Bounding singular surfaces of general type. in in articles [2], [3], for log surfaces of general type with semi log canonical singularities.

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1 Bounding singular surfaces of general type Valery Alexeev 1 and Shigefumi Mori 2 1 Department of Mathematics, University of Georgia, Athens GA 30605, USA 2 Research Institute for Mathematical Sciences Kyoto University, Kyoto , JAPAN Abstract. We provide simpler proofs of several boundedness theorems, contained in in articles [2], [3], for log surfaces of general type with semi log canonical singularities. At the same time, we make these proofs eective, with explicit upper bounds. 0 Introduction In this work, we present several eective boundedness results for various invariants of singular surfaces. The most important application of these results is the existence of the moduli space of semi-log canonical surfaces of general type. The construction of this moduli space was started in [7], and the boundedness provides the nal necessary step to complete it. This paper was written in 1995 for a planned volume on moduli of surfaces of general type, which has not yet been published. Therefore, with the editors' consent, we are publishing the paper independently. Time, meanwhile, was not standing still, and there have been new developments in the eld, none of which supersede the results of our work. The most signicant is [4]. In it, Karu proves the boundedness of smoothable semi-log canonical n-folds assuming the Minimal Model Program in dimension n + 1. The Minimal Model Program in dimension 3, of course, has been a very signicant and deep development in algebraic geometry of 1980s and 1990s, due to eorts of many people. As a corollary, Karu's theorem gives boundedness of smoothable semi-log canonical surfaces of general type. Using this, one can prove existence of the coarse moduli space for a restricted moduli functor of stable surfaces, dened on the category of reduced schemes only. The usual moduli functor, dened on the category of all schemes over the base, still requires looking at non-smoothable surfaces. That is the case because smoothable surfaces can be deformed to non-smoothable ones, and there is no known good denition for an innitesimal family of smoothable varieties. Hence, the more general boundedness result is still necessary. Our approach also is much more elementary and uses only two-dimensional geometry The main purpose of this paper is to give simpler proofs for several theorems contained in articles [2], [3] of one of the authors, and at the same time make these proofs eective. We give explicit formulas for the bounds in

2 2 Valery Alexeev and Shigefumi Mori the theorems. It has to be admitted, however, that these bounds are quite high. Notation 0.2. Let X be a normal surface dened over an algebraically closed eld of arbitrary characteristic, and let B = P b j B j be an R-divisor on X with b j 0 such that K X + B is R-Cartier. For a resolution of singularities f : Y! X we, as it is customary, call the coecients a i in the following formula the log discrepancies. K Y + X b j f 1 B j + X F i = f (K X + X b j B j ) + X a i F i ; where F i are the exceptional divisors of the morphism f. We say that the pair (X; P b j B j ) is "-log canonical (resp. "-log terminal) if all a i " and b j 1 " (resp. if the inequalities are strict) for any resolution f : Y! X. For " = 0 we get the usual denition of a log canonical (resp. Kawamata log terminal) pair. We will also use the abbreviations lc, lt and klt. If we are given a normal surface Z birational to X, we can dene B Z as follows. Take a resolution f : Y! X which dominates Z, say via g : Y! Z. Then write K Z + B Z = g f (K X + B). It is easy to see that B Z does not depend on the choice of Y. We will write A B (resp. A B) if A is numerically (resp. linearly) equivalent to B. 1 Bounds for Picard groups 1.1. The main result of this section is (1.8). We start with a few easy introductory lemmas. The next one is used very often in this paper. Lemma 1.2. Let X be a nonsingular projective surface and B = P b j B j be an R-divisor on X with 0 b j 1 " < 1. If E is an irreducible curve on X with (K + B) E 0 and E 2 < 0 then E ' P 1 and E 2 2=". Proof. Follows from 2 2p a (E) 2 = (K + E)E = "E 2 + (K + (1 ")E) E "E 2 + (K + B)E "E 2 < 0 ut The following is a special case of (2.8). However (1.8) needs only this easy form. PLemma 1.3. Let X be the ruled rational surface F n (n 0) and B = bj B j be an R-divisor on X with 0 b j 1. Assume that (K + B) is nef. Then P b j 4.

3 Bounding singular surfaces of general type 3 Proof. Let F; S be a ber and a section of X such that S 2 = n. We can always add a curve to B with coecient 0 without changing the situation. Thus we may set B 0 = S. From 0 (K + B) F, we see 2 X b j (B j F ) X (B jf )6=0 Since (K S) = n 2, we get the following from 0 (K + B) S. b j : 2 n(1 b 0 ) + X b j (B j S) X (B js)>0 Since (E F ) = 0 implies (E S) > 0 for an arbitrary irreducible curve E, we have the lemma. ut The following is a special case of (4.2.1) in [12], which is however enough as the starting point of (1.8) and allows us to to focus on the main case (3). Lemma 1.4. Let X be a nonsingular projective surface and B = P b j B j be an R-divisor on X with 0 b j 1 " < 1. Assume that K + B 0. Then one of the following is true: 1. B = 0, K X 0 and X is either a K3, or an Enriques, or an Abelian, or a hyperelliptic surface (in particular (X) 22 by the classication), 2. X ' P C (E), where C is an elliptic curve and E is a locally free sheaf which is an extension of an invertible sheaf L by O C such that 0 degl 2, 3. X is rational, and either X ' P 2 or there exists a birational morphism g : X! F n with n 2=". Remark 1.5. For the study of K + B 0, it is essential by (1.4) to study the birational map g : X! X = F n and B = g (B). Let Y be any nonsingular projective surface decomposing g as : X! Y and : Y! X. We note that (K X + B) = (K X + B) by K X + B 0 and (K X + B) has an eective boundary. Proof. If B = 0 and K 0 then X is one of (1) by the classication of surfaces. Otherwise, K B 6= 0. Assuming X 6= P 2, we contract ( 1)- curves on X and its contractions until we get a birational morphism g : X! X to a model X which is a P 1 -bundle over a nonsingular curve C. (Since K X g g K X, we have g B 6= 0 and we see the above assertion.) If g(c) = 0 then X = F n and n 2=" by Lemma 1.2 applied to X. This is the case (3). Assume therefore that g(c) 1. Let us denote by B j (resp. B) the images of B j (resp. B) on X, and omit B j 's (and B j 's) with b j = 0 in the rest of this proof. Case 1. There exists a curve D on X with D 2 < 0: By Lemma 1.2 D is a nonsingular rational curve. Since it does not lie in a ber of X! C, g(c) = 0. So this does not occur. b j :

