2+ 1

2k+1

2k =

mld(X ∋ x,B′). We get the desired counterexample for Conjecture 7.2when the germ X ∋ x is not fixed. We remark that we can not even find af -exceptional divisor Ei such that ai = mld(X ∋ x,B) and a′i = mld(X ∋x,B′) in this example.

It would also be interesting to ask the following:

Conjecture 7.7. Let Γ ⊆ [0, 1] be a finite set and X ∋ x a germ. Then thereexists a positive real number τ depending only on Γ and X ∋ x satisfyingthe following.

Assume that

(1) B′ ≤ B, ||B −B′|| < τ,B ∈ Γ, and(2) KX +B′ is R-Cartier.

Then KX +B is R-Cartier.

Appendix A. ACC for MLDs for surfaces according to

Shokurov

In this appendix, we give a simple proof of the ACC for MLDs for surfacesfollowing Shokurov’s idea [14]. As one of the key steps, we will show theACC for pld’s for surfaces (Theorem 2.2) which we apply to prove Theorem1.2. The proof of Theorem 2.2 only depends on Lemma 3.11 in the previoussections. The other results in this appendix are not used elsewhere in thispaper.

Definition A.1. Let DG be a fixed dual graph. PDG denotes the set of alllc surface germs (X ∋ x,B) with the following properties:

• there exist a smooth surface Y , and a projective birational morphismf : Y → X ∋ x with the dual graph DG, and

• a(E,X,B) ≤ 1 for any E ∈ DG.

Lemma A.2. Fix a DCC set Γ ⊆ [0, 1] and a dual graph DG whose vertices{Ei}1≤i≤n are smooth rational curves. Then the set {a(E,X,B) | (X ∋x,B) ∈ PDG , B ∈ Γ, E ∈ DG} satisfies the ACC.

Proof. Let {Ei}1≤i≤n be the set of vertices of DG, and wi := −Ei · Ei for1 ≤ i ≤ n.

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32 J.Han, and Y.Luo

For any surface germ (X ∋ x,B) ∈ PDG , let f : Y → X ∋ x be aprojective birational morphism, such that the dual graph of f is DG. Wemay write

KY + f−1∗ B +

i

(1− ai)Ei = f∗(KX +B),

For each 1 ≤ j ≤ n, we have

(KY + f−1∗ B +

i

(1− ai)Ei) · Ej = 0,

which implies f−1∗ B ·Ej − 2− Ej

2 =∑n

i=1(ai − 1)Ei ·Ej , orE1 ·E1 · · · En ·E1

.... . .

...E1 ·En · · · En · En

a1 − 1

...an − 1

=

w1 − 2 + f−1

∗ B ·E1...

wn − 2 + f−1∗ B ·En

,

where ai = a(Ei,X ∋ x,B) for 1 ≤ i ≤ n. Since B ∈ Γ, {f−1∗ B · Ej |

(X ∋ x,B) ∈ PDG, Ej ∈ DG, B ∈ Γ} is a set which satisfies the DCC. Thus{f−1

∗ B ·Ej+wj−2 | (X ∋ x,B) ∈ PDG , Ej ∈ DG, B ∈ Γ} is also a set whichsatisfies the DCC.

By [10, Lemma 3.40], (Ei · Ej)1≤i,j≤n is a negative definite matrix. Let(sij)n×n be the inverse matrix of (Ei · Ej)1≤i,j≤n. By [10, Lemma 3.41],sij < 0 for any 1 ≤ i, j ≤ n, and this shows that a(E,X ∋ x,B) belongs tothe set

{n∑

j=1

sij(wj − 2 + f−1∗ B · Ej) + 1 | (X ∋ x,B) ∈ PDG , Ej ∈ DG, B ∈ Γ},

which satisfies the ACC. �

Proof of Theorem 2.1. Let f : Y → X ∋ x be a birational morphism whichsatisfies Lemma 3.18(1)–(4). By Lemma 3.13(1) and Lemma 3.19, the num-ber of vertices of the dual graph DG of f is bounded, and by Lemma 3.11(1),DG belongs to a finite set {DGi}1≤i≤N which only depends on ǫ0 and Γ. ByLemma A.2, we are done. �

We are going to prove Theorem 2.2. Our proof avoids calculations tocompute pld’s explicitly.

We introduce some notations here. We say a surface germ (X ∋ x,B)is of type GIm if the dual graph DG of the minimal resolution of X ∋ x isa chain with m vertices, and we say (X ∋ x,B) is of type GT 1,1,m if thedual graph DG is a tree with only one fork E0 and three branches of length(1, 1,m).

Proof of Theorem 2.2. It suffices to show that Pld(2,Γ)∩ [ǫ0, 1] satisfies theACC for any given ǫ0 ∈ (0, 1). Let γ := mint∈Γ,t>0{t, 1} > 0, and δ :=min{ ǫ0

3 ,γ2} > 0.

Let {(Xi ∋ xi, Bi)}i be a sequence of lc surface germs, such that

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On boundedness of divisors computing MLDs for surfaces 33

• Bi ∈ Γ,• pld(Xi ∋ xi, Bi) ∈ [ǫ0, 1], and• the sequence {pld(Xi ∋ xi, Bi)}i is increasing.

