ON SINGULAR CUBIC SURFACES

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1 ASIAN J. MATH. c 2009 International Press Vol. 13, No. 2, pp , June ON SINGULAR CUBIC SURFACES IVAN CHELTSOV Abstract. We study global log canonical thresholds of singular cubic surfaces. Key words. Cubic surfaces, singularities, log canonical thresholds, del Pezzo fibrations, birational maps, Kahler-Einstein metric, alpha-invariant of Tian, orbifolds. AMS subject classifications. 14J26, 14J45, 14J70, 14Q10, 14B05, 14E05, 32Q20 All varieties are assumed to be defined over C. 1. Introduction. Let X be a variety with at most log terminal singularities, let Z X be a closed subvariety, and let D be an effective Q-Cartier Q-divisor on X. the number lct Z X, D = sup {λ Q the log pair X, λd } is log canonical along Z is said to be the log canonical threshold of D along Z see [8]. Example 1.1. Let φ C[z 1,, z n ] be a nonzero polynomial, let O C n be the origin. lct O C n, φ = 0 { } 1 = sup c Q the function φ 2c is locally integrable near O. For the case Z = X we use the notation lctx, D instead of lct X X, D. lct X, D = inf {lct } P X, D P X { = sup λ Q the log pair X, λd } is log canonical. Suppose, in addition, that X is a Fano variety. Definition 1.2. We define the global log canonical threshold of X by the number lct X = inf {lct X, D } D is effective Q-divisor on X such that D KX. The number lctx is an algebraic counterpart of the α-invariant introduced in [11]. Example 1.3. Let X be a smooth cubic surface in P 3. it follows from [4] that lct X { 2/3 when X has an Eckardt point, = 3/4 when X does not have Eckardt points. Received October 28, 2008; accepted for publication December 2, School of Mathematics, University of Edinburgh, Edinburgh EH9 3JZ, UK I.Cheltsov@ ed.ac.uk. The author would like to thank I. Dolgachev, J. Kollár, V. Shokurov, J. Park and the referee for useful comments. 191

2 192 I. CHELTSOV In this paper we prove the following result 1. Theorem 1.4. Let X be a singular cubic surface in P 3 with canonical singularities. 2/3 when Sing X = { } A 1, 1/3 when Sing X { } A 4, 1/3 when Sing X = { } D 4, lct X 1/3 when Sing X { } A 2, A 2, = 1/4 when Sing X { } A 5, 1/4 when Sing X = { } D 5, 1/6 when Sing X = { } E 6, 1/2 in other cases. Let us consider one birational application of Theorem 1.4. Theorem 1.5. Let Z be a smooth curve. Suppose that there is a commutative diagram V ρ V 1.6 π Z such that π and π are flat morphisms, and ρ is a birational map that induces an isomorphism Z π ρ V \X : V \ X V \ X, 1.7 where X and X are scheme fibers of π and π over a point O Z, respectively. Suppose that the varieties V and V have terminal Q-factorial singularities, the divisors K V and K V are π-ample and π-ample, respectively, the fibers X and X are irreducible. ρ is an isomorphism if one of the following conditions hold: the varieties X and X have log terminal singularities, and lctx+lct X > 1; the variety X has log terminal singularities, and lctx 1. The assertion of Theorem 1.5 is sharp see [10, Example ]. Example 1.8. Let V be V subvarieties in C 1 P 3 given by the equations x 3 + y 3 + z 2 w + t 6 w 3 = 0 and x 3 + y 3 + z 2 w + w 3 = 0, respectively, where t is a coordinate on C 1, and x, y, z, w are coordinates on P 3. The projections cone. π: V C 1 and π: V C 1 1 A cubic surface in P 3 with isolated singularities has canonical singularities it is not a

3 ON SINGULAR CUBIC SURFACES 193 are fibrations into cubic surfaces. Let O be the point on C 1 given by t = 0. X is smooth, the surface X has one singular point of type D 4. Put Z = C 1. the map x, y, z, w t 2 x, t 2 y, t 3 z, w induces a birational map ρ: V V such that the diagrams 1.6 and isomorphism 1.7 exist, and ρ is not biregular. But lctx = 1/3 and lct X = 2/3 see Example 1.3 and Theorem 1.4. Example 1.9. Let V be V subvarieties in C 1 P 3 given by the equations x 3 + y 2 z + z 2 w + t 12 w 3 = 0 and x 3 + y 2 z + z 2 w + w 3 = 0, respectively, where t is a coordinate on C 1, and x, y, z, w are coordinates on P 3. The projections π: V C 1 and π: V C 1 are fibrations into cubic surfaces. Let O be the point on C 1 given by t = 0. X is smooth, the surface X has one singular point of type E 6. Put Z = C 1. the map x, y, z, w t 2 x, t 3 y, z, t 6 w induces a birational map ρ: V V such that the diagrams 1.6 and isomorphism 1.7 exist, and ρ is not biregular. But lctx = 1/6 and lct X = 2/3 see Example 1.3 and Theorem 1.4. Example Let V be V subvarieties in C 1 P 3 given by the equations wz 2 + zx 2 + y 2 x + t 8 w 3 = 0 and wz 2 + zx 2 + y 2 x + w 3 = 0, respectively, where t is a coordinate on C 1, and x, y, z, w are coordinates on P 3. The projections π: V C 1 and π: V C 1 are fibrations into cubic surfaces. Let O be the point on C 1 given by t = 0. X is smooth, the surface X has one singular point of type D 5. Put Z = C 1. the map x, y, z, w t 2 x, ty, z, t 4 w induces a birational map ρ: V V such that the diagrams 1.6 and isomorphism 1.7 exist, and ρ is not biregular. But lctx = 1/4 and lct X = 2/3 see Example 1.3 and Theorem 1.4. The number lctx is closely related to the existence of a Kähler Einstein metric see [6], but we can not use Theorem 1.4 to prove the existence of such a metric on singular cubic surfaces. Remark If a singular normal cubic surface in P 3 admits an orbifold Kähler Einstein metric, then its singular locus must consist of singular points of type A 1 and A 2 see [7].