4 4 Valery Alexeev and Shigefumi Mori Case 2. D 2 0 for all curves D on X. By B K X, we have 0 8 8g(C) = K 2 X = B2 0: 2 It follows that all B j = K X B j = B j B k = 0 and that g(c) = 1. By the arithmetic genus formula, p a (B j ) = 1. The normalization of each B j is irrational, otherwise C would be rational too. Hence, each B j is a nonsingular elliptic curve, and dierent B j do not intersect. It is then easy to see that g : X! X is an isomorphism by Remark 1.5 and Lemma 1.6. Let S be a section of : X! C such that S 2 ( 0) is the smallest. From the standard exact sequence 0! O C! O(S)! O S (S)! 0; we see that h 0 (S) S 2. We claim that S 2 2. Indeed if S 2 > 2 then js F j 6= ; by h 0 (O F (S)) = h 0 (O F (1)) = 2, which would produce a section S 0 with (S 0 ) 2 < S 2. Thus S 2 2 and we are done by X ' P( O(S)). ut Th next two results are technical remarks for the reader's convenience. Lemma 1.6. Let P : Y! X be a blow up of a nonsingular surface X at P 2 X and let B X. Then P (K X + B) = K Y + 1 P (B) + (m P (B) 1)E; where E is the exceptional divisor and m P (B) is the multiplicity of B at P. Furthermore m Q ( 1 P (B)) m P (B) for every Q 2 E. Proof. The formula is a direct computation. The second assertion follows from E P 1 (B) = m P (B). ut The last assertion can be used to switch the order of two successive blow ups if m i < m i+1 in the next corollary. Corollary 1.7. Let g : X! X be a composition of point blow ups of nonsingular surfaces. Let B X. Then we can decompose g as g : X = Y r+1! Y r!! Y 2! Y 1 = X so that Y i+1! Y i is the blow up at P i 2 Y i and the multiplicities m i at P i of the birational transform of B to Y i satisfy the condition m 1 m 2 m r. Here is the main result of this section. Theorem 1.8. Let X be a nonsingular projective surface and B = P b j B j be an R-divisor on X with 0 b j 1 " < 1. Assume that K + B 0 and " < 1= p 3. Then the following are true:

5 Bounding singular surfaces of general type 5 1. (X) 128=" 5, 2. if, in addition, b j > 0 then (X) max(22; 8=" 3 2 ). Proof. By (1.4) we can assume that X is rational and there exists a birational morphism g from X to X = F n. As before, denote by B j the images g(b j ). We x a positive number 1 and divide g into two parts: g 1 : X 1! X and g 2 : X! X 1 by Corollary 1.7. The morphism g 1 is a composition of blowups at points P where the birational transform of B has multiplicity P mult P B = b j mult P B j 1, and the morphism g 2 is a composition of blowups at points with multiplicity < 1. Let 2 2 [0; 1 ] be such that for blowups of g 2 we have all the multiplicities 2. Later the values 1 ; 2 will be specied in such a way that 2 < ". P We rst bound (X 1 =X). At each blowup the expression ( b j B j ) 2 decreases by at least 2 1. On X one has P P bj B j b j Bj P P 2 bj ( 2=") b j (1 ")( 2=") Lemma 1.3 immediately implies that if KFn + P b j B j is nef, then P bj 4. Therefore, on the surface X X bj B j 2 4(1 ")( 2=") = 8 8 " On the other hand, on the surface X: X bj B j 2 = K 2Fn = 8 We conclude that X 1 =X 8=" 2 1: We next bound (X=X 1 ). Let Y be an arbitrary intermediate blowup of X! X 1. We write the image B Y of B as B Y = B Y + F Y, where B Y is the birational transform of B and F Y is the sum of curves which are exceptional for X 1! X. Let F i be an irreducible component of F Y, and the corresponding coecients in K Y + B Y by f i. Note that F Y is a simple normal crossing divisor. Since m P (B Y ) 2 < " and f i < 1 ", the blowups of X! X 1 are the blowups at the nodes of Supp F Y by Lemma 1.6. Let P = F 1 \ F 2 be one of such points. We have f 1 ; f 2 1 ", mult P B 2. The coecient of the new curve F 3 appearing after the blowup will be at most (1 ") (" 2 ). So, by 2 < ", the new coecients f i appearing at future blowups are strictly decreasing. At the same time they all have to be nonnegative by Remark 1.5, so the number of blowups over P can be easily bounded. The following two lemmas were suggested to us by J.Kollar.

6 6 Valery Alexeev and Shigefumi Mori Lemma 1.9. The number of blowups in the case f 1 = 1 a, f 2 = 1 b and arbitrary 2 (< ") is at most 1=(a 2 )(b 2 ) 1. Proof. Easy by descending induction. Note that 1 (a 2 )(b 2 ) 1 = 1 (a + b 2 2 ) (a 2 ) (a + b 2 2 ) (b 2 ) Thus the estimate for the number at P = F 1 \F 2 is reduced to those for F 0 1 \F 3 and F 0 2 \F 3. Thus by Remark 1.5, it is enough to check the lemma in the case where no further blowups are allowed. In this case, we have a 2 ; b 2 2 (0; 1] and hence the lemma holds by 0 1=(a 2 )(b 2 ) 1. ut Applying these lemmas we see that (X=X 1 ) (1= (a 2 ) (b 2 ) 1) N; where N is the number of nodes of [F i on X 1, and N (X 1 =X) 1. Adding (X 1 =X) and (X=X 1 ) together we obtain the following estimate (X) 8 " " 2 1 = 8 " 2 1 (" 2) (" 2) (" 2) " 2 1 (" 2)2 by 0 < " 2 " 1= p 3. For the statement (1) of the theorem we take 1 = 2 = "=2. For (2) we can take 1 = and 2 = 0 because in this case m P (B) < means m P (B) = 0. ut Corollary Let X be a projective surface and B = P b j B j be an R- divisor on X. Let " be a real number such that 0 < " < 1= p 3. Assume that the pair (X; B) is "-lc and that (K X + B) is ample. Denote by : e X! X the minimal resolution of singularities. Then ex 128=" 5 : Proof. Ampleness is an open condition, therefore changing the coecients b j slightly we can assume that they are rational, rather than only real, numbers. Let D be a general member of the linear system N(K X + B) for a large divisible N. Then K X + B + (1=N)D 0 and the pair (X; B + (1=N)D) is again "-lc. Now apply (1.8.1) to K e X + B ex = (K X + B + (1=N) D) :

7 Bounding singular surfaces of general type 7 Remark Theorem 1.8 was proved in [3] 6.3 in more general situation, when (K + B) is only nef instead of being numerically trivial. The explicit bound was not given. It can be deduced from [3] but turns out to be worse than that of (1.8). More importantly, in [3] it was proved using a much more combinatorially involved technique: the Diagram Method. The Diagram Method was also applied for proving bounds for Picard numbers of surfaces that do not follow in any obvious way from (1.8), see for example [1]. In other situations where both techniques are applicable the Diagram Method sometimes produces better bounds. 2 DCC sets and Shokurov's Log Adjunction 2.1. In this section, we will explain DCC sets and Shokurov's log adjunction and prove an application. Below we give the necessary denitions and list several facts about sets satisfying the descending chain condition (DCC). Denition 2.2. A subset A of R is said to satisfy the descending chain condition if any strictly decreasing subsequence of elements of A is nite. We also say that A is a DCC set. Lemma 2.3. Let A R be an arbitrary subset. Then A satises the DCC i every innite sequence fa n g of elements of A contains an innite nondecreasing subsequence. Proof. Elementary. ut Denition 2.4. A sum of n sets A 1 ; A 2 :::A n is dened as Next, we dene nx i=1 A i = fa 1 + a 2 + :::a n j a i 2 A i g A 1 = f0g [ 1[ nx n=1 i=1 If each A i satises DCC, then so does P n i=1 A i. The proof immediately follows from Lemma 2.3. If, in addition, A consists only of nonnegative numbers then clearly A 1 also satises DCC since positive numbers in a DCC set have a minimum. Denition 2.5. For a set A [0; 1] we dene the derivative set A 0 = f n 1 + a 1 j n 2 N; a 1 2 A 1 \ [0; 1]g [ f1g n It is easy to see that if A is a DCC set then so is A 0. A

