LOG DEL PEZZO SURFACES WITH LARGE VOLUMES 4 KENTO FUJITA 1 0 2 n a Abstract. WeclassifyallofthelogdelPezzosurfacesS ofindex J a such that the volume (−K2) is larger than or equal to 2a. S 8 ] 1. Introduction G A The purpose of this paper is to classify all of the log del Pezzo sur- . faces of fixed index with large anti-canonical volumes. A normal pro- h t jective variety S is called a log Fano variety if S is log-terminal and the a m anti-canonical divisor −KS is ample. We call 2-dimensional log Fano varieties as log del Pezzo surfaces. For a log Fano variety S, the index [ of S is defined as the smallest positive integer a such that −aK is 1 S v Cartier. Log del Pezzo surfaces S with fixed index a have been treated 8 by many authors in various viewpoint. 8 We recall the viewpoint in terms of the classification problem of log 5 1 del Pezzo surfaces. If a = 1 then all such S are classified by several . 1 authors, see [Bre80, Dem80, HW81]; if a = 2 then all such S are 0 classified in [AN88, AN89, AN06, Nak07]; and if a = 3 then all such 4 S are classified in [FY14]. However, it is not realistic to classify all of 1 : the log del Pezzo surfaces of (fixed) index a ≥ 4 since even for the case v i a = 3 there are 300 types of such log del Pezzo surfaces (see [FY14]). X On the other hand, the problem of the boundedness for log Fano va- r a rieties of fixed index a is also important in the area of minimal model program. The problem is equivalent to bound the anti-canonical volume (−Kn) for such S if the characteristic of the base field is zero. S Such problem is called the Batyrev conjecture. See [Bor01, HMX12]. We remark that the authors in [HMX12] showed that the Baryrev conjecture is true (in any dimension, if the characteristic of the base field is zero). Nowadays, a generalized version of the Batyrev conjecture, called the BAB conjecture, is considered by many authors, see [BB92, Ale94, AM04, Lai12, Jia13]. The BAB conjecture is true for Date: January 9, 2014. 2010 Mathematics Subject Classification. Primary 14J26;Secondary 14E30. Key words and phrases. del Pezzo surface, rational surface, extremal ray. 1 2 KENTO FUJITA log del Pezzo surfaces but is open in higher-dimensional case. Note that, as a corollary of [Jia13], any log del Pezzo surface S of index a ≥ 2 satisfies that (−K2) ≤ (2a2+4a+2)/a and equality holds if and S only if S is isomorphic to the weighted projective plane P(1,1,2a). In this paper, we classify the log del Pezzo surfaces S of index a ≥ 4 such that (−K2) ≥ 2a. The motivation is related to both the classi- S fication problem and the boundedness problem. We note that log del Pezzo surfaces S of index a ≤ 3 are completely classified. Theorem 1.1. Let S be a log del Pezzo surface of index a ≥ 4. Then (−K2) ≥ 2a holds if and only if S is isomorphic to the log del Pezzo S surface associated to the a-fundamental multiplet of length b with b = ⌊(a+1)/2⌋ whose type is one of the following: (1) hOi ((−K2) = (2a2 +4a+2)/a), a S (2) hIi ((−K2) = (2a2 +3a+2)/a), a S (3) hIIi ((−K2) = (2a2 +2a+2)/a), a S (4) hIIIi ((−K2) = (2a2 +a+2)/a), a S (5) hAi (if a = 5) ((−K2) = 54/5), 5 S (6) hIVi ((−K2) = (2a2 +2)/a), a S (7) hBi (if a = 4) ((−K2) = 8), 4 S (8) hCi (if a = 4) ((−K2) = 8). 