4 194 I. CHELTSOV Nevertheless, we can use an equivariant analogue of the number lctx to prove the existence of an orbifold Kähler Einstein metric on some symmetric singular cubic surfaces. Example Let X 1 be the Cayley cubic surface in P 3, i.e. a singular surface given by xyz + xyt + xzt + yzt = 0 P 3 = Proj C[x, y, z, t], and let X 2 be a cubic surface in P 3 that is given by the equation xyz = t 3. Put lct } the log pair X 1, λd has log canonical singularities X 1, S 4 = sup {λ Q, for every S 4 -invariant effective Q-divisor D K X1 where we consider S 4 as a subgroup of AutX 1. Similarly, we define lctx 2, S 3 Z 3. lct X 1, S 4 = lct X 2, S 3 Z 3 = 1 > 2 3 by [4, Lemma 5.1]. X 1 and X 2 admit Kähler Einstein metrics 2 by [6] cf. [5, Appendix A]. We prove Theorem 1.4 in Section 3, and we prove Theorem 1.5 in Section Basic tools. Let S be a surface with canonical singularities, and D be an effective Q-divisor on it. Remark 2.1. Let B be an effective Q-divisor on S such that S, B is log canonical. 1 S, D αb 1 α is not log canonical if S, D is not log canonical, where α Q such that 0 α < 1. Let LCSS, D S be a subset such that P LCSS, D if and only if S, D is not log terminal at the point P. The set LCSS, D is called the locus of log canonical singularities. P. Lemma 2.2. Suppose that K S + D is ample. LCSS, D is connected. Proof. See Theorem 17.4 in [9]. Let P be a point of the surface S such that S, D is not log canonical at the point Remark 2.3. Suppose that S is smooth at P. mult P D > 1. Let C be an irreducible curve on the surface S. Put D = mc + Ω, where m Q such that m 0, and Ω is an effective Q-divisor such that C SuppΩ. 2 The existence of orbifold Kähler Einstein metrics on X 1 and X 2 is obvious, because both X 1 and X 2 are quotients branched over singular points of smooth Kähler Einstein del Pezzo surfaces see [2] and [7, Example 1.4].

5 ON SINGULAR CUBIC SURFACES 195 Remark 2.4. Suppose that C LCSS, D. m 1. Lemma 2.5. Suppose that P C, the surface S is smooth at P, and m 1. C Ω > 1. Proof. It follows from Theorem 17.6 in [9] that C Ω mult P Ω C > 1. Let π: S S be a birational morphism such that the surface S has canonical singularities, and D is a proper transform of D via π. K S + D + r a i E i π K S + D, where E i is a π-exceptional curve, and a i is a rational number. i=1 Remark 2.6. The log pair S, D is log canonical if and only if S, D+ r i=1 a ie i is log canonical. Suppose that r = 1, πe 1 = P, and P is an ordinary double point. Lemma 2.7. Suppose that S is smooth along E 1. a 1 > 1/2. Proof. The inequality a 1 > 1/2 follows from Theorem 17.6 in [9]. Most of the described results are valid in much more general settings see [9]. 3. Main result. Let S be a singular cubic surface in P 3 with canonical singularities. Put Σ = SingS and lct n S = sup {µ Q the log pair S, µ n D is log canonical for every D } nkx for every n N. it follows from [12] that 2/3 when Σ = { } A 1, 1/3 when Σ { } A 4, 1/3 when Σ = { } D 4, lct S = inf n N lct 1/3 when Σ { } A 2, A 2, n S lct 1 S = 1/4 when Σ { } A 5, 1/4 when Σ = { } D 5, 1/6 when Σ = { } E 6, 1/2 in other cases. Let D be an arbitrary effective Q-divisor on the surface S such that D K S O P 3 1 S, and let λ be an arbitrary positive rational number such that λ < lct 1 S. Lemma 3.1. Suppose that lct 1 S 1/3. LCSS, λd Σ. Proof. Suppose that S, λd is not log terminal at a smooth point P S. 3 = K S D mult P D > 1/λ > 3,

6 196 I. CHELTSOV which is a The obtained contradiction implies that LCSS, λd Σ. Lemma 3.2. Suppose that LCSS, λd < +. LCSS, λd Σ. Proof. The required assertion follows from [4]. Let O be a singular point of the surface S, and α: S S be a partial resolution of singularities that contracts smooth rational curves E 1,..., E k to the point O such that k S \ E i = S \ O, i=1 the surface S is smooth along k i=1 E i, and E 2 i = 2 for every i = 1,...,k. D α D k a i E i, where D is the proper transform of D on the surface S, and a i Q. Let L 1,..., L r be lines on the surface S such that O L i, and L i be the proper transform of L i on the surface S. i=1 K S L 1 = = K S L r = 1. Remark 3.3. To prove Theorem 1.4, we must show that the equality lct S = lct 1 S holds. Hence, it follows from the choice of the divisor D and λ Q that to prove Theorem 1.4 it is enough to show that the singularities of the log pair S, λd are log canonical. In the rest of the section, we prove Theorem 1.4 case by case using [1]. Lemma 3.4. Suppose that Σ = {A 1 }. lcts = 2/3. Proof. Suppose that the log pair S, λd is not log canonical. Let us derive a Suppose that there is an irreducible curve Z S such that D = µz + Ω, where µ is a rational number such that µ 1/λ, and Ω is an effective Q-divisor such that Z SuppΩ. 3 = K S D = µdeg Z K S Ω µdeg Z > 3deg Z /2, which implies that Z is a line. Let C be a general conic on S such that K S Z +C. 2 = C D = µc Z + C Ω µc Z 3 2 µ, which is a LCSS, λd = O by Lemma 3.2. We have 3 2a 1 = H D 0, where H is a general curve in K S E 1. It follows from K S + λ D + λa 1 E 1 α K S + λd

