LOG CANONICAL THRESHOLDS OF SMOOTH FANO THREEFOLDS

LOG CANONICAL THRESHOLDS OF SMOOTH FANO THREEFOLDS IVAN CHELTSOV AND CONSTANTIN SHRAMOV WITH AN APPENDIX BY JEAN-PIERRE DEMAILLY Abstract. We compute global log canonical thresholds of some smooth Fano threefolds.

Contents 1. Introduction 1 2. Preliminaries 19 3. The Mukai–Umemura threefold 31 4. Cubic surfaces 34 5. Del Pezzo surfaces 41 6. Toric varieties 45 7. Del Pezzo threefolds 47 8. Fano threefolds with ρ = 2 50 9. Fano threefolds with ρ = 3 63 10. Fano threefolds with ρ > 4 86 11. Upper bounds 95 Appendix A. By Jean-Pierre Demailly. On Tian’s invariant and log canonical thresholds 100 Appendix B. The Big Table 105 References 109

1. Introduction The multiplicity of a polynomial φ ∈ C[z1 , . . . , zn ] in the origin O ∈ Cn is the number ) ( ∂ m φ z1 , . . . , z n O 6= 0 ∈ Z>0 ∪ + ∞ . min m ∈ Z>0 m1 m m n 2 ∂ z1 ∂ z2 . . . ∂ z n There is a similar but more subtle invariant ) ( 1 n ∈ Q>0 ∪ {+∞}, c0 φ = sup ε ∈ Q the function 2ε is locally integrable near O ∈ C φ which is called the complex singularity exponent of the polynomial φ at the point O. Example 1.1. Suppose that n = 2, and φ = 0 defines an irreducible curve in C2 . Then 1 1 c0 (φ) = min 1, + m n by [90], where (m, n) is the first pair of Puiseux exponents of φ. On the other hand, the equality ! 1 1 + 1 1 m m 1 2 c0 z1n1 z2n2 z1km1 + z2km2 = min , , n1 n2 n1 n2 k + m + m2 1 holds (see [109]), where n1 , n2 , m1 , m2 , k are non-negative integers. The first author was supported by the grants NSF DMS-0701465 and EPSRC EP/E048412/1, the second author was supported by the grants RFFI No. 08-01-00395-a, N.Sh.-1987.2008.1 and EPSRC EP/E048412/1. 1

Example 1.2. Let m1 , . . . , mn be positive integers. Then ! ! ! n n n X Y X 1 1 1 1 mi mi = c0 zi > c0 zi = min , ,..., . min 1, mi m1 m2 mn i=1

i=1

i=1

The set of complex singularity exponents has interesting properties. Put n o Hn = c0 (φ) φ ∈ C[z1 , . . . , zn ] ⊂ Q ∪ + ∞ which implies that Tn ⊂ 0, 1 ∪ {+∞}. Then • • • •

the set Hn is closed in R ∪ {+∞} (see [60]), we expect that Hn satisfies ascending chain condition (ACC) for every n (see [106]), the set Hn satisfies ACC for n 6 3 (see [169], [1], [109], [141], [58]), it follows from [60] that the following assertions are equivalent: – the set Hn satisfies ACC for every n ∈ Z>0 ; – for every n ∈ Z>0 , there is δn ∈ (0, 1) such that Hn ∩ (δn , 1) = ∅; • it follows from [106] that Hn−1 ⊂ Hn and Hn−1 \ 1, +∞ ⊆ ∂Hn ⊆ Hn−1 \ + ∞ ,

where ∂Hn is the set of all accumulation points of Hn , • it follows from [109] and [110] that the set H2 is the union   − km1 < n1 − n2 < km2 ,  [  [ 2 m1 + m2 gcd m1 , m2 = 1, 0, +∞ , 2 6 m ∈ Z>0   m  km1 m2 + n1 m2 + n2 m1  k, m1 , m2 , n1 , n2 ∈ Z>0 which implies that ∂H2 = H1 \ {1, +∞}, where H1 = {1/n | n ∈ Z>0 } ∪ {0}, • it follows from [110] that the intersection H3 ∩ [ 56 , 1) is the union [ [ [ 5 1 5 2 5 4 19 15 12 25 15 5 + m > 6 + m > 4 + m > 2 , , , , , , 6 m 6 3m 6 9m + 6 20 16 13 28 17 6 where m ∈ Z>0 , which implies that 5/6 is the largest accumulation point of H3 (cf. [145], [146]), • it follows from [124] that ∂H3 = H2 \ {1, +∞} (cf. [106]), • it follows from [104] that 41/42 is the maximal element of the set H3 ∩ [0, 1). Remark 1.3. For a non-constant φ, the complex singularity exponent c0 (φ) coincides with the absolute value of the biggest root of the Bernstein–Sato polynomial of φ (see [10], [105]). Let X be a variety1 with at most log canonical singularities (see [102]), let Z ⊆ X be a closed subvariety, and let D be an effective Q-Cartier Q-divisor on the variety X. Then the number n o lctZ X, D = sup λ ∈ Q the log pair X, λD is log canonical along Z ∈ Q ∪ + ∞ is called a log canonical threshold of the divisor D along Z. It follows from [105] that lctO Cn , φ = 0 = c0 φ , so that lctZ (X, D) is an algebraic counterpart of the number c0 (φ). One has n o n o lctX X, D = inf lctP X, D P ∈ X = sup λ ∈ Q the log pair X, λD is log canonical , and, for simplicity, we put lct(X, D) = lctX (X, D). 1All varieties are assumed to be complex, algebraic, projective and normal if the opposite is not stated explicitly. 2