8 8 Valery Alexeev and Shigefumi Mori 2.6. The derivative set appears very naturally in the following situation. Lemma 2.7 (Shokurov's P Log Adjunction Formula). Let X be a projective surface and B = B 0 + b j B j be an R-divisor on X. Assume that the pair (X; B) is lc. Denote by : X e! X the minimal resolution of singularities and by B0 e the birational transform 1 (B 0 ). Then 1. there exists a natural formula (K + B) j e B 0 K e B 0 + X d k P k ; where P k are nonsingular points on e B0 and d k = 1 or n k 1 + P j a j;kb j n k for some integers n k > 0 and a j;k 0. Thus if b j 2 A for all j then d k 2 A K e B 0 + P k d kp k is lc. 3. If P k 2 e B0 \ 1 (B`), then d k = (n k 1 + P j a j;kb j )=n k and a`;k > 0. Proof. It follows from the classication of log canonical surface singularities that e B0 intersects the exceptional divisors F i of transversally. Now using the adjunction formula for the divisor e B0 on e X we see the existence of the above formula with some 0 d k 1. The precise form of the coecients is an exercise in linear algebra. It can be found e.g. in [13]. ut The essential case of (X) = 1 in following result holds true in all dimensions in characteristic 0 ([8],18.24). Here we derive it as a corollary to (2.7). Lemma 2.8. Let X be a normal projective Q-factorial surface such that (X; B) is lc and K + B 0, where B = P j b jb j. Then P b j (X) + 2. Proof. Set P (X; B) = P b j (X). Assume that B 6= 0. Since K X B 6= 0, we can apply K-MMP to get g : X! Y such that P (X; B) P (Y; g B) and K Y is lc and such that 1. (Y ) = 2 and there is a surjective morphism : Y! C such that a general ber F is P 1, or 2. (Y ) = 1 and K Y is ample. (Since B K X g g K Y, we have g B 6= 0 and hence we have the above cases.) We treat the two cases. Case 1. This case is reduced to two easy cases.

9 Bounding singular surfaces of general type 9 Subcase 1. Supp g B contains no bers of. By the adjunction, we have 0 = (K + B) F = 2 + X b j (B j F ) 2 + X b j : Subcase 2. Supp g B F 0, a ber of. Then by (K X + g B) F = 0, we have g B F 0 > 0 and hence g B + F 0 is ample if 0 < 1. Then D = K + (1 2 )g B F 0 is lc and D is ample. Thus Y has two D- extremal rays R 1 and R 2 such that contracts R 1. If the contraction : Y! P Z of R 1 is of ber type, then let G be a general ber of and we have b j 4 by (K + B) (F + G) = 0. If is birational, then we have P (Y; g B) P (Z; b B). So this case is reduced to the next case. Case 2. X is Q-factorial, (X) = 1, K X is lc, K X is ample. We treat two cases. Subcase 1. There exists an i (say i = 1) such that b i = 1. We apply Lemma 2.7 to B 1 X, and get X d k 2 + X j6=1 b j ; which proves P b j 3. Subcase 2. b j < 1 for all j. By renumbering B 1 ; B 2 ;, we may assume ( K B i ) is non-decreasing in i. We try to P replace B i (i 2) with a multiple of B 1 keeping K + B 0. By doing so, (X; B) does not decrease and it is enough to prove (X; B) 2 after the replacement. One of the following changes occurs. 1. The number of irreducible components of Supp B decreases. 2. b 1 becomes There is a birational morphism f : Y! X such that E = f 1 (P ) red is an irreducible divisor, f induces Y E ' X fp g and f (K X + B) = K Y + f 1 B + E 0: The rst case is settled by the induction and the second is already treated in the above subcase. So we only need to treat the third case. If we set D = K Y + f 1 B + (1 )E with 0 < 1, then D is lc, D E and Y has a D-extremal ray R. Since (D E) > 0, the contraction g : Y! Z of R does not contract E. Since K Z + g f 1B + g E 0 is lc, Z is treated in the above subcase. ut 3 Chains of coecients 3.1. The main purpose of this section is to prove the following, which is an eective version of (5.3) in [2].

10 10 Valery Alexeev and Shigefumi Mori Theorem 3.2. Let A [0; 1] be an arbitrary DCC set. Then there exists a constant = (A) depending only on A (dened below) so that the following holds. Let X be a normal projective surface, B j be divisors on X, and let b j, x j be positive real numbers. Assume that 1. X is a (singular) Q-factorial Del Pezzo surface (i.e. K X is ample) and (X) = 1, 2. b j > 0 and b j 2 A, 3. 1 < x j 1, 4. at least one x j is strictly less than 1, 5. the pair (X; P x j b j B j ) is lc. Then the divisor K X + P x j b j B j We rst make a basic computation for the derivative set. Then we introduce the functions necessary to dene (A) used in the theorem. Lemma 3.4 (Basic computation for the derivative set). Let d i = n i 1 + P n i;j b j n i be an element of the derivative set A 0. Assume that all b j > 0 and that they appear in d i with positive coecients. Consider real numbers x j with 1 < x j 1, with at least one x j being strictly less than 1. Then (1 ) d i < n i 1 + P n i;j x j b j n i < d i Proof. Evident. ut 3.5. We now introduce the functions that will be used in the statements below. For a subset A R and t 2 R, we set A >t = fx 2 Ajx > 0g. We assume everywhere that A satises the DCC and contains only nonnegative numbers. t is a nonnegative real number. mf 1 (A; t) = min(ca 1 ) >t t mf 2 (A; t) = minfx > 0 j (1 x) a 1 = t; a 1 2 A 1 g = mf 1 (A; t) t + mf 1 (A; t) We set by denition mf 1 (;; t) = mf 2 (;; t) = +1. (A; ") = (") = mf 2 A [ f1g; [2="] [128="5 ] ([2="] + 2) 2 (A) = ( ( ( (mf 2 (A 0 ; 2)))))

11 Bounding singular surfaces of general type 11 The following are the very basic properties of the functions we introduced above. Lemma 3.6. Assume that a DCC set A contains 1. Then the following hold < mf 1 (A; t) < 1 and 0 < mf 2 (A; t) < 1=(1 + t) for all t > If t 1 t 2 2 N, then mf k (A; t 1 ) mf k (A; t 2 ) for k = 1; Let a; > 0 and 0 be such that mf 2 (A; a)=(1 + ). Then for any nite number of arbitrary b j ; x j ; y 2 R such that 0 y, 0 < b j 2 A and 1 < x j 1 for all j, x j < 1 for some j, we have P xj b j 6= a + ay. 4. If " < 1, then (") < " 128 =16. Proof. (1) is obvious. By the denition of mf 1, we have mf 1 (A; t + 1) mf 1 (A; t), whence mf 2 (A; t + 1) = mf 1 (A; t + 1) t mf 1 (A; t + 1) mf 1(A; t) t + mf 1 (A; t) = mf 2(A; t): Thus (2) is proved. For (3), assume that P x j b j = a + ay. Then a + a > (1 ) P b j by y and x j > 1. Thus ( X b j + a) > X bj a: By x j 1 for all j and x j < 1 for some j, we have P b j > a + ay a. Whence (1 + ) X b j > X b j a > 0; and > mf 2 (A; a)=(1 + ), a contradiction. It remains to prove (4). By (1), we have mf 2 (A; t) < 1=t and hence by [2="] 2=" 1 > 1=". (") < ut 1 16 [2="] 128 < 1 16 "128 ; The following two results treat easy cases of (3.2). Lemma 3.7. In the situation of (3.2) assume, in addition, that " < 1= p 3, that the pair (X; 0) is "-lc for a xed " > 0 and that = (A; "). Then K + P x j b j B j 6 0. Proof. Let : e X! X be the minimal resolution of singularities of X and Fi be the corresponding exceptional curves. For a suciently divisible m > 1=", mk X is very ample and let D 2 j mk X j be a general member so that D is nonsingular and disjoint from the singular locus of X. The log divisor (K X + D) = K e X + X f i F i + 1 m D