7 ON SINGULAR CUBIC SURFACES 197 that there is a point Q E 1 such that S, λ D+λa 1 E 1 is not log canonical at the point Q. It follows from [1] that r = 6. Let π: S P 2 be a contraction of the curves L 1,..., L 6. Suppose that Q 6 i=1 L i. π D + a1 E 1 π K S KP 2, and π is an isomorphism in a neighborhood of Q. Let L be a general line on P 2. the locus LCS P 2, L + π λ D + λa 1 E 1 is not connected, which is impossible by Lemma 2.2. Therefore, we may assume that Q L 1. Put D = al 1 + Υ, where a is a nonnegative rational number, and Υ is an effective Q-divisor, whose support does not contain the line L 1. Ῡ α Υ ǫe 1, where ǫ = a 1 a/2, and Ῡ is the proper transform of the divisor Υ on the surface S. The log pair S, λa L 1 + λῡ + λa/2 + ǫe 1 is not log canonical at Q. 1 + a/2 ǫ = L 1 Ῡ > 1/λ a/2 ǫ by Lemma 2.5, because λa 1. Hence, we have a > 1/2. It follows from [12] that there is a conic C 1 S such that the log pair S, lct1 S L 1 + C 1 is not log terminal. But it must be log canonical. Therefore, in the case when C 1 SuppD, we can use Remark 2.1 to find an effective divisor D on the surface S such that the equivalence D K S holds, the log pair S, λd is not log canonical at the point P, and C 1 SuppD. To complete the proof, we may assume that C 1 SuppD. Let C 1 be the proper transforms of the conic C 1 on the surface S. 2 3a/2 ǫ = C 1 Ῡ mult QῩ > 1/λ a/2 ǫ, which implies that a < 1/2. But a > 1/2. The obtained contradiction completes the proof. Lemma 3.5. Suppose that Σ = {A 1,..., A 1 } and Σ 2. lcts = 1/2. Proof. Suppose that the log pair S, λd is not log canonical. Let us derive a Suppose that there is an irreducible curve Z on the surface S such that D = µz + Ω,

8 198 I. CHELTSOV where µ is a rational number such that µ 1/λ, and Ω is an effective Q-divisor, whose support does not contain the curve Z. Z is a line see the proof of Lemma 3.4. We have 2 = C D = µc Z + C Ω µc Z µ 1/λ > 2, where C is a general conic on S that intersects Z in two points. We may assume that LCSS, λd = O by Lemmas 2.2 and 3.2. a 1 > 1 by Lemma 2.7. Arguing as in the proof of Lemma 3.4, we see that there is a point Q E such that the singularities of the log pair S, λ D+λa 1 E 1 are not log canonical at the point Q. Let P be a point in Σ such that P O. We may assume that P L 1. 2L 1 + L K S for some line L S. Suppose that Q L 1. Let a be a non-negative rational number such that D = al 1 + Υ, where Υ is an effective Q-divisor, whose support does not contain the line L 1. Ῡ α Υ ǫe 1, where Ῡ is the proper transforms of Υ on the surface S, and ǫ = a 1 a/2. The log pair S, λa L1 + λῡ + λa/2 + ǫe 1 is not log canonical at the point Q. We have L 2 1 = 1/2. 1 ǫ = L 1 Ῡ > 1/λ a/2 ǫ by Lemma 2.5. We have a > 1/λ, which is impossible. Hence, we see that Q L 1. There is a unique reduced conic Z S such that O Z P and Q Z, where Z is the proper transform of the conic Z on the surface S. L 1 SuppZ, because Q L 1. Suppose that Z is irreducible. Put D = ez +, where e Q, and is an effective Q-divisor such that C Supp. α δe 1, where is the proper transforms of on the surface S, and δ = a 1 e/2. 2 e δ = Z > 1/λ e/2 δ > 2 e/2 δ by Lemma 2.5, because C 2 = 1/2. We have e < 0, which is impossible. We see that the conic Z is reducible. Z = L 2 + L 2,

9 ON SINGULAR CUBIC SURFACES 199 where L 2 is a line on S such that P L 2 and L 2 L 2. The intersection L 2 L 2 consists of a single point. The impossibility of the case Q L 1 implies that the surface S is smooth at the point L 2 L 2. There is a rational number c 0 such that D = cl 2 + Ξ, where Ξ is an effective Q-divisor, whose support does not contain the line L 2. Ξ α Ξ υe 1, where Ξ is the proper transforms of Ξ on the surface S, and υ = a 1 c/2. The log pair S, λc L2 + λ Ξ + λc/2 + υe 1 is not log canonical at Q. We have Q L 2 and L 2 2 = c/2 υ = L 2 Ξ > 1/λ c/2 υ > 2 c/2 υ by Lemma 2.5. Therefore, the inequality c > 1 holds. There is a unique hyperplane section T of the surface S such that T = C 2 + L 2 and Q = C 2 L 2 = O, where C 2 is a conic, and C 2 is the proper transforms of C 2 on the surface S. The conic C 2 is irreducible. We may assume that C 2 SuppD see Remark c/2 υ = C 2 Ξ mult Q Ξ > 1/λ c/2 υ, which implies that c < 0. The obtained contradiction completes the proof. Lemma 3.6. Suppose that Σ = {D 4 }. lcts = 1/3. Proof. Suppose that the log pair S, λd is not log canonical. Let us derive a It follows from [1] that r = 3. The lines L 1, L 2, L 3 lie in a single plane. Thus, we may assume that L 3 SuppD due to Remark 2.1 and Lemma 3.1. Let β : S S be a birational morphism such that the morphism α contracts one irreducible rational curve E that contains three singular points O 1, O 2, O 3 of type A 1. Let D and L i be the proper transforms of D and L i on the surface S, respectively. D β D µe, where µ is a positive rational number. We have L i β L i E. 0 D L 3 = β D µe L 3 = 1 µe L 3 = 1 µ/2, which implies that µ 2. Therefore, we may assume that there is a point Q E such that the singularities of the log pair S, λ D + µe are not log canonical at the point Q see Lemma 3.1.

10 200 I. CHELTSOV Suppose that S is smooth at Q. The log pair S, λ D + E is not log canonical at Q. 1 µ/2 = µe 2 = E D > 1/λ > 3 by Lemma 2.5. We see that Q = O j for some j. The curves L 1, L2 and L 3 are disjoined, and each of them passes through a singular point of the surface S. Therefore, we may assume that O i L i for every i. Let γ : Ŝ S be a blow up of the point O j, and G be the exceptional curve of γ. ˆL j γ Lj 1 2 G β γ L1 Ê G, where ˆL j and Ê are proper transforms of the curves L j and E on the surface Ŝ, respectively. Let ˆD be the proper transform of the divisor D on the surface Ŝ. ˆD γ D ǫg β γ D µ Ê µ/2 + ǫ G, where ǫ is a rational number, because 2Ê γ 2E G. By Lemma 2.7, we have λǫ + λµ/2 > 1/2. Suppose that j = 3. 1 µ/2 ǫ = ˆD ˆL 3 0. But ǫ + µ/2 > 3/2. We may assume that Q = O 1, and the support of the divisor D contains the line L 1. Put D = al 1 + Ω, where a Q and a 0, and Ω is an effective Q-divisor such that L 1 SuppΩ. ˆΩ β γ Ω m Ê m/2 + b G, where ˆΩ is the proper transform of Ω, and m and b are non-negative rational numbers. β γ D µ Ê µ/2 + ǫ G ˆD = aˆl 1 + ˆΩ β γ al1 + Ω a + m Ê a + m/2 + b G, which implies that µ = a + m 2 and ǫ = a/2 + b. We have ˆL 2 1 = 1, Ê2 = 1, G 2 = 2, ˆL Ê = 0, ˆL G = Ê G = 1 on the surface Ŝ. The surface Ŝ is smooth along the curve G. a a + ˆΩ ˆL 1 = aˆl 1 + ˆΩ ˆL 1 = 1 a m/2 b, which implies that m/2+b 1 and a+m/2+b 1+a 3. Thus, by the equivalence KŜ + λaˆl 1 + λˆω + λ a + m Ê + λ a + m/2 + b G β γ KS + λal 1 + λω,