It suffices to show that passing to a subsequence, {pld(Xi ∋ xi, Bi)}i is aconstant sequence.

Let fi : Yi → Xi ∋ xi be the minimal resolutions, DGi the dual graph of fi,{Ek,i}k the vertices of DGi, wk,i := −(Ek,i · Ek,i), and ak,i := a(Ek,Xi, Bi).

We may write KYi+ f−1

i∗ Bi +∑

k(1 − ak,i)Ek,i = f∗i (KXi

+ Bi). Thus forany Ej,i ∈ DG,

0 = −2 + f−1i∗ Bi · Ej,i + wj,iaj,i +

k 6=j

(1− ak,i)Ek,i · Ej,i.(A.1)

By [10, Theorem 4.7] and [10, Theorem 4.16], except for finitely manygraphs, any dual graph DGi is either of type GIm or of type GT 1,1,m (m ≥ 3).Possibly passing to a subsequence of {(Xi ∋ xi, Bi)}i, by Lemma A.2, weonly need to consider the following cases.

Case 1: The dual graphs DGi are of type GImi, mi ≥ 3.

We may index each DGi as {Ek}−n1,i≤k≤n2,i, where Ek,i is adjacent to

Ek+1,i for −n1,i ≤ k ≤ n2,i − 1. Then for each i, possibly shifting k, thereexist a vertex E0,i ∈ DG with a(E0,i,X,B) = pld(Xi ∋ xi, Bi).

If a(Ek,i,Xi, Bi) = pld(Xi ∋ xi, Bi) for any E ∈ DG, then possibly shift-ing k, we may assume that n1,i = 0, and we may set Ii := {E0,i}.

Otherwise, we may assume that there exists a vertex E0,i ∈ DGi, suchthat a0,i = pld(Xi ∋ xi, Bi), and E0,i is adjacent to some vertex Ek0,i, suchthat ak0,i > a0,i. We show that there are two possibilities, either

• E0,i is adjacent to only one vertex E1,i, thus a1,i > a0,i, or• E0,i is adjacent to two vertices E1,i, E−1,i, such that a−1,i− a0,i ≥ δ.

It suffices to show a−1,i − a0,i ≥ δ in the latter case. Let j = 0 in (A.1), wehave

0 < (a−1,i − a0,i) + (a1,i − a0,i) = f−1i∗ Bi ·E0,i + (w0,i − 2)a0,i.

Thus either (w0,i − 2)a0,i ≥ a0,i ≥ ǫ0 ≥ 2δ or w0 = 2 and f−1i∗ Bi ·E0,i ≥ γ ≥

2δ. Possibly switching {Ek,i}k<0 and {Ek,i}k>0, we may assume that a−1,i−a0,i ≥ δ. Let Ii be the vertex-induced subgraph of DGi by {Ek,i}k≤0. ByLemma 3.11(1)–(2), {Ii}i>0 is finite set. In particular, {n1,i}i is a boundedsequence.

w−n1,i

Ii

w−1,i w0,i w1,i wn2,i

Figure 10. Dual graph of DGi

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34 J.Han, and Y.Luo

We may assume that {n2,i}i is an unbounded sequence, otherwise the set{DGi}i is a finite set, and by Lemma A.2, we are done. Possibly passing toa subsequence of {(Xi ∋ xi, Bi)}i, we may assume that:

• I := Ii is a fixed graph with weights wk := wk,i for any k ≤ 0 andany i,

• {si := a1,i − a0,i}i is decreasing as n2,isi ≤ 1, and

• for each k ≤ 0, the sequence {f−1i∗ Bi · Ek,i}i is increasing.

For convenience, set a−n1−1,i := 1 for any i, where n1 + 1 is the numberof vertices of I. By (A.1), for any −n1 ≤ k ≤ 0 and any i, we have

ak−1,i − ak,i = f−1i∗ Bi · Ek,i + (wk − 2)ak,i + (ak,i − ak+1,i).(A.2)

In particular, when k = 0, we have

(a1,i − a0,i) + (a−1,i − a0,i) = f−1i∗ Bi ·E0,i + (w0 − 2)a0,i.(A.3)

Since a0,i = pld(Xi ∋ xi, Bi) is increasing, the right-hand side of (A.3) isincreasing respect to i. Note that {si = a1,i − a0,i}i is decreasing, {a−1,i −a0,i}i as well as {a−1,i}i are increasing respectively.

By induction on k using (A.2), for any −n1 ≤ k ≤ 0, we conclude thatthe sequences

{ak−1,i − ak,i}i and {ak,i}i

are increasing respectively. Since

1 = a−n1−1,i =0∑

k=−n1

(ak−1,i − ak,i) + a0,i,

each summand must be a constant sequence. In particular, {a0,i = pld(Xi ∋xi, Bi)}i is a constant sequence, and we are done.

Case 2: The dual graphs DGi are of type GT 1,1,mi, mi ≥ 3.