12 12 Valery Alexeev and Shigefumi Mori has nonnegative coecients f i and it is numerically trivial. The additional condition of our lemma means precisely that all f i 1 ". Therefore, the pair e X; D=m + P f i F i belongs to one of the types listed in Lemma 1.4. Since P x j b j B j 6= ;, it cannot be a surface as in (1.4.1). If it is in (1.4.2), then we restrict the numerically trivial divisor K X + P x j b j B j to a general ber F of the P 1 -bration to get P x j b j (B j F ) = 2. By (3.6.2) we have mf 2 (A [ f1g; 2), and by (3.6.3) we have a contradiction. Therefore we can assume that X is rational and there is a birational morphism e X! F n, n 2=". Let t be a positive integer that makes tk X into an ample Cartier divisor. Then and, consequently, tk X (K X + X x j b j B j ) = 0 X tkx B j xj b j = tk 2 X (1) The coecients tk 2 X, tk X B j in the latter formula are positive integers. The integer t is bounded from above by the determinant of the matrix ( F i F k ). This square matrix has the dimension at most ( e X) < 128="5 by corollary (1.10) applied with B = ;. Each diagonal element is positive and bounded by 2=" by Lemma 1.2, other elements are non-positive. It follows that t [2="] [128="5 ] We also have the following bound for K 2 X : K 2 X ([2="] + 2)2 Indeed, e X has a birational morphism to F n with n 2=". The preimages of the curves in the linear system js n +nfj on F n form a free system of curves fc t g on e X. We have K X + X x j b j B j Ct K e X C t = KFn(s n + nf) = n + 2 Therefore, by Fano's argument (see for example [10]) one has KX + X x j b j B j 2 (n + 2) 2 ([2="] + 2) 2 Thus tk 2 X [2="][128="5] ([2="] + 2) 2 : By (3.6.2), we have mf 2 (A; tk 2 X ) and (3.6.3) contradicts the equation (1). ut

13 Bounding singular surfaces of general type 13 Lemma 3.8. In the situation of (3.2) assume that = mf 2 (A 0 ; 2). Then the following hold. 1. If x j b j = 1 for some j then K + P x j b j B j Let P 2 X and J 1 = fjjb j 3 P g. If x a < 1 for some a 2 J 1 and if x j b j < 1 for all j 2 J 1, then K + P x j b j B j is not maximally lc at P i.e. for some small > 0 the divisor K + P j2j 1 (x j b j + )B j is still log canonical at P. Proof. Assume P rst that x 0 b 0 = 1. We apply the log adjunction to B 0 and get KB e0 + y i d i Q i 0, where 0 < d i 2 A 0, 1 P < y i 1 for all i and y b < 1 for some b (3.4). Then deg KB e0 = 2 and y i d i = 2 contradicting (3.6.3). This P proves (1). If K X + x j b j B j is maximally lc at P, then by (1) there exists a partial resolution f : (Y; E)! (X; P ) such that the exceptional set of f is an irreducible curve E with log discrepancy 0, i.e. X X f K X + x j b j B j = K Y + x j b j f 1(B j) + E Restricting this inequality to E and applying the log adjunction formula (2.7), one gets K e E + X y i d i Q i 0; where 0 < d i 2 A 0, 1 < y i 1 for all i by (3.4). P Furthermore for any Q k lying over E \ f 1(B a) we have y k < 1. So y i d i = 2, and it is a contradiction by (3.6.3). ut The following is the key lemma for the proof of (3.2). Lemma 3.9. In the situation of (3.2) assume, in addition, that b 0 = 1 and that = mf 2 (A 0 ; 2)=2. Then if K + P x j b j B j 0, then for the birational transform of B 0 on the minimal desingularization : e X! X one has eb Proof. First of all, x j b j < 1 for all j (3.8.1), and we see that the pair P (X; B 0 + P x j>0 jb j B j ) is lc. Otherwise, we would get a maximally lc surface (X; y 0 B 0 + x j>0 jb j B j ) with x 0 < y 0 < 1 contradicting the previous lemma 3.8. P We apply the log adjunction to K + B 0 + x j>0 jb j B j (1 x 0 )B 0 and get 0 0 (1 x 0 ) B 2 X 1 0 KX + B 0 + x j b j B j A jb0 j>0 K e B 0 + X y i d i Q i ; (2)

14 14 Valery Alexeev and Shigefumi Mori where 0 < d i 2 A 0, 1 < y i 1 for all i and y a < 1 for some a by (X) = 1 (3.4). Note that we have x 0 < 1. If e B 2 0 < 0 then we claim that e B0 is a ( 1)-curve and B 2 0 < 1=x 0. Indeed, 0 > K X B = K X eb0 K e X e B 0 : Therefore, e B0 is a ( 1)-curve and the claim follows from x 0 B 2 0 = K X + P j>0 x jb j B j B 0 K X B 0 K ~X ~ B0 = 1: From the inequality (2) we see that X yj d j = 2 + (B 2 0 ) (1 x 0) and (B 2 0) (1 x 0 ) (1 x 0 )=x 0 2(1 x 0 ) by x 0 > 1 > 1=2. This contradicts (3.6.3). ut Proof. Proof of (3.2) Set " n = n (mf 2 (A 0 ; 2)) for n 0. We note " n " 0 1=3 by (3.6.1). Let n 1 and assume that a surface (X; B j ; b j ; x j ) with K+ P b j x j B j 0 and Minfx j g > 1 " n as in (3.2) exists (suppressing = " n ). We will keep variable n for the clarity of argument and set n = 4 at the end of the proof. Then by (3.7) and " 0 1=3, (X; 0) is not " n 1 -lc. Thus let f : Y! X be a partial resolution f : (Y; E)! (X; P ) dominated by the minimal desingularization for which the exceptional set of f is an irreducible curve E with the minimal log discrepancy We have a(e) = a(e; K X + X x j b j B j ) a(e; K X ) < " n 1 : f K X + X x j b j B j = K Y + X x j b j B j + (1 a (E)) E The Picard number (Y ) = 2, and there exists a second extremal ray g : Y! X 1. For the new curve B 0 = E set b 0 = 1 and x 0 = 1 a(e). The morphism g can S collapse some curves. However, we claim that there will be a component of B j [ E with x j < 1 which is not contracted to a point. Indeed, if a(e) 6= 0, this is the curve E. If one has a(e) = 0 then the the corresponding pair is maximally lc. Lemma 3.8 guarantees that in this case a curve with x j < 1, for example B 1, does not pass through P, because