11 ON SINGULAR CUBIC SURFACES 201 there exists a point A G such that the log pair Ŝ, λaˆl1 + λˆω + λ a + m Ê + λ a + m/2 + b G is not log canonical at the point A. Suppose that A ˆL 1 Ê. Ŝ, λˆω + λa + m/2 + bg is not log canonical at A, and 2b + a = aˆl 1 + ˆΩ G = a + ˆΩ G > a + 3, by Lemma 2.5. We see that b > 3/2. But m/2 + b 1. We see that A ˆL 1 Ê. Suppose that A ˆL 1. The log pair Ŝ, λˆω + λ a + m Ê + λ a + m/2 + b G is not log canonical at the point A. Arguing as in the previous case, we see that m/2 b = ˆΩ Ê > 3 a m/2 b, which implies that a + m > 3. But a + m 2. We see that A ˆL 1. The log pair Ŝ, λaˆl 1 + λˆω + λa + m/2 + bg is not log canonical at A. 1 a m/2 b = aˆl 1 + ˆΩ ˆL 1 = a + ˆΩ ˆL 1 > a + 3 a + m/2 + b by Lemma 2.5. We have a > 2. But a + m 2, which is a Lemma 3.7. Suppose that Σ = {D 5 }. lcts = 1/4. Proof. Suppose that the log pair S, λd is not log canonical. Let us derive a We see that LCSS, λd = {O} by Lemma 3.1. It follows from [1] that r = 2 and the surface S contains a line L such that O L. Projecting from L, we see that there is a conic C S such that the equivalence K S C + L holds, O C and C L = 1. Put P = C L. 3 P O LCS S, C + L + λd P O C L, 4 which is impossible by Lemma 2.2. The obtained contradiction completes the proof. Lemma 3.8. Suppose that Σ = {E 6 }. lcts = 1/6 Proof. Suppose that the log pair S, λd is not log canonical. Let us derive a It follows from [1] that r = 1. The log pair S, lct 1 SL 1 is not log terminal. But it must be log canonical. The surface S contains a plane cuspidal cubic curve C such that O C. Arguing as in the proof of Lemma 3.6, we obtain a

12 202 I. CHELTSOV Using the classification of possible singularities of the surface S obtained in [1], we see that it follows from Lemmas 3.4, 3.5, 3.6, 3.7 and 3.8 that we may assume that } Σ = {A i1,...,a is to complete the proof of Theorem 1.4. We assume that i 1 i s and O is of type A is. Lemma 3.9. Suppose that Σ = {A 2 }. lcts = 1/2. Proof. Suppose that the log pair S, λd is not log canonical. Let us derive a It follows from [1] that r = 6. We may assume that the equivalences K S L 1 + L 2 + L 3 L 4 + L 5 + L 6 hold. The log pairs S, lct 1 SL 1 + L 2 + L 3 and S, lct 1 SL 4 + L 5 + L 6 are log canonical. Arguing as in the proof of Lemma 3.4, we see that LCS S, λd = O. Let H be a proper transform on S of a general hyperplane section that contains O. 0 H D = 3 a 1 a 2, 2a 1 a 2 = E 1 D 0, 2a 2 a 1 = E 2 D 0, which implies that a 1 2 and a 2 2. There is a point Q E 1 E 2 such that the log pair S, λ D + a1 E 1 + a 2 E 2 is not log canonical at Q. We may assume that Q E 1, and L 1 E 1 = L 2 E 1 = L 3 E 1 = L 4 E 2 = L 5 E 2 = L 6 E 2 = 1, which implies that L 1 E 2 = L 2 E 2 = L 3 E 2 = L 4 E 1 = L 5 E 1 = L 6 E 1 = 0. It follows from Remark 2.1 that we may assume that L 1 SuppD L 4. { 1 a1 = D L 1 0, 1 a 2 = D L 4 0, which implies that a 1 1 and a 2 1. Suppose that Q E 2. S, λ D + E 1 is not log canonical at Q. We have 2a 1 a 2 = D E 1 > 1/λ > 2, by Lemma 2.5. a 1 4/3, which is impossible, because a 1 1. Hence, we see that Q E 2. The log pairs S, λ D +E 1 +a 2 E 2 and S, λ D +a 1 E 1 +E 2 are not log canonical at Q. { 2a1 a 2 = D E 1 > 1/λ a 2 > 2 a 2, 2a 2 a 1 = D E 2 > 1/λ a 1 > 2 a 1,