For each i, let E0,i be the fork in DGi. By (2) and (3) of Lemma 3.11,a0,i = pld(Xi ∋ xi, Bi). Let E1,i, E−1,i, E−2,i be the vertices adjacent toE0,i, where E1,i belongs to the chain DGi\{E1,i ∪ E2,i}. We decompose thedual graph of fi into three parts Ii,Mi, Ti, where Ii, Mi Ti are the vertex-induced subgraphs of DGi by {Ek,i}−2≤k≤0, {Ek,i}1≤k≤n0,i

, {Ek,i}n0,i<k≤mi

respectively, such that n0,i ≥ 0, ak+1,i − ak,i < δ for 1 ≤ k ≤ n0,i, andak+1,i − ak,i ≥ δ for k > n0,i (see Figure 11). We remark that if n0,i = 0,then Mi = ∅.

By Lemma 3.11(4), Mi only consists of (−2)-curves whose intersectionswith f−1

i∗ Bi are all zero, thus by (A.1), for each i, si := ak+1,i − ak,i is aconstant for any 0 ≤ k ≤ n0,i. By Lemma 3.11(1)–(2), {Ti}i>0 is a finiteset.

By Lemma A.2 again, we may assume that {n0,i}i is an unbounded se-quence. Possibly passing to a subsequence, we may assume that:

• {n0,i}i is strictly increasing, n0,i ≥ 1 for any i, and {si = a1,i−a0,i}iis decreasing as n0,isi ≤ 1,

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On boundedness of divisors computing MLDs for surfaces 35

w−1

Ii

w0

w−2

22

Mi

2wn0,i+1

Ti

wmi

Fk,i = En0,i+k,i

Figure 11

• T := Ti is a fixed graph with weights wk := wn0,i+k,i for any 0 <k ≤ mi − n0,i and any i,

• I := Ii is a fixed graph with weights wk := wk,i for any −2 ≤ k ≤ 0and any i, and

• for each 0 < k ≤ mi − n0,i, the sequence {f−1i∗ Bi · En0,i+k,i}i is

increasing.

Let j = 0 in (A.1), we have

(a−1,i −1

w−1a0,i) + (a−2,i −

1

w−2a0,i)

=1 + f−1i∗ Bi ·E0,i + (w0 − 1−

1

w−1−

1

w−2)a0,i − (a1,i − a0,i).

(A.4)

Suppose that possibly passing to a subsequence, {si = a1,i−a0,i}i is strictlydecreasing, then possibly passing to a subsequence, the right-hand side of(A.4) is strictly increasing, so is {(a−1,i −

1w−1

a0,i) + (a−2,i −1

w−2a0,i)}i.

Thus possibly passing to a subsequence and switching E−1 with E1, we mayassume that {a−1,i −

1w−1

a0,i}i is strictly increasing. Let j = −1 in (A.1),

we have

w−1a−1,i − a0,i = 1− f−1i∗ Bi ·E−1,i.

It follows that {1−f−1i∗ Bi ·E−1,i}i is strictly increasing, a contraction. Hence

possibly passing to a subsequence, we may assume that {si}i is a constantsequence, and there exists a non-negative real number s, such that s = sifor all i. Since 0 ≤ n0,is ≤ 1 for all i and {n0,i}i is strictly increasing, wehave s = 0. In particular, an0,i,i = an0,i+1,i = pld(Xi ∋ xi, Bi) is increasing.

The rest part of the proof of Case 2 is very similar to that of Case 1. Forthe reader’s convenience, we give the proof in details.

Let m′ := mi − n0,i be the number of vertices of T . We denote Fk,i :=En0,i+k,i (see Figure 11), a′k,i := an0,i+k,i, w

′k := wk, and set a′m′+1,i =

ami+1,i = 1. By (A.1), for any 1 ≤ k ≤ m′ and any i, we have

a′k+1,i − a′k,i = f−1i∗ Bi · Fk,i + (w′

k − 2)a′k,i + (a′k,i − a′k−1,i).(A.5)

In particular, when k = 1, we have

a′2,i − a′1,i = f−1i∗ Bi · F1,i + (w′

1 − 2)a′1,i.(A.6)

It follows that {a′2,i − a′1,i} is increasing. By induction on k using (A.5), for

any 1 ≤ k ≤ m′, we conclude that both sequences {a′k+1,i−a′k,i}i and {a′k,i}i

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36 J.Han, and Y.Luo

are increasing respectively. Since

1 = a′m′+1,i =

m′∑

k=1

(a′k+1,i − a′k,i) + a′1,i,

each summand must be a constant sequence. In particular, {a′1,i = pld(Xi ∋xi, Bi)}i is a constant sequence, and we are done. �

Proof of Theorem 1.5. By Lemma 3.15, it suffices to prove that Mld(2,Γ)∩[ǫ0, 1] satisfies the ACC for any ǫ0 ∈ (0, 1). This follows from Theorem 2.1and Theorem 2.2. �

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Jingjun Han, Department of Mathematics, Johns Hopkins University, Bal-

timore, MD 21218, USA

E-mail address: [email protected]

Yujie Luo, Department of Mathematics, Johns Hopkins University, Balti-

more, MD 21218, USA

E-mail address: [email protected]