15 Bounding singular surfaces of general type 15 mf 2 (A 0 ; 2) > by (3.6.4). Therefore the preimage of B 1 on Y has a positive self-intersection, and this curve is not contracted by the morphism g. We claim that dimx 1 = 2. Otherwise the morphism g is a generically P 1 -bration. Restricting the divisor K Y + X x j b j B j + 1 a(e) E 0 on a general ber, we get a contradiction because Thus from a surface mf 1 (A [ f1g; 2) > mf 2 (A 0 ; 2) > : (X; B j ; b j ; x j ) with K + X b j x j B j 0 and minfx j g > 1 " n as in (3.2), we obtained another X (X (1) ; B (1) j ; b (1) j ; x (1) j ) with K (1) + b (1) j x (1) j B (1) j 0 and minfx (1) j g > 1 " n 1 as in (3.2) such that X (1) is dominated by the minimal resolution X e of X, b (1) 1 = 1 and ( B e(1) 1 )2 0. If we have such an (X (a) ; B (a) j ; b (a) j ; x (a) j ), we can obtain the next as long as a < n (so that minfx (a) k g > 1 " 1). The point in the procedure is that we have a sequence of birational morphisms e X! e X (1)!! e X (a)! X (a+1) so that the birational transform of an arbitrary curve E X (i) (i a) with ( e E2 ) 0 is a curve on X (a+1). Thus we have obtained (X (n) ; B (n) j X ; b (n) ; x (n) ) with K (n) + b (n) j j j x (n) j B (n) j 0 such that b (n) i = 1 (i = 1; ; n), x (n) i > 1 " 1 > 1 1=128 (i = 1; n 1), and 1 x n (n) > 1 " 0 2=3. On the other hand, X 3 b (n) i x (n) i nx i=1 b (n) i x (n) i > ( )(n 1) by (2.8). Now we set n = 4 and get a contradiction. Thus if we set = " 4 = (A), then (3.2) holds. ut 4 A lower bound for (K + B) 2 and the boundedness In this section, we give versions of a few theorems of [3]. Among them, (4.7) and (4.8) are the main results. We begin with a result from the log minimal model theory of surfaces, followed by two easy lemmas.

16 16 Valery Alexeev and Shigefumi Mori Lemma 4.1. Assume that K + B is lc and big on a normal Q-factorial projective surface X. Let B 0 B be an eective R-divisor, and let t 0 > 0 be the largest real number such that B t 0 B 0 0. Then one of the following holds true. 1. K + B t 0 B 0 is big, 2. there exists t (0; t 0 ] such that K+B xb 0 is big i x < t 0 0. There exists a birational morphism f : X! X 0 to a normal Q-factorial projective surface X 0 such that D = f (K + B t 0 0 B0 ) is nef and K + B 0 t0 0 B0 0 f D, and one of the following holds. (a) ((D 6 0 or f (B t 0 0 B0 ) 6= 0) and (X 0 ) 2) There exist a morphism : X 0! C onto a nonsingular projective curve C and an R-divisor L on C such that deg L 0 and D L and such that every irreducible curve G f (B t 0 0B 0 ) in a ber of satises G = 1 (G) as sets. (b) ((D 6 0 or f (B t 0 0B 0 ) 6= 0) and (X 0 ) = 1) K X 0 is ample and D 0. (c) (D 0 and f (B t 0 0 B0 ) = 0). Furthermore if we have t 0 0 < t 0 in Case (a), then a general ber of is P 1. Lemma 4.2. Let g : X! Y be a birational morphism of normal projective surfaces and let B = P j2j b jb j B 0 be eective R-divisors on X. Assume that 1. K X + B is big and lc, 2. g (K X + B) is nef and lc, 3. K X + B g g (K X + B). Let C >0 and C <0 be eective R-divisors without common components such that K X + C >0 C <0 = g g (K X + B), and let J 1 = fjjb j 6 Supp C <0 g. Then K X + P j2j 1 b j B j is big and lc. Proof. First we note that the irreducible components of Supp B are all Q- Cartier because K + B is lc. Since the sum of a big divisor and a nef divisor is big, (K + B) + (1 )g g (K + B) = K + B + (1 )C >0 (1 )C <0 is big for all real 2 (0; 1]. If 0 < 1, then K + B + (1 )C >0 (1 )C <0 K + X j2j 1 b j B j : ut Lemma 4.3. Let f : X! X 0 be a birational morphism of a nonsingular projective surface X to a normal surface X 0 and let C >0 and C <0 be two eective R-divisors without common components such that f C <0 = 0, f (K + C >0 ) is Q-Cartier, and K + C >0 C <0 = f f (K + C >0 ). Then f factors through a birational morphism g : X! Y to a nonsingular surface Y such that Supp C <0 = Exc(g), the exceptional set of g.

17 Bounding singular surfaces of general type 17 Proof. It is enough to prove that Supp C <0 contains a ( 1)-curve if C <0 6= 0. We have C <0 K = (C <0 ) 2 (C >0 ) (C <0 ) (C <0 ) 2 < 0: Hence there exists an irreducible component E of C <0 such that E K < 0. Since E 2 < 0 by f E = 0, E is a ( 1)-curve. ut The following is an eective version of (7.4) of [3]. Theorem 4.4. For ; " > 0, let N = [128=" 5 + 4=] Then we have the following. Let K + B = P j2j b jb j be big and lc on a nonsingular projective surface X such that b j 1 " for all j. Then there is a subset J 0 J such that jj 0 j N and K + P j2j 0 b jb j is big and lc. Proof. We use the induction on the number of irreducible components of Supp B. We apply Lemma 4.1 to subtract B 1 from B. In the case (4.1.1), we have K + P j6=1 b jb j is big and lc, and we are done by the induction. So let t (0; b 0] and a birational morphism f : X! X 0 be as in Case (4.1.2). In particular, we have D = f (K + B t 0 0B 0 ) is nef and K + B t 0 0 f D. If we write f f (K + B t 0 0B 0 ) = K + C >0 C <0 as in (4.2), then K + P j2j 1 b j B j is big, where J 1 = fjjb j 6 Supp C <0 g. We treat two cases. Case 1. D 0 (4.1.2). We will give a uniform bound of jj 1 j. By Lemma 4.3, there exists a birational morphism g : X! Y to a nonsingular projective surface Y such that Supp C <0 = Exc(g), the exceptional set of g. Let h : Y! X 0 be the induced morphism. Since K Y + g C >0 = g (K + C >0 ) = h D 0 and since C >0 B, we have (Y ) 128=" 5 (1.8). By we have J 1 = f0g [ fjjb j : f-exc., not g-exc.g [ fjjb j not f-exc.g; by (2.8). jj 1 j 1 + ((Y ) (X 0 )) + ((X 0 ) ) = (Y ) " Case 2. There exist a surjection : X 0! C to a curve C and an R-divisor L on C such that deg L > 0 and D L (4.1.2). We treat two subcases. Subcase 1. ( f) (B j ) 6= 0 for every j 2 J 1. Let F be a general ber of f. By (K + g C >0 ) F = 0, we have t 0 0 (B 0 F ) + P b j (B j F ) 2. Hence jj 1 j 1 + 2=. We are done in this subcase.