13 ON SINGULAR CUBIC SURFACES 203 by Lemma 2.5. a 1 > 1 and a 2 > 1. But a 1 1 and a 2 1, which is a Lemma Suppose that Σ = {A 3 }. lcts = 1/2 Proof. Suppose that the log pair X, λd is not log canonical. Let us derive a It follows from [1] that r = 5. We may assume that L 1 E 1 = L 2 E 1 = L 3 E 2 = L 4 E 3 = L 5 E 3 = 1, which implies that L 3 E 1 = L 3 E 2 = 0 and L 1 E 2 = L 2 E 2 = L 1 E 3 = L 2 E 3 = L 4 E 2 = L 5 E 2 = L 4 E 1 = L 5 E 1 = 0. The inequalities L 2 i = 1 and L i L j = 0 hold for i j. We have K S L 1 + L 2 + L 3. Suppose that there are a line L S and a rational number µ 1/λ such that D = µl+ω, where Ω is an effective Q-divisor, whose support does not contain the line L. 2 = C D = µc L + C Ω µc L > 2C L, where C is a general conic on the surface S such that the divisor C+L is a hyperplane section of the surface S. L C = 1 and C L < 1, which implies that L = L 3. But L 3 C = 1. Arguing as in the proof of Lemma 3.2, we see that LCSS, λd = O by Lemmas 2.2. Let H be a general curve in K S 3 i=1 E i. a 1 + a 3 3, 2a 1 a 2, 2a 2 a 1 + a 3, 2a 3 a 2, because H D 0, E 1 D 0, E 2 D 0, E 3 D 0, respectively. We may assume that either L 1 SuppD or L 3 SuppD by Remark 2.1. But L 1 D = 1 a 1, L3 D = 1 a 2, which implies that either a 1 1 or a 2 1. Similarly, we assume that either a 3 1 or a 2 1. We have a 1 2, a 2 2, a 3 2. There is a point Q E 1 E 2 E 3 such that the log pair S, λ D + a1 E 1 + a 2 E 2 + a 3 E 3 is not log canonical at Q. We may assume that Q E 3. Suppose that Q E 2. S, λ D+E 1 is not log canonical at Q, which implies that 2a 1 a 2 = D E 1 > 2 by Lemma 2.5. a 1 > 3/2 and a 2 > 1. But either a 1 1 or a 2 1. Suppose that Q E 2 E 1. Arguing as in the proof of of Lemma 3.9, we see that { 2a1 a 2 = D E 1 > 1/λ a 2 > 2 a 2, 2a 2 a 1 a 3 = D E 2 > 1/λ a 1 > 2 a 1,

14 204 I. CHELTSOV by Lemma 2.5. a 1 > 1 and 2a 2 > 2 + a 3, which is impossible. We see that Q E 2 and Q E 1. S, λ D + E 2 is not log canonical at Q. We have 2a 2 a 1 a 3 = D E 2 > 1/λ > 2, which implies that a 1 > 3/2 and a 2 > 2. The obtained contradiction completes the proof. Lemma Suppose that Σ = {A 4 }. lcts = 1/3 Proof. Suppose that the log pair X, λd is not log canonical. Let us derive a It follows from [1] that r = 4. We may assume that L 1 E 1 = L 2 E 1 = L 3 E 3 = L 4 E 4 = 1, which implies that L 3 E 1 = L 3 E 2 = L 3 E 4 = 0 and L 1 E 2 = L 2 E 2 = L 1 E 3 = L 2 E 3 = L 1 E 4 = L 2 E 4 = L 4 E 1 = L 4 E 2 = L 4 E 3 = 0. We have LCSS, λd = O by Lemma 3.1. Let H be a general curve in K S 4 i=1 E i. 3 a 1 + a 4, 2a 1 a 2, 2a 2 a 1 + a 3, 2a 3 a 2 + a 4, 2a 4 a 3, because H D 0, E 1 D 0, E 2 D 0, E 3 D 0, E 4 D 0, respectively. One can easily check that the equivalences K S L 1 + L 2 + L 3 2L 3 + L 4 hold. Therefore, we may assume that either L 1 SuppD L 4 or L 3 SuppD by Remark 2.1 and Lemma 3.1. But L 3 D = 1 a 3, L1 D = 1 a 1, L4 D = 1 a 4, which implies that there is a point Q 4 i=1 E i such that the log pair 4 S, λ D + a i E i is not log canonical at the point Q. Arguing as in the proof of Lemma 3.10, we see that Q E 1 \ E 1 E 2 2a 1 > a 2 + 3, Q E 1 E 2 2a 1 > 3 and 2a 2 > 3 + a 3, Q E 2 \ E 1 E 2 E 2 E 3 2a 2 > a 1 + a 3 + 3, Q E 2 E 3 2a 2 > 3 + a 1 and 2a 3 > 3 + a 4, Q E 3 \ E 2 E 3 E 3 E 4 2a 3 > 3 + a 2 + a 4, Q E 3 E 4 2a 3 > 3 + a 2 and 2a 4 > 3, Q E 4 \ E 4 E 3 2a 4 > 3, i=1

15 ON SINGULAR CUBIC SURFACES 205 which leads to a contradiction, because either a 3 1 or a 1 1 and a 4 1. Lemma Suppose that Σ = A 5. lcts = 1/4. Proof. Suppose that the log pair X, λd is not log canonical. Let us derive a It follows from [1] that r = 3. We may assume that L 1 E 1 = L 2 E 1 = L 3 E 4 = 1 and L 1 E 2 = L 2 E 2 = L 1 E 3 = L 2 E 3 = L 1 E 4 = L 2 E 4 = L 1 E 5 = L 2 E 3 = 0 and L 3 E 1 = L 3 E 2 = L 3 E 3 = L 3 E 5 = 0. LCSS, λd = O by Lemma 3.1. Let H be a proper transform on S of a general hyperplane section that contains O. 3 a 1 +a 5, 2a 1 a 2, 2a 2 a 1 +a 3, 2a 3 a 2 +a 4, 2a 4 a 3 +a 5, 2a 5 a 4, 3.13 because H D 0, E 1 D 0, E 2 D 0, E 3 D 0, E 4 D 0, E 5 D 0, respectively. We have K S 3L 3. Thus, we may assume that L 3 SuppD by Remark 2.1. a 1 5/2, a 2 2, a 3 3/2, a 4 1, a 5 5/4, because 1 a 4 = L 3 D 0. Arguing as in the proof of Lemma 3.10, we see that there is a point Q 5 i=1 E i such that Q E 1 \ E 1 E 2 2a 1 > a 2 + 4, Q E 1 E 2 2a 1 > 4 and 2a 2 > 4 + a 3, Q E 2 \ E 1 E 2 E 2 E 3 2a 2 > a 1 + a 3 + 4, Q E 2 E 3 2a 2 > 4 + a 1 and 2a 3 > 4 + a 4, Q E 3 \ E 2 E 3 E 3 E 4 2a 3 > 4 + a 2 + a 4, 3.14 Q E 3 E 4 2a 3 > 4 + a 2 and 2a 4 > 4 + a 5, Q E 4 \ E 3 E 4 E 4 E 5 2a 4 > 4 + a 3 + a 5, Q E 4 E 5 2a 4 > 4 + a 3 and 2a 5 > 4, Q E 5 \ E 4 E 5 2a 5 > a The inequalities 3.13 and 3.14 imply that either Q = E 3 E 4 or Q = E 4 E 5, because a 4 1. Let H 1 and H 3 be general divisors in K S that contain L 1 and L 3, respectively. H 1 = L 1 + C 1, H 3 = L 3 + C 3, where C 1 and C 3 are irreducible conics such that C 1 SuppD C 3. Let C 1 and C 3 be the proper transforms of C 1 and C 3 on the surface S, respectively. { 2 a5 = C 1 D 0, 2 a 2 = C 3 D 0,