Example 1.4. Suppose that X = P2 and D ∈ |OP2 (3)|. Then  1 if D is a smooth curve,     1 if D is a curve with ordinary double points,        5/6 if D is a curve with one cuspidal point,  lct X, D = 3/4 if D consists of a conic and a line that are tangent,    2/3 if D consists of three lines intersecting at one point,      1/2 if Supp D consists of two lines,     1/3 if Supp D consists of one line. Example 1.5. Suppose that X = P2 and D ∈ |OP2 (d)| for d > 3. The papers [103] and [112] show that the curve D is semistable (stable, respectively) if lct(X, D) > 3/d (> 3/d, respectively). The set of log canonical thresholds of Weil divisors has interesting properties, which are similar to the properties of the set Hn (cf. [8]). Put the variety X has at most log canonical singularities, Tn = lct X, D ⊂Q∪ +∞ , dim(X) = n and D is effective Q-Cartier Weil divisor which implies that Tn ⊂ 0, 1 ∪ + ∞ . Then • the set Tn satisfies ACC for n 6 3 (see [169], [1], [109], [141], [58]), • we expect that Tn satisfies ACC for every n (see [105, Conjecture 8.8]), • it follows from [105, Proposition 8.8] that that Tn−1 ⊂ Tn and Tn−1 \ 1, +∞ ⊆ ∂Tn , where ∂Tn is the set of all accumulation points of Tn , • it follows from [169] and [109] that ∂T2 = T1 \{1, +∞}, where T1 = {1/n | n ∈ Z>0 }∪{0}, • it follows from [145] and [146] that 1 1 1 1 + 3 6 n ∈ Z>0 , ∂T3 ∩ , 1 = T2 ∩ , 1 = 2 2 2 n which implies that 5/6 is the largest accumulation point of T3 (cf. [110]), • it follows from [124] that ∂T3 = T2 \ {1, +∞} (cf. [106]), • it follows from [104] that 41/42 is is the maximal element of the set T3 ∩ [0, 1). Remark 1.6. If X is smooth, and D is a Weil divisor, then it follows from [131] that ( ) dim Dm lct X, D = dim X − sup m ∈ Z>0 , m+1 where Dm is the m-th jet scheme of the divisor D (see [131]). Suppose that X is a Fano variety with at most log terminal singularities (see [98]). Definition 1.7. Global log canonical threshold of the Fano variety X is the number n o lct X = inf lct X, D D is an effective Q-divisor on X such that D ∼Q −KX > 0. Remark 1.8. To define the number lct(X) ∈ R, we only need to assume that − nKX 6= ∅ for some n 0. This property is shared by many varieties (toric varieties, weak Fano varieties), but all the currently known applications are related to the case when −KX is ample. The number lct(X) is an algebraic counterpart of the α-invariant introduced in [179]. One has   ε   the log pair X, D is log canonical for n lct X = sup ε ∈ Q .  every divisor D ∈ − nKX and all n ∈ Z>0  3

Recall that every Fano variety X is rationally connected (see [170], [193]). Thus, the group Pic(X) is torsion free. Then ) ( the log pair X, λD is log canonical . lct X = sup λ ∈ Q for every effective Q-divisor D ∼Q −KX Example 1.9. Let X be a smooth hypersurface in Pn of degree m < n. Then 1 lct X = n+1−m as shown in [20] (see Corollary 2.16). In particular, the equality lct(Pn ) = 1/(n + 1) holds. Example 1.10. Let X be a rational homogeneous space such that −KX ∼ rD and Pic X = Z D , where D is an ample divisor and r ∈ Z>0 . Then lct(X) = 1/r (see [78], [79]). Example 1.11. Let X be a general intersection of hypersurfaces F1 , . . . , Fk ⊂ Pn such that k X

deg Fi = n > 5k + 1 > 11,

i=1

where deg(Fk ) > . . . > deg(F1 ) > 2 and deg(Fk ) > 8. Then lct(X) = 1 (see [158]). In general, the number lct(X) depends on small deformations of the variety X. Example 1.12. Let X be a smooth hypersurface in P(1, 1, 1, 1, 3) of degree 6. Then 5 43 13 33 7 33 8 9 11 13 15 17 19 21 29 lct X ∈ , , , , , , , , , , , , , , ,1 6 50 15 38 8 38 9 10 12 14 16 18 20 22 30 by [157] and [37], and all these values are attained. Example 1.13. Let X be a smooth hypersurface in P(1n+1 , n) of degree 2n. The inequalities 2n − 1 1 > lct X > 2n hold (see [37]). But the equality lct(X) = 1 holds if X is general and n > 3. Example 1.14. Let X be a smooth hypersurface in Pn of degree n > 2. Then the inequalities n−1 1 > lct X > n hold (see [20]). Then it follows from [157] and [37] that  1 if n > 6,     22/25 if n = 5, lct X >  16/21 if n = 4,    3/4 if n = 3, whenver X is general. But lct(X) = 1 − 1/n if X contains a cone of dimension n − 2. P Example 1.15. Let X be a quasismooth hypersurface in P(1, a1 , a2 , a3 , a4 ) of degree 4i=1 ai such that X has at most terminal singularities (see [102]), where a1 6 a2 6 a3 6 a4 . Then −KX ∼ OP(1, a1 , a2 , a3 , a4 ) 1 , X

and there are 95 possibilities for the quadruple (a1 , a2 , a3 , a4 ) (see [89], [82]). Then  16/21 if a1 = a2 = a3 = a4 = 