18 18 Valery Alexeev and Shigefumi Mori Subcase 2. B 1 6 Supp C <0 and ( f) (B 1 ) = 0. Let m 1 2 R be such that m 1 B 1 D (4.1.2). If m 1 > b 1, then let B = B t 0 0B 0 t 0 1B 1 for any t (b 1 ; m 1 ). Then g (K + B) is nef and K X + B g g (K X + B). Then by (4.2), K X + P j6=1 b jb j is big and we are done by induction. If m 1 b 1, then set B = B t 0 0B 0 m 1 B 1. Then g (K + B) 0 and we can use the same argument as Case 1 except that jj 1 j 128=" 5 + 4=. ut 4.5. We rephrase Theorem 3.2 using (4.1) into the following eective version of (7.5) of [3]. Theorem 4.6. Let A [0; 1] be an arbitrary DCC set such that 1 2 A. Let X be a normal projective surface, B j be divisors on X, and let b j be positive real numbers. Assume that 1. b j 2 A, 2. the pair (X; P b j B j ) is lc, 3. K X + P b j B j is big. Then the divisor K X + (1 (A)) P b j B j is big (cf (3.5)). P Proof. Under the notation and assumptions of the theorem, let B = b j B j. Let P : Y! X be a projective birational morphism such that B Y = 1 B + Ei and [E i = Exc() in the formula (K + B) P = K Y + B Y, and such that K Y + B Y is lt. Since (K Y + (1 x)( 1 B + E i )) K + (1 x)b, it is enough to prove the theorem for K Y + B Y. Hence we may assume that our X is Q-factorial ignoring Y. Assume that there is a positive real number (A) such that K + (1 x)b is big i x <. We note that (A) < 1=16 (3.6.4). We apply (4.1) to K + B with B 0 = B and we are in case (4.1.2). Thus there is a birational morphism f : X! X 0 to a normal Q-factorial surface X 0 such that D = f (K + (1 )B) is nef and K + (1 )B f D, and one of the following holds. 1. there exist a morphism : X 0! C onto a nonsingular projective curve C with a general ber F ' P 1 such that (D F ) = D 0. P In Case (1), we have (1 )( b j f (B j ) F ) = 2 which contradicts (3.6.3) because (A) < mf 2 (A; 2) (3.6.4). Finally Case (2) is disproved by (3.2). ut The following takes the same form as (7.6) of [3]. However it can be made eective in the sense of (4.8). Theorem 4.7. Fix C > 0 and a DCC set A such that 1 2 A. Then there exists a bounded class of surfaces with divisors (Z; D) such that for every surface X with K + B = K + P b j B j nef big lc such that 0 < b j 2 A and (K + B) 2 C there exists a diagram where X f Y g! Z;

19 Bounding singular surfaces of general type Y is the minimal resolution of X, 2. D = g(supp B Y [ Exc(f)) where Exc(f) is the exceptional set of f and B Y is dened by K Y + B Y = f (K + B). Proof. We make auxiliary constructions of R-divisors B 0 ; B 00 ; B 000 on Y. (Construction of B 0 ) By (4.6), there exists a natural number m = m(a) Psuch that m P P > 3= mina >0 and K + (b j 3=m)B j is big. Let B Y = j2j b jbj 0 + i2i e ie i with [E i = Exc(f) and 0 e i 1. Take b 0 j ; e0 i 2 fk=mjk = 1; 2; mg be such that b 0 j 2 [b j 3 m ; b j 2 m ); e0 i 2 [e i ; e i + 1 m ]: We note that Exc(f) is an SNC divisor by the classication of lc surface singularities. Thus there is a log resolution : Y 0! Y of K Y + B Y such that ( X (b j b 0 j)b 0 j + X (e i e 0 i)e i ) 1 ( X (b j b 0 j)b 0 j + X (e i e 0 i)e i ): (The proof is left to the reader.) Hence if we set B 0 = P b 0 j B0 j + P e 0 i E i then (K Y + B Y ) P (K Y + B 0 ) + 1 (B Y B 0 ) and K Y + B 0 is lc. Since K Y + B 0 f (K + (b j 3=m)B j ), K Y + B 0 is big. (Construction of B 00 ) Again by (4.6), there exist a natural number ` = `(m) > m depending only on m and b 00 j ; e00 i P 2 fk=`jk = 1; 2; ` 1g with b 00 j < b0 j and e00 i < e0 i such that if we set B00 = b 00 j B0 +P j e 00 i E i then K Y +B 00 is lc big. (Construction of B 000 ) By (4.4), there exist a natural number N = N(`) (depending only on `) and subsets I 0 I P and J 0 J P such that ji 0 j+jj 0 j N and K Y + B 000 is lc big, where B 000 = J b00 0 j B0 j + I e00 0 i E i. We note that K Y + B 000 has the properties: 1. the coecients of B 000 are in f1=`; ; (` 1)=`g, 2. #firreducible component of B 000 g N, and 3. f (K Y + B 000 ) K + B. Now let g : Y! Z be the log canonical model of K Y + B 000. Since g (K Z + g B 000 ) K Y + B 000, K Z is lt and hence Z is rational. Let H = K Z + g B 000, which is ample. Let : Z e! Z be the minimal resolution. Then by (1), (2) above, the boundary of (K Z + g B 000 ) has only coecients in [0; 1 1=`] and it has at most N components. Thus if U Z is the complement of the set of Du Val singular points 62 g B 000, then there are at most N g-exceptional irreducible curves E k 1 (U) and each E k satises E k ' P 1 and (Ek 2 ) 2` (1.2). Claim. There exists a natural number t `(2`) N such that tk Z =` and td=` are Cartier for every irreducible component D of g B 000. In particular th is Cartier.

20 20 Valery Alexeev and Shigefumi Mori For the claim we can ignore points not in U. We work only on H since P the argument is the same. On 1 (U), we can write 1 (D) 1 (D)+ a k E k with a k 2 Q. P If we set t = ` jdet(e k E k 0)j, then we have t (2`) N and t( 1 (D) + a k E k ) is a Cartier divisor relatively trivial for. Since Z has only rational singularities, induces a Cartier divisor td. This proves the claim. Claim. H 2 (f g H) 2 C and 3tC K Z H C. Since f g H K + B are both nef we have H K Z H 2 (g H) 2 (f g H) 2 (K + B) 2 C: Assume that (K Z + 3tH) H < 0. Then (KZ e + 3t H) H < 0 and KZ e + 3t H is not nef. If we take an extremal rational curve C of it then (KZ e + 3t H) C < 0 and it contradicts KZ e C 3. Thus the claim is proved. Let D 00 Y be the sum of ( 2)-curves which are f-exceptional, and let D 0 be the closure of (Supp(B Y ) [ Exc(f)) Supp(F 2 ). We note that D = g(d 0 ) [ g(d 00 ). Claim. (g D 0 H) (1 + 3t)C= minf1=3; Ag, P ED 00(g E H) 2 2C Let B Y P = f 1 B + E a EE. Then by (K Y + B Y ) E = 0, we have 0 (K Y + a E E) E and a E 1=3 by (E 2 ) 3. Thus the rst inequality follows from (g B Y H) (1 + t)c (4). Let F = g (H) + 1=2 P ED 00(H g E)E. Since (F 2 ) (f F ) 2 = (f g H) 2 C (4), we have the second by (F 2 ) (H 2 ) + 1=2 P (H g E) 2. Thus the claim is proved. Now by [9], [5] and [11], there exists a uniform M = M(C; t) so that MtH is very ample (4). By (4), (Z; D) is also bounded. ut Theorem 4.8 (An eective bound of (K + B) 2 ). Let X be a normal projective surface, B j divisors on X, and let b j be positive real numbers. Assume that 1. K X + P b j B j is nef big lc, 2. b j belong to a DCC set A. Then K X + X b j B j 2 1 ` (2`) N ; where N = 128`5 + 4` and ` = d1=((a) mina >0 )e. Proof. Let f : Y! X be a log resolution of K + B. We set B 0 = f 1B + Pf-exc. E E. Then we follow the proof of (4.7) from the construction of B00 till the proof of Claim (4), where we set ` = d1=((a) mina >0 )e. Thus we get (K + B) 2 (H 2 ) `=t 2 1=`(2`) N with N = 128`5 + 4`. ut

21 5 A DCC set for klt surfaces Bounding singular surfaces of general type Once we obtain a bounded family f(z; D)g as in (4.7), we would like to reconstruct (X; B) from (Z; D) in the bounded family. For this it is enough to obtain Y by blowing up in some bounded manner since then Exc(f) g 1 (D). In order to do this systematically, we study the family of the maps g : Y! Z as follows Let A be a DCC set and let = f(z; D)g be a bounded family of normal surfaces Z with reduced Weil divisors D. Consider a set of the set (Y; B Y ; g; Z; D) consisting of a pair (Y; B Y ) and a birational morphism g : Y! Z to a (Z; D) 2 such that 1. Y is a nonsingular surface and K Y + B Y is klt, 2. Supp B Y g 1 (D) [ Exc(g), 3. (K Y + B Y ) 2 > 0 and (K Y + B Y ) is nef on C g 1 (D) [ Exc(g), 4. there are no ( 1)-curves C g 1 (D) [ Exc(g) with (K Y + B Y ) C = 0, 5. the coecient in B Y of every curve C g 1 (D) [ Exc(g) with (K Y + B Y ) C > 0 is in A [ f0g. We note that we have this situation in (4.7). The main purpose of this section is to prove the following. Theorem 5.3 (Alexeev [3] (8.5)). Under the notation and the assumptions of (5.2), let 0 be an arbitrary innite sequence of. Then there exist an innite subsequence f s = (Y s ; B Y s ; g s ; Z s ; D s )g 0 and a bounded (at) family of blowups s : V s! Z s dominated by Y s via h s : Y s! V s such that for every s < t we have (K Y s + B Y s ) 2 (K V s + B V s ) 2 (K Y t + B Y t ) 2 : (3) Since in (5.2) arises from klt surfaces (X; B = P b j B j ) with K + B ample and b j 2 A by (4.7), we have the following. Corollary 5.4. Let A be a DCC set. Consider klt surfaces (X; B = P b j B j ) such that K + B is ample and b j 2 A. Then f(k + B) 2 g is a DCC set. The rest of this section is devoted to the proof of the theorem. Readers mainly interested in knowing how this is used can skip to the next section. In order to prove (5.3), we would like to simplify the maps g by changing (Z; D) uniformly. Lemma 5.5. Let the notation and the assumptions be as in (5.2). We can change the bounded family without changing (Y; B Y ) in so that we can assume that D is an SNC divisor at every point P blown up by g.

22 22 Valery Alexeev and Shigefumi Mori Proof. Take an arbitrary (Y; B Y ; g; Z; D) 2. Then Y dominates the minimal resolution Z 0 of Z by g 0 : Y! Z 0. Let D 0 Z 0 be the union of the exceptional set of Z 0! Z and the inverse image of D. Furthermore as long as there is a point in Z 0 blown up by g 0 at which D 0 is not an SNC divisor, we blow up such a point. Repeating this until we have a birational morphism g 00 : Y! Z 00 such that the inverse image D 00 of D 0 is an SNC divisor at every point Z 00 blown up by g 00. It is obvious that such (Z 00 ; D 00 ) again form a bounded family 0. ut We say that a reduced curve C on a nonsingular surface has a reducible ODP at P if C = C 1 [ C 2 in a neighborhood of P for some curves C 1 and C 2 smooth at P and intersecting transversally at P. Lemma 5.6. Let the notation and the assumptions be as in (5.2) and assume that has the extra property in (5.5) Then for each (Y; B Y ; g; Z; D) 2, 6. g : Y! Z is obtained by repeatedly blowing up a point at which the total transform of D has a reducible ODP as a set. Proof. Let P 2 Z be a point blow up by g. Assume that D is smooth at P. Then Supp(B Z ) is also smooth at P. Hence by (1.6), we have g (K Z +B Z ) K Y + g 1(BZ ) in a neighborhood of g 1 (P ). On the other hand we have K Y + B Y g (K Z + B Z ) because K Y + B Y is nef on g 1 (D). This means K Y + B Y = g (K Z + B Z ) in a neighborhood of g 1 (P ). Then g 1 (P ) (K Y + B Y ) = g 1 (P ) g (K Z + B Z ) = 0 Hence g 1 (P ) does not contain any ( 1)-curves by the condition (4), which is a contradiction. Since this argument applies to any subsequent blowups, the lemma is proved. ut Remark 5.7 (Reduction to the case of a xed (Z; D)). breaks up into a nite number of families i (i = 1; ; r) which are projective at families parameterized by irreducible algebraic varieties so that each irreducible component of D comes from an irreducible component of the total space and each ordinary double point of D forms a section of the at family. Then breaks up into i (i = 1; ; r) correspondingly. For the innite sequence 0 of in (5.3), 0 i = 0 \ i is still innite for some i (say i = 1). Take a generic (Z; D) 2 1. Then for each = (Y; B Y ; g; Z; D), we can make the blow up g : Y! Z corresponding to g : Y! Z as above, and we can write B Y using the same coecients as B Y. Consider 0 1 = f(y ; B Y ; g; Z; D)j 2 0 1g. If we prove (5.3) for 0 1, then from : V! Z we can construct a bounded family of blowups s : V s! Z s and (5.3) holds for 0. In other words, we can assume consists of one member to prove (5.3). So from now on, (Z; D) will be xed during the proof and one blow up V will be constructed which works as V s for every s. We note however that the notation like K V s + B V s is used to denote h s (K Y s + B T s ).

23 Bounding singular surfaces of general type 23 With these notation and assumptions, the following is the key technical result. Theorem 5.8 (Alexeev [3] (8.5)). There exist an innite subsequence f s = (Y s ; B Y s ; g s ; Z; D)g 0 1 and a blowup : V! Z dominated by Y s, say via h s : Y s! V s such that for every s < t we have h t (KV s + B V s ) K Y t + B Y t : (4) Proof. Proof of (5.3) using (5.8) By the denition of B V s, we have h s (K Y s + B Y s ) = K V s + B V s, this implies (K Y s + B Y s ) 2 (K V s + B V s ) 2. Since K Y t + B Y t and K V s + B V s are nef over D by the condition (3), so is (K Y t + B Y t ) + h t (KV s + B V s ) on (g t ) 1 (D). By inequality (4), (K Y t + B Y t ) h t (KV s + B V s ) is an eective divisor supported on (g t ) 1 (D). Hence (K Y t +B Y t ) 2 h t (KV s +B V s ) 2 0, which is inequality (3). ut Remark 5.9 (Reformulation of inequality (4)). For any valuation v of the function eld of Z we can talk about the log discrepancy a`(v; Y ) = a`(v; K Y + B Y ) for any log divisor in the usual way. We also set a`(v; Y 1 ) = lim sup a`(v; K Y + B Y ) 2 R[ f1g: (Y; )2 1 0 We identify a divisor and its valuation for simplicity of notation. P Note that if v is the valuation of an irreducible component B j of B Y = b j B j then a`(v; Y ) = 1 b j. We say that a valuation v is D-toric if either v is the valuation of an irreducible component of D or an exceptional divisor which is obtained by successively blowing up a point at which the total total transform of D has a reducible ODP as a set. In the proof, we consider only D-toric valuations. Then inequality (4) is equivalent to \a`(v; V s ) a`(v; Y t ) for all divisors v on Y t " and even to a`(v; V s ) a`(v; Y t ) for all divisors v on Y t with 1 a`(v; Y t ) 2 A [ f0g: Indeed let us assume inequality (5.9). If we write K Y t +B Y t (h t ) (K V s + B V s ) = L 1 L 2 with eective divisors L 1 and L 2 without common components, then inequality (5.9) means that 1 a`(c; Y t ) 62 A[f0g for every curve C Supp L 2. From the conditions (3) and (5), we see that the intersection matrix of components of Supp L 2 is negative denite and (K Y t+b Y t )L 2 = 0. Then (L 2 2) (L 2 L 2 L 1 ) = L 2 (h t ) (K V s + B V s ) 0; and L 2 = 0. Thus we have inequality (4).

24 24 Valery Alexeev and Shigefumi Mori Remark 5.10 (A consequence of klt). Suppose V is chosen. To check inequality (5.9) for a given s, we need to check the inequality only for a nite number of v's. To be precise, there exists a nite set s of valuations such that a(v; V s ) > 1 if v 62 s. (Note that K Y s + B Y s is klt and h s : Y s! V s is an isomorphism except at reducible ODP's of h s (B Y s ) by the condition (6). Hence K V s + B V s is klt.) Therefore if v 62 s then inequality (5.9) holds for these s and v no matter what t is. Remark 5.11 (Finite number of divisors on V can be ignored). When we have : V! Z and an innite subsequence of 0 with all Y s dominating V, we need to check inequality (5.9) only for V -exceptional valuations (i.e. valuations whose center on V is a point). More generally given a nite set of divisors on V, we can replace the sequence by an innite subsequence so that inequality (5.9) holds for all s < t and v 2. Indeed we can assume consists of one valuation v 0. By replacing the sequence by an innite subsequence, we can treat inequality (5.9) in two cases a`(v 0 ; Y s ) 62 A for all s, 2. 1 a`(v 0 ; Y s ) 2 A for all s. In the rst case inequality (5.9) holds for v 0. In the second case, we take an innite subsequence again and assume that a`(v 0 ; Y s ) is a non-increasing sequence since A is a DCC set. Then inequality (5.9) holds for v 0. In the proof, we will mean an innite subsequence by a subsequence The construction of will be done locally for each reducible ODP P of D. Let D 1 and D 2 be the irreducible components of D such that D = D 1 [D 2 near P. Each D i is dened by a local equation i = 0 near P. It is easy to see that D-toric valuations v centered at P corresponds to e(v) = (v 1 ; v 2 ) 2 N 2 such that gcd(v 1 ; v 2 ) = 1 bijectively by v i = v( i ). We make the additional correspondence D 1 = (1; 0); D 2 = (0; 1). We note that = ( 1 ; 2 ) : (Z; P )! (A 2 ; 0) is etale at P and everything near P can be computed by the torus embedding method. Set N = Z 2 and A 2 G 2 m corresponds to the cone NR= N ZR R 2. 0 The theory of torus embeddings says the following. For any nite set E R P 2, let < E > be the cone 0 e2e R 0 e spanned by E. For any nite set E of primitive elements of N \ R 2 0, let (E) be the subdivision of R 2 0 by the rays R 0e (e 2 E). That is, if all the elements e 1 ; ; e r of E are numbered in the decreasing order of slopes then < e i ; e i+1 > (i = 0; ; r) are the 2-dimensional cones of (E) where e 0 = (0; 1) and e r+1 = (1; 0) and the slope of a non-zero v = (c; d) 2 R 2 0 is d=c 2 R[ f1g.

25 Bounding singular surfaces of general type 25 To the decomposition (E), associated an algebraic variety T (E) and a proper birational morphism (E) : T (E)! A 2. If we set E(Y s ) = fe(v)jv is an irreducible component of (g s ) 1 (P )g; then Y s ' T (E(Y s )) A2 Z over a neighborhood of P. (This can be proved by reconstructing the blow-ups of g s on A 2.) In this sense, we identify (Z; P ) with (A 2 ; 0) and also v with e(v) from now on. The following formulas for log discrepancies a` are the basis of our proof. Lemma Let u i 2 Z 2 0 (i = 1; 2; 3) be primitive vectors such that u 3 = 1 u u 2 for some i 2 Q 0. then for all s 1 we have a`(u 3 ; Y s ) 1 a`(u 1 ; Y s ) + 2 a`(u 2 ; Y s): 2. Let E = fv 1 ; ; v q g Z 2 >0 be primitive vectors in the decreasing order of slopes and let v 2 Z 2 >0 be a primitive vector such that v = v a + v a+1 for some a 2 [1; q 1] and ; 2 Q 0. Then for all s 1 we have a`(v; T (E) s ) = a`(v a ; Y s ) + a`(v a+1 ; Y s ): Proof. If 1 = 0 or 2 = 0 in (1), then u 3 = u 1 or u 2 and the formula is trivial. So we can assume 1 ; 2 > 0 in (1) and similarly ; > 0 in (2). We can also assume s < 1. For (1), we look at W = T (u 1 ; u 2 ; u 3 ). It contains three Q-Cartier divisors C i corresponding to R 0 u i, and C 3 is proper by 1 ; 2 > 0. From the relation 1 u u 2 u 3 = 0, we see that ([Torus Embedding]) (C 1 C 3 ) : (C 2 C 3 ) : (C 2 3) = 1 : 2 : ( 1): We have K W + P C i 0 in a neighborhood of C 3 since W is a toric variety. Since K Y s + B Y s is nef, K W + B W s = K W + X (1 a`(u i ; Y s ))C i (in a nbd of C 3 ) is also nef. Hence we have 0 (K W + X (1 a`(u i ; Y s ))C i ) C 3 = ( C 2 3) (a`(u 3 ; Y s ) 2X i=1 i a`(u i ; Y s )): For (2), we repeat the argument in (1) with u 1 = v a ; u 2 = v a+1 and u 3 = v. We note that if we set : W = T (u 1 ; u 2 ; u 3 )! T (E), then (K T (E) + B T (E)s ) = K W + X (1 a`(u i ; T (E) s ))C i (in a nbd of C 3 ) is trivial on the -exceptional C 3. Hence we get the equality. ut