16 206 I. CHELTSOV which is impossible due to the inequalities 3.13 and Lemma Suppose that Σ = {A 1, A 5 }. lcts = 1/4. Proof. Suppose that the log pair X, λd is not log canonical. Let us derive a It follows from [1] that r = 2. We have LCSS, λd Σ by Lemma 3.1. Let P be a point in Σ of type A 1. We may assume that P L 1. L 2 E 1 = L 2 E 2 = L 2 E 3 = L 2 E 5 = L 1 E 2 = L 1 E 3 = L 1 E 4 = L 1 E 5 = 0, and L 1 E 1 = L 2 E 4 = 1. The equivalence K S 3L 2 holds. Suppose that S, λd is not log canonical at P. Let β : S S be a blow up of P. D β K S mf, where F is the β-exceptional curve, D is the proper transform of the divisor D, and m Q. 0 H D = β K S mf β K S F = 3 2m, where H is general curve in K S F. Thus, we have m 3/2. But m > 2 by Lemma 2.7. We see that LCSS, λd = O. Let C 1 and C 2 be general conics on the surface S such that L 1 + C 1 L 2 + C 2 K S, and let C 1 and C 2 be the proper transforms of C 1 and C 2 on the surface S, respectively. { 2 a1 = C 1 D 0, 2 a 5 = C 2 D 0, because C 1 SuppD C 2. We may assume that L 2 SuppD due to Remark 2.1. Arguing as in the proof of Lemma 3.12, we obtain the inequalities 3 a 1 + a 5, 2a 1 a 2, 2a 2 a 1 + a 3, 2a 3 a 2 + a 4, 2a 4 a 3 + a 5, 2a 5 a 4, 2 a 2, 2 a 5, 1 a 4, which imply that there is a point Q 5 i=1 E i such that the log pair 5 S, λ D + a i E i is not log canonical at Q. Arguing as in the proof of Lemma 3.10, we obtain a Lemma Suppose that Σ = {A 1, A 4 }. lcts = 1/3. Proof. Suppose that the log pair X, λd is not log canonical. Let us derive a i=1

17 ON SINGULAR CUBIC SURFACES 207 Let P be a point in Σ of type A 1. We may assume that P L 1. It follows from [1] that r = 3. L 1 E 1 = 1, L1 E 2 = L 1 E 3 = L 1 E 4 = 0, and we may assume that L 3 E 3 = L 2 E 4 = 1. K S L 2 + 2L 3 and L 3 E 1 = L 3 E 2 = L 3 E 4 = L 2 E 1 = L 2 E 2 = L 2 E 3 = 0. We may assume that either L 3 SuppD or L 1 SuppD L 2 see Remark 2.1. Arguing as in the proof of Lemma 3.15, we see that LCS S, λd = O, and arguing as in the proof of Lemma 3.11, we obtain a Lemma Suppose that Σ = {A 1, A 3 }. lcts = 1/2. Proof. Suppose that the log pair X, λd is not log canonical. Let us derive a Let P be a point in Σ of type A 1. We may assume that P L 1. It follows from [1] that r = 4 and S contains lines L 5, L 6, L 7 such that L 5 P L 6, O L 7 P, L 3 L 5, L 4 L 6, L 7 L 2, L 7 L 5, L 7 L 6, which implies that L 7 L 1 = L 7 L 3 = L 7 L 4 =. L 1 + L 3 + L 5 L 1 + L 4 + L 6 L 5 + L 6 + L 7 L 2 + 2L 1 L 2 + L 3 + L 4 2L 2 + L 7 and K S L 1 + L 3 + L 5. Put D = µ i L i + Ω i, where µ i is a non-negative rational number, and Ω i is an effective Q-divisor, whose support does not contain the line L i. Let us show that that µ i < 1/λ for i = 1,..., 7. Suppose that µ 2 1/λ. We may assume that L 1 SuppD by Remark = L 1 D = L 1 µ 2 L 2 + Ω 2 µ2 L 1 L 2 = µ 2 /2 > 1, which is a Similarly, we see that µ i < 1/λ for i = 1,...,7. Arguing as in the proof of Lemma 3.4, we see that LCS S, λd Σ, which implies that LCSS, λd = O or LCSS, λd = P by Lemma 2.2. Suppose that LCSS, λd = P. Put D = µ 5 L 5 + µ 6 L 6 + Υ, where Υ is an effective Q-divisor such that L 5 SuppΥ L 6. µ 5 > 0 and µ 6 > 0. But 1 = L 7 D = L 7 µ 5 L 5 + µ 6 L 6 + Υ L 7 µ 5 L 5 + µ 6 L 6 = µ5 + µ 6,

18 208 I. CHELTSOV because we may assume that L 7 SuppΥ. Let β : S S be a blow up of the point P. µ 5 L5 + µ 6 L6 + Υ β µ 5 L 5 + µ 6 L 6 + Υ µ 5 /2 + µ 6 /2 + ǫ G, where ǫ is a rational number, G is the exceptional curve of β, and L 5, L 6, Υ are proper transforms of the divisors L 5, L 6, Υ on the surface S, respectively. Υ 0 µ 5 L5 + µ 6 L6 + H = 3 µ5 µ 6 2ǫ, where H is a general curve in K S G. There is a point Q G such that the log pair S, λ µ5 L5 + µ 6 L6 + Υ + λ µ 5 /2 + µ 6 /2 + ǫ G are not log canonical at Q. We have 2 2ǫ = Υ L5 + L 6 0, which implies that ǫ 1. it follows from Lemma 2.5 that 2ǫ = Ω G > 2 if Q L 5 L 6, which implies that we may assume that Q L µ 5 /2 µ 6 ǫ = Ω L 5 > 2 µ 5 /2 µ 6 /2 ǫ, by Lemma 2.5. Thus, we see that µ 5 > 1. But µ 5 µ 5 + µ 6 1, which is a The obtained contradiction shows that LCSS, λd P. We see that LCSS, λd = O. We may assume that L 1 E 2 = L 1 E 3 = L 2 E 1 = L 2 E 3 = L 3 E 1 = L 3 E 2 = L 4 E 1 = L 4 E 2 = 0 and L 1 E 1 = L 2 E 2 = L 3 E 3 = L 4 E 3 = 1. But the log pair S, lct 1 S 2L 1 + L 2 has log canonical singularities. Similarly, the log pair S, lct 1 S L 2 + L 3 + L 3 is log canonical. By Remark 2.1 and Lemma 3.1, we may assume that either L 1 Supp D L 3 or L 2 SuppD. Arguing as in the proof of Lemma 3.10, we obtain a Lemma Suppose that Σ = {A 1, A 2 }. lcts = 1/2. Proof. Suppose that the log pair X, λd is not log canonical. Let us derive a

19 ON SINGULAR CUBIC SURFACES 209 It follows from [1] that r = 5. We may assume that L 1 E 1 = L 2 E 1 = L 3 E 2 = L 4 E 2 = L 5 E 2 = 1 and L 1 E 2 = L 2 E 2 = L 3 E 1 = L 4 E 1 = L 5 E 1 = 0. Let P be a point in Σ of type A 1. We may assume that P L 1. It follows from [1] that S contains lines L 6, L 7, L 8, L 9, L 10, L 11 such that P = L 1 L 6 L 7 L 8, L 9 L 6, L 9 L 7, L 9 L 6 and L 9 L 7, L 10 L 7, L 10 L 8, L 11 L 6, L 11 L 8. L 2 P L 3, L 4 P L 5, L 6 O L 7, L 8 O L 9, L 10 O L 11, which implies that K S L 3 + L 4 + L 5 2L 1 + L 2 L 3 + L 4 + L 5 and K S 2L 1 + L 2 L 1 + L 3 + L 6 L 1 + L 4 + L 7 L 1 + L 5 + L 8 L 6 + L 7 + L 9 and K S L 7 + L 8 + L 10 L 6 + L 8 + L 11. Arguing as in the proof of Lemma 3.17, we see that LCS S, λd = O. By Remark 2.1, we may assume that either L 1 SuppD or L 2 SuppD, because 2L 1 + L 2 K S and the log pair S, lct 1 S2L 1 + L 2 has log canonical singularities. Similarly, we may assume that SuppD does not contain at least one of the lines L 3, L 4, L 5, because the equivalence L 3 + L 4 + L 5 K S holds. Arguing as in the proof of Lemma 3.9, we obtain a Lemma Suppose that Σ = {A 2,..., A 2 } and Σ 2. lcts = 1/3. Proof. Suppose that the log pair X, λd is not log canonical. Let us derive a Let P be a point in Σ such that P O. We may assume that P L 1. K S 3L 1. We may assume that S, λd is not log canonical at O by Lemma 3.1, and we assume that L 1 SuppD by Remark 2.1 and Lemma 3.1. We may assume that L 1 E 2. a 2 1, because D L 1 0. Arguing as in the proof of Lemma 3.9, we see that 3 a 1 + a 2, 2a 1 a 2, 2a 2 a 1, 1 a 2.

20 210 I. CHELTSOV There is a point Q E 1 E 2 such that the log pair S, λ D + a1 E 1 + a 2 E 2 is not log canonical at the point Q. Arguing as in the proof of Lemma 3.9, we see that Q E 1 \ E 1 E 2 2a 1 > a 2 + 3, Q E 1 E 2 2a 1 > 3 and 2a 2 > 3, Q E 2 \ E 2 E 1 2a 2 > a 1 + 3, which easily leads to a contradiction, because 3 a 1 +a 2, 2a 1 a 2, 2a 2 a 1, 1 a 2. Lemma Suppose that Σ = {A 1, A 2, A 2 }. lcts = 1/3. Proof. Suppose that the log pair X, λd is not log canonical. Let us derive a It follows from Lemma 3.1 that LCSS, λd Σ. Let P O be a point in Σ of type A 2. We may assume that P L 1. K S 3L 1, which implies that we may assume that L 1 SuppD due to Remark 2.1 and Lemma 3.1. Arguing as in the proof of Lemma 3.15, we see that LCS S, λd O P, which easily leads to a contradiction see the proof of Lemma Lemma Suppose that Σ = {A 1, A 1, A 3 }. lcts = 1/2. Proof. Suppose that the log pair X, λd is not log canonical. Let us derive a It follows from [1] that r = 3. Let P 1 and P 2 be points in Σ of type A 1. we may assume that P 1 L 1 and P 2 L 2. It follows from [1] that S contains lines L 4 and L 5 such that P 1 L 4 P 2, O L 4, P 1 L 3 P 2, L 5 Σ =, which implies that L 5 L 3, L 5 L 4, L 5 L 1 =, L 5 L 2 =. K S L 1 + L 2 + L 4 L 3 + 2L 1 L 3 + 2L 2 2L 3 + L 5 2L 4 + L Let us show that LCSS, λd does not contains the lines L 1,..., L 5. Put D = µ i L i + Ω i, where µ i Q, and Ω i is an effective Q-divisor such that L i SuppΩ i. Suppose that µ 1 1/λ. it follows from the equivalence 3.22 and Remark 2.1 that we may assume that L 3 SuppD. Therefore, we have 1 = L 3 D = L 3 µ 1 L 1 + Ω 1 µ1 L 3 L 1 = µ 1 /2 > 1,

21 ON SINGULAR CUBIC SURFACES 211 which is a Similarly, we see that µ 2 < 1/λ, µ 3 < 1/λ, µ 4 < 1/λ, µ 5 < 1/λ. Arguing as in the proof of Lemma 3.4, we see that LCSS, λd = 1 and LCS S, λd Σ. Suppose that LCSS, λd = P 1. Let β : S S be a blow up of the point P 1. µ 4 L4 + Ω β µ 4 L 4 + Ω µ 4 /2 + ǫ G, where G is the exceptional curve of the birational morphism β, L 4 and Ω are proper transforms of the divisors L 4 and Ω on the surface S, respectively, and ǫ is a positive rational number. 0 µ 4 L4 + Ω H = β µ 4 L 4 +Ω µ 4 /2+ǫ G β K S G = 3 µ 4 2ǫ, where H is a general curve in K S G. Thus, there is a point P G such that the log pair S, µ4 L4 + Ω + µ 4 /2 + ǫ G is not log canonical at P. 1 ǫ = Ω L 4 0. It follows from Lemma 2.5 that 2ǫ = Ω G > 2 in the case when P L 4. Therefore, we see that P L 4. 1 ǫ = Ω L 4 > 2 µ 4 /2 ǫ by Lemma 2.5. Thus, we see that µ 4 > 2, which is a Similarly, we see that P 2 LCSS, λd. LCSS, λd = O. We may assume that L 1 E 1 = L 2 E 3 = L 3 E 2 = 1, L 1 E 2 = L 1 E 3 = L 2 E 1 = L 2 E 2 = L 3 E 1 = L 3 E 3 = 0. It follows from the equivalences 3.22 that we may assume that either L 3 SuppD or L 1 SuppD L 2 by Remark 2.1. Arguing as in the proof of Lemma 3.10, we obtain a Lemma Suppose that Σ = {A 1, A 1, A 2 }. lcts = 1/2. Proof. Suppose that the log pair X, λd is not log canonical. Let us derive a It follows from [1] that r = 4. Let P 1 P 2 be points in Σ of type A 1. we may assume that P 1 L 1 and P 2 L 4. It follows from [1] that S contains lines L 5, L 6, L 7, L 8 such that P 1 L 5, P 2 L 6, P 1 L 7 P 2, O L 8, P 1 L 8 P 2,

22 212 I. CHELTSOV which implies that L 8 L 7, L 8 L 2, L 8 L 3, L 2 L 7 =, L 3 L 7 =. L 1 + L 4 + L 7 L 2 + 2L 1 L 3 + 2L 4 2L 7 + L 8 L 2 + L 3 + L 8 L 1 + L 3 + L 5 L 4 + L 2 + L 6, and K S L 1 + L 4 + L 7. Without loss of generality, we may assume that L 1 E 1 = L 2 E 1 = L 3 E 2 = L 4 E 2 = 1, L1 E 2 = L 2 E 2 = L 3 E 1 = L 4 E 1 = 0. Arguing as in the proof of Lemma 3.21, we see that LCSS, λd = O. By Remark 2.1, we may assume that either L 1 SuppD or L 2 SuppD, because 2L 1 + L 2 K S O P 3 1 S and the log pair X, lct 1 S2L 1 +L 2 is log canonical, where lct 1 S = 1/2. Similarly, we may assume that either L 3 SuppD or L 4 SuppD, because K S L 3 +2L 4. Arguing as in the proof of Lemma 3.9, we obtain a It follows from [1], that the equalities 2/3 when Σ = { } A 1, 1/3 when Σ { } A 4, 1/3 when Σ = { } D 4, lct S 1/3 when Σ { } A 2, A 2, = lct 1 S = 1/4 when Σ { } A 5, 1/4 when Σ = { } D 5, 1/6 when Σ = { } E 6, 1/2 in other cases. are proved for all possible values of the set Σ. Hence, the assertion of Theorem 1.4 is proved. 4. Fiberwise maps. Let us use the assumptions and notation of Theorem 1.5. Proof of Theorem 1.5. Suppose that X is log terminal and lctx 1, but ρ is not an isomorphism. Let D be a general very ample divisor on Z. Put Λ = nkv + π nd, Γ = nk V + π nd, Λ = ρλ, Γ = ρ 1 Γ, where n is a natural number such that Λ and Γ have no base points. Put M V = 2ε n Λ + 1 ε n Γ, M V = 2ε n Λ + 1 ε n Γ, where ε is a positive rational number. The log pairs V, M V and V, M V are birationally equivalent, and K V + M V and K V +M V are ample. The uniqueness of canonical model see [3, Theorem ] implies that ρ is biregular if the singularities of both log pairs V, M V and V, M V are canonical.

23 ON SINGULAR CUBIC SURFACES 213 The linear system Γ does not have base points. Thus, there is a rational number ε such that the log pair V, M V is canonical. So, the log pair V, M V is not canonical. the log pair V, X + 1 ε n Γ is not log canonical, because Λ does not have not base points, and Γ does not have base points outside of the fiber X, which is a Cartier divisor on the variety V. The log pair X, 1 ε n Γ X is not log canonical by Theorem 17.6 in [9], which is impossible, because lctx 1. To conclude the proof we may assume that the varieties X and X have log terminal singularities, the inequality lctx + lct X > 1 holds, and ρ is not an isomorphism. Let Λ, Γ, Λ, Γ and n be the same as in the previous case. Put M V = lct X ε n Λ + lctx ε n Γ, M V = lct X ε n Λ + lctx ε n where ε is a sufficiently small positive rational number. it follows from the uniqueness of canonical model that ρ is biregular if both log pair V, M V and V, M V are canonical. Without loss of generality, we may assume that the singularities of the log pair V, M V are not canonical. Arguing as in the previous case, we see that the log pair X, lctx ε n Γ X is not log canonical, which is impossible, because Γ X nk X. The assertion of Theorem 1.5 is a generalization of the Main Theorem in [10]. Γ, REFERENCES [1] J.W. Bruce and C.T.C. Wall, On the classification of cubic surfaces, Journal of the London Mathematical Society, , pp [2] A. del Centina and S. Recillas, Some projective geometry associated with unramified double covers of curves of genus 4, Annali di Matematica Pura ed Applicata, , pp [3] I. Cheltsov, Birationally rigid Fano varieties, Russian Mathematical Surveys, , pp [4] I. Cheltsov, Log canonical thresholds of del Pezzo surfaces, Geometric and Functional Analysis, , pp [5] I. Cheltsov and C. Shramov, Log canonical thresholds of smooth Fano threefolds. With an appendix by Jean-Pierre D ly, Russian Mathematical Surveys, , pp [6] J.-P. D ly and J. Kollár, Semi-continuity of complex singularity exponents and Kähler- Einstein metrics on Fano orbifolds, Annales Scientifiques de l École Normale Supérieure, , pp [7] W. Ding and G. Tian, Kähler Einstein metrics and the generalized Futaki invariant, Inventiones Mathematicae, , pp [8] J. Kollár, Singularities of pairs, Proceedings of Symposia in Pure Mathematics, , pp [9] J. Kollár et al., Flips and abundance for algebraic threefolds, Astérisque, [10] J. Park, Birational maps of del Pezzo fibrations, Journal fur die Reine und Angewandte Mathematik, , pp

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