4 S We describe the above types in Section 4.1. By Theorem 1.1 and the arguments in Section 4, we have the following result. We note that we do not use the results in [Jia13] in order to prove Corollary 1.2. Corollary 1.2. Let S be a log del Pezzo surface of index a ≥ 2. Then (−K2) > 2(a + 1) holds if and only if S is isomorphic to one of the S following (We describe the following varieties in Section 4.2.) : (1) P(1,1,2a) ((−K2) = (2a2 +4a+2)/a), S (2) S ((−K2) = (2a2 +3a+2)/a), I,a S (3) S ((−K2) = (2a2 +2a+2)/a), II1,a S (4) S ((−K2) = (2a2 +2a+2)/a), II2,a S (5) P(1,1,3) (if a = 3) ((−K2) = 25/3). S We remark that all of them are toric varieties. The strategy for the classification is essentially same as the strategy in [FY14] based on the earlier work in [Nak07]. First, we reduce the classification problem of log del Pezzo surfaces of index a to the classification problem of the pair of its minimal resolution and (−a) times the discriminant divisor. We call such pair an a-basic pair (see Section 3.1). Next, by contracting (−1)-curves very carefully, we get LOG DEL PEZZO SURFACES WITH LARGE VOLUMES 3 an a-fundamental multiplet from an a-basic pair (see Section 3.2). See also the flowcharts in [FY14, §1]. From the assumption (−K2) ≥ 2a, S thestructureoftheassociateda-fundamentalmultiplet hasveryspecial structure. Thus we can get the multiplets in Section 4.1. Acknowledgments. The author is partially supported by a JSPS Fel- lowship for Young Scientists. Notation and terminology. We work in the category of algebraic (separated and of finite type) scheme over a fixed algebraically closed field k of arbitrary characteristic. A variety means a reduced and irreducible algebraic scheme. A surface means a two-dimensional variety. For a normal variety X, we say that D is a Q-divisor (resp. divisor or Z-divisor) if D is a finite sum D = a D where D are prime P i i i divisors and a ∈ Q (resp. a ∈ Z). For a Q-divisor D = a D , i i P i i the value a is denoted by coeff D and set coeffD := {a } . For an i Di i i effective Q-divisor or a scheme D on X, let |D| be the support of D. A normal variety X is called log-terminal if the canonical divisor K is X Q-Cartier and the discrepancy discrep(X) of X is bigger than −1 (see [KM98, §2.3]). For a proper birational morphism f: Y → X between normal varieties such that both K and K are Q-Cartier, we set X Y KY/X := X a(E0,X)E0, E0⊂Y f-exceptional where a(E ,X)is thediscrepancy ofE with respects toX (see[KM98, 0 0 §2.3]). (We note that if aK and aK are Cartier for a ∈ Z , then X Y >0 aK is a Z-divisor.) Y/X ForanonsingularsurfaceS andaprojectivecurveC whichisaclosed subvariety of S, the curve C is called a (−n)-curve if C is a nonsingular rational curve and (C2) = −n. For a birational map M 99K S between normal surfaces and for a curve C ⊂ S, the strict transform of C on M is denoted by CM. If the birational map is of the form M 99K M , i j then the strict transform of C ⊂ M on M is denoted by Ci. j i Let S be a nonsingular surface and let D = a D be an effective P j j divisor on S (a > 0) such that |D| is simple normal crossing and D , j i D are intersected at most one point. (It is sufficient in our situation.) j The dual graph of D is defined as follows. A vertex corresponds to a component D . Let v be the vertex corresponds to D . The v and j j j i v are joined by a (simple) line if and only if D , D are intersected. j i j In the dual graphs of divisors, a vertex corresponding to (−n)-curve is expressed as (cid:13)n . On the other hand, an arbitrary irreducible curve is expressed by the symbol ⊘ when it is not necessary a (−n)-curve. 4 KENTO FUJITA Let F → P1 be a Hirzebruch surface P (O⊕O(n)) of degree n with n P1 the P1-fibration. A section σ ⊂ F with (σ2) = −n is called a minimal n section. If n > 0, then such σ is unique. We usually denote a fiber of F → P1 by l. n For a real number t, let ⌊t⌋ be the greatest integer not grater than t. 2. Elimination of subschemes In this section, we recall the results in [Nak07, §2] (see also [Fuj14, §2]). Let X be a nonsingular surface and ∆ be a zero-dimensional subscheme of X. The defining ideal sheaf of ∆ is denoted by I . ∆ Definition 2.1. Let P be a point of ∆. (1) Let ν (∆) := max{ν ∈ Z |I ⊂ mν}, where m is the maxi- P >0 ∆ P P mal ideal sheaf in O defining P. If ν (∆) = 1 for any P ∈ ∆, X P then we say that ∆ satisfies the (ν1)-condition. (2) The multiplicity mult ∆ of ∆ at P is given by the length of P the Artinian local ring O . ∆,P (3) The degree deg∆ of ∆ is given by mult ∆. PP∈∆ P Definition 2.2. Assume that ∆ satisfies the (ν1)-condition. Let V → X be the blowing up along ∆. The elimination of ∆ is the birational projective morphism ψ: Y → X which is defined as the composition of the minimal resolution Y → V of V and the morphism V → X. For any divisor E on X and for any positive integer s, we set E∆,s := ψ∗E −sK . Y/X Proposition 2.3 ([Nak07, Proposition 2.9]). (1) Assume that the subscheme ∆ satisfies the (ν1)-condition and let ψ: Y → X be the elimination of ∆. Then the anti-canonical divisor −K Y is ψ-nef. More precisely, for any P ∈ ∆ with mult ∆ = P k, the set-theoretic inverse image ψ−1(P) is the straight chain k Γ of nonsingular rational curves and the dual graph of Pj=1 P,j ψ−1(P) is the following: Γ Γ Γ Γ P,1 P,2 P,k−1 P,k (cid:13)2 (cid:13)2 (cid:13)2 (cid:13)1 (2) Conversely, for a proper birational morphism ψ: Y → X between nonsingular surfaces such that −K is ψ-nef, the mor- Y phismψ is the eliminationof ∆whichsatisfiesthe (ν1)-condition defined by the ideal I := ψ O (−K ). ∆ ∗ Y Y/X LOG DEL PEZZO SURFACES WITH LARGE VOLUMES 5 Definition 2.4. Under the assumption of Proposition 2.3 (1), we al- ways denote the exceptional curves of ψ over P by Γ ,...,Γ . The P,1 P,k order is determined as Proposition 2.3 (1). Now we see some examples of the dual graphs of E∆,s. Example 2.5 ([Fuj14, Example 2.5]). Assume that ∆ satisfies the (ν1)-condition such that |∆| = {P}. Let E = eC be a divisor on X such that P ∈ C and C is nonsingular. Let m := deg∆ and k := mult (∆∩C). Then we have P k m E∆,s = eCY +Xi(e−s)ΓP,i + X (ek −si)ΓP,i, i=1 i=k+1 where ψ: Y → X is the elimination of ∆. Moreover, the dual graph of ψ−1(E) is the following: Γ Γ Γ Γ P,1 P,k P,m−1 P,m (cid:13)2 (cid:13)2 (cid:13)2 (cid:13)1 ⊘ CY Example 2.6. [Fuj14, Example 2.6] Assume that ∆ satisfies the (ν1)- condition such that |∆| = {P}. Let E = e C + e C be a non-zero 1 1 2 2 effective divisor such that C and C are nonsingular and intersect 1 2 transversally at a unique point P = C ∩ C . Let m := deg∆ and 1 2 k := mult (∆∩C ). By [Nak07, Lemma 2.12], we may assume that j P j k = 1. Then we have 1 k2 m E∆,s = e1C1Y +e2C2Y +X(i(e2−s)+e1)ΓP,i+ X (e1+k2e2−is)ΓP,i, i=1 i=k2+1 where ψ: Y → X is the elimination of ∆. Moreover, the dual graph of ψ−1(E) is the following: CY Γ Γ Γ Γ 1 P,1 P,k2 P,m−1 P,m ⊘ (cid:13)2 (cid:13)2 (cid:13)2 (cid:13)1 ⊘ CY 2 3. Log del Pezzo surfaces We define the notion of log del Pezzo surfaces, a-basic pairs, and a-fundamental multiplets, and we see the correspondence among them. 6 KENTO FUJITA 3.1. Log del Pezzo surfaces and a-basic pairs. Definition 3.1. (1) A normal projective surface S is called a log del Pezzo surface if S is log-terminal and the anti-canonical divisor −K is an ample Q-Cartier divisor. S (2) Let S be a log del Pezzo surface. The index of S is defined as min{a ∈ Z |−aK is Cartier}. >0 S Remark 3.2. AnylogdelPezzo surfaceisarationalsurfaceby[Nak07, Proposition3.6]. Inparticular, thePicardgroupPic(S)ofS isafinitely generated and torsion-free Abelian group. Definition 3.3 ([FY14, Definition 3.3]). Fix a ≥ 2. A pair (M,E ) M is called an a-basic pair if the following conditions are satisfied: (C1) M is a nonsingular projective rational surface. (C2) E is a nonzero effective divisor on M such that coeffE ⊂ M M {1,...,a−1} and |E | is simple normal crossing. M (C3) A divisor L ∼ −aK −E (called the fundamental divisor of M M M (M,E ))satisfiesthatK +L isnefand(K +L ·L ) > 0. M M M M M M (C4) For any irreducible component C ≤ E , (L ·C) = 0 holds. M M We see the correspondence between log del Pezzo surfaces and a- basic pairs. Proposition 3.4 ([FY14, Proposition 3.4]). Fix a ≥ 2. (1) Let S be a non-Gorenstein log del Pezzo surface such that −aK S is Cartier. Let α: M → S be the minimal resolution of S and let E := −aK . Then (M,E ) is an a-basic pair and the M M/S M divisor α∗(−aK ) is the fundamental divisor of (M,E ). S M (2) Let (M,E ) be an a-basic pair and L be the fundamental M M divisorof (M,E ). Thenthere exists aprojective andbirational M morphism α: M → S such that S is a non-Gorenstein log del Pezzo surface with −aK Cartier and L ∼ α∗(−aK ) holds. S M S Moreover, the morphism α is the minimal resolution of S. In particular, under the correspondence between S and (M,E ), we M have a(−K2) = (1/a)(L2 ), where L is the fundamental divisor. S M M Definition 3.5. Let a ≥ 2. (1) For a log del Pezzo surface S of index a, the corresponding a-basic pair (M,E ) which is given in Proposition 3.4 (1) is M called the associated a-basic pair of S. (2) For an a-basic pair (M,E ), the corresponding log del Pezzo M surface S which is given in Proposition 3.4 (2) is called the associated log del Pezzo surface of (M,E ). We note that a is M divisible by the index of S. LOG DEL PEZZO SURFACES WITH LARGE VOLUMES 7 We discuss that when the log del Pezzo surface S associated to an a-basic pair (M,E ) is of index a. M Lemma 3.6. Let S be a log del Pezzo surface of index a ≥ 2 and (M,E ) be the associated a-basic pair. Pick any irreducible component M C ≤ E and set e := coeff E and d := −(C2). Then 2 ≤ d ≤ M C M 2a/(a−e). In particular, d ≤ 2a. Proof. By Proposition 3.4, d ≥ 2. We know that −ed ≤ (E · C) = M (−aK ·C) = a(2−d) since (L ·C) = 0, where L is thefundamental M M M divisor. Thus d ≤ 2a/(a−e). Since e ≤ a−1, we have d ≤ 2a. (cid:3) Corollary 3.7. Let (M,E ) be an a-basic pair with a ≥ 2. Assume M that there exists an irreducible component C ≤ E such that either M (C2) < −a, or coeff E and a are coprime. Then the associated log C M del Pezzo surface S is of index a. Proof. Let a′ be the index of S. Then a′ > 1 and a is divisible by a′. Moreover, coeff E is divisible by a/a′. Thus the assertion follows C M (cid:3) from Lemma 3.6. 3.2. a-fundamental multiplets. In order to determine a-basic pairs, we define the notion of a-fundamental multiplets given in [FY14, §1]. Definition 3.8. Fix a ≥ 2 and 1 ≤ i ≤ a − 1. We define the following notion inductively. A multiplet (M ,E ;∆ ,...,∆ ) is called an i i 1 i a-pseudo-fundamental multiplet of length i if the following conditions are satisfied: (F1) M is a nonsingular projective rational surface and E is a i i nonzero effective divisor on M . i (F2) A divisor L ∼ −aK − E (called the fundamental divisor) i Mi i satisfies that iK +L is nef and ((i+1)K +L ·γ) ≥ 0 for Mi i Mi i any (−1)-curve γ ⊂ M . i (F3) ∆ ⊂ M is a zero-dimensional subscheme which satisfies the i i (ν1)-condition. (F4) Let π : M → M be the elimination of ∆ and E := i i−1 i i i−1 E∆i,a−i. Then the following holds: i • If i = 1, then (M ,E ) is an a-basic pair. 0 0 • If i ≥ 2, then (M ,E ;∆ ,...,∆ ) is an a-pseudo- i−1 i−1 1 i−1 fundamental multiplet of length i−1. Moreover, if (i+1)K +L is not nef, then we call the multiplet an Mi i a-fundamental multiplet of length i. Definition 3.9. (1) Let(M ,E ;∆ ,...,∆ )beana-pseudo-funda- i i 1 i mentalmultiplet. Wecallthea-basicpair(M ,E )(constructed 0 0 from the multiplet inductively) as the associated a-basic pair. 8 KENTO FUJITA (2) We sometimes call a-basic pairs as a-pseudo-fundamental multiplets of length zero for convenience. The following proposition is important. Proposition3.10. Fix a ≥ 2and 0 ≤ i ≤ a−1. Let(M ,E ;∆ ,...,∆ ) i i 1 i be an a-pseudo-fundamental multiplet of length i, π : M → M be j j−1 j the elimination of ∆ , E := E∆j,a−j and L be the fundamental j j−1 j j divisor of (M ,E ;∆ ,...,∆ ) for any 0 ≤ j ≤ i. j j 1 j (1) Assume that (i+1)K +L is nef. Then i ≤ a−2 and iK +L Mi i Mi i is nef and big. Moreover, there exists a projective and birational morphism π : M → M between nonsingular surfaces such i+1 i i+1 that the following conditions are satisfied: • There exists a zero-dimensional subscheme ∆ ⊂ M i+1 i+1 which satisfies the (ν1)-condition such that π is the elim- i+1 ination of ∆ . i+1 • Set E := (π ) E and L := (π ) L . Then E = i+1 i+1 ∗ i i+1 i+1 ∗ i i E∆i+1,a−i−1 and L = L∆i+1,i+1. i+1 i i+1 • (M ,E ;∆ ,...,∆ ) is ana-pseudo-fundamentalmul- i+1 i+1 1 i+1 tiplet of length i+1 and L is the fundamental divisor. i+1 (2) Assume that (i + 1)K + L is not nef, that is, the multiplet Mi i (M ,E ;∆ ,...,∆ ) is an a-fundamental multiplet. Then M is i i 1 i i isomorphic to either P2 or F , and ((i + 1)K + L · l) < 0, n Mi i where l is a line (if M ≃ P2); a fiber (if M ≃ F ). i i n (3) L is nef and big. Moreover, we have the following: i • (L ·E ) = i j(a−j)deg∆ . i i Pj=1 j • (K +L ·L )−(K +L ·L ) = i j(j −1)deg∆ . Mi i i M0 0 0 Pj=1 j • (L ·C) = i jdeg(∆ ∩Cj) for any nonsingular com- i Pj=1 j ponent C ≤ E . i • (1/a)(L2) = (−K ·L )− i jdeg∆ . 0 Mi i Pj=1 j Proof. We can assume by induction on i that L is nef and big and i−1 L = L∆i,i if i ≥ 1. Thus L is nef and big. i−1 i i (1) Assume that (i+1)K +L is nef. Since a((i+1)K +L ) ∼ Mi i Mi i −(i+1)E +(a−(i+1))L , we must have a−(i+1) > 0. Moreover, i i since (i+1)(iK +L ) = i((i+1)K +L )+L , iK +L is nef and Mi i Mi i i Mi i big. From now on, we run ((i+2)K +L )-minimal model program Mi i and let π : M → M be the composition of the morphisms in i+1 i i+1 the program. More precisely, we obtain the morphism π which is i+1 the composition of monoidal transforms such that each exceptional (−1)-curve intersects the strict transform of (i+2)K +L negatively. Mi i Furthermore, ((i + 2)K + L · γ) ≥ 0 holds for any (−1)-curve Mi+1 i+1 LOG DEL PEZZO SURFACES WITH LARGE VOLUMES 9 γ ⊂ M . Since (i+1)K +L is nef, any (−1)-curve in each step of i+1 Mi i themonoidaltransformintersects thestrict transformof(i+1)K +L Mi i trivially. Thus(i+1)K +L = π∗ ((i+1)K +L ). Inparticular, Mi i i+1 Mi+1 i+1 −K is π -nef. By Proposition 2.3, there exists a zero-dimensional Mi i+1 subscheme ∆ ⊂ M which satisfies the (ν1)-condition such that i+1 i+1 π is the elimination of ∆ . Furthermore, we have L = L∆i+1,i+1 i+1 i+1 i i+1 and E = E∆i+1,a−i−1. Since each step of the ((i+2)K +L )-minimal i i+1 Mi i model programcan be seen as a step of (−E )-minimal model program, i E is nonzero effective. Thus the assertion (1) follows. i+1 (2) Follows from [Mor82, Theorem 2.1]. (3) We know that (−K ·L ) = (−K ·L )−i(K2 ) = Mi i Mi−1 i−1 Mi−1/KMi (−K · L ) +ideg∆ . Thus we have (1/a)(L2) = (−K ·L ) − Mi−1 i−1 i 0 Mi i i jdeg∆ by induction on i. Other assertions follow similarly. (cid:3) Pj=1 j As a direct corollary of Proposition 3.10 (1), we have the following. Corollary 3.11. Fix a ≥ 2. Let (M,E ) be an a-basic pair. Then M there exists a positive integer 1 ≤ b ≤ a − 1 and an a-fundamental multiplet (M ,E ;∆ ,...,∆ ) of length b such that the associated a- b b 1 b basic pair is isomorphic to (M,E ). M From the next proposition, we can replace an a-fundamental multi- pletwithanother one. Theproofissameasthatof[Nak07, Proposition 4.4]. See also [Fuj14, Proposition 3.14] and [FY14, Theorem 3.12]. Proposition 3.12. Fix a ≥ 2 and 1 ≤ b < a. Let (M ,E ;∆ ,...,∆ ) b b 1 b be an a-fundamental multiplet of length b and L be the fundamental b divisor. Assume that M = F and bK +L is not big and not trivial. b n Mb b Thenthere existsan a-fundamentalmultiplet(M′,E′;∆ ,...,∆ ,∆′) b b 1 b−1 b such that the associated a-pseudo-fundamental multiplets of length b−1 are same, Mb′ = Fn′ for some n′ ≥ 0 and ∆′b ∩σ′ = ∅, where σ′ ⊂ Fn′ is a section with (σ′2) = −n′. The next proposition is useful to know when a given multiplet is an a-pseudo-fundamental multiplet. Proposition 3.13. Fix 1 ≤ i < a. Let X be a nonsingular projective surface, E be a divisor on X, L ∼ −aK − E be a divisor such that X iK +L is nef, ∆ ⊂ X be a zero-dimensional subscheme which satisfies X the (ν1)-condition, ψ: Y → X be the elimination of ∆, L := L∆,i and Y E := E∆,a−i. If E is effective and (L ·C) ≥ 0 for any irreducible Y Y Y component C ≤ E , then jK +L is nef for any 0 ≤ j ≤ i. Y Y Y Proof. Assume that there exists an integer 0 ≤ j ≤ i such that jK + Y L is non-nef. Since iK + L = ψ∗(iK + L), we have j < i. Pick Y Y Y X 10 KENTO FUJITA an irreducible curve B ⊂ Y such that (jK +L ·B) < 0. Then Y Y 0 > (a−i)(jK +L ·B) Y Y = (a−j)(iK +L ·B)+(i−j)(E ·B) ≥ (i−j)(E ·B). Y Y Y Y Thus B ≤ E . Hence (L ·B) ≥ 0. However, Y Y 0 > i(jK +L ·B) = j(iK +L ·B)+(i−j)(L ·B) ≥ 0, Y Y Y Y Y (cid:3) which leads to a contradiction. 3.3. Local properties. In this section, we fix the following notation: Let 1 ≤ i < a, (M ,E ;∆ ,...,∆ ) be an a-pseudo-fundamental multi- i i 1 i plet of length i, L be its fundamental divisor, π : M → M be the i j j−1 j elimination of ∆ , E := E∆j,a−j and L := L∆j,j for 1 ≤ j ≤ i. j j−1 j j−1 j Moreover, we fix a point P ∈ ∆ . i Lemma 3.14. (1) The multiplicity of E at P is bigger than or i equal to a−i. (2) Pick any π -exceptional curve Γ ⊂ M . If (Γ2) = −1, then i i−1 (L ·Γ) = i. If (Γ2) = −2, then (L ·Γ) = 0. i−1 i−1 (3) Assume that E = eC around P, where P ∈ C and C is non- i singular and e ≤ a−i. Then e = a−i, ∆ ⊂ C around P and i E is the strict transform of E around over P. i−1 i (4) Assume that E = (a−1)C around P, where P ∈ C and C is i nonsingular. If2i ≤ a+1, mult (∆ ∩C) = 1 andmult ∆ ≥ 2, P i P i then 2i = a+1 and mult ∆ = 2. P i Proof. (1) The value coeff E must be nonnegative. ΓP,1 i−1 (2) Follows from the fact (iK +L ·Γ) = 0. Mi−1 i−1 (3) Set m := mult ∆ and k := mult (∆ ∩ C). By Example P i P i 2.5, coeff E = e − (a − i). Thus e = a − i. If k < m, then ΓP,1 i−1 coeff E = (a−i)(k −m) < 0 by Example 2.5, a contradiction. ΓP,m i−1 (4) We know that coeff E = a−1−2(a−i) ≤ 0 by Example ΓP,2 i−1 2.5. Thus 2i = a+1 and mult ∆ = 2. (cid:3) P i Lemma 3.15. Assume that a ≥ 4, i = 1, 1 ≤ e ≤ 2, E = (a−1)C + 1 1 eC around P, where C , C are nonsingular and intersect transver- 2 1 2 sally at P. Then e = 2 and mult ∆ = mult (∆ ∩ C ). Moreover, P 1 P 1 2 (a,mult ∆ ) = (5,2) or (4,3). P 1 Proof. Set m := mult ∆ and k := mult (∆ ∩C ). By Lemma 3.6, P 1 j P 1 j coeff E = 0. If k > 1, then k = 1 by Example 2.6. However, if ΓP,m 0 1 2 m = k , then coeff E = e; if m > k , then coeff E = e−(a− 1 ΓP,m 0 1 ΓP,k1+1 0 1) < 0. This leads to a contradiction. Thus k = 1. If m > k , then 1 2 coeff E = a−1+ek −m(a−1) ≤ 2k −k (a−1) < 0 by Example ΓP,m 0 2 2 2

Log del Pezzo surfaces with large volumes

Similar Log del Pezzo surfaces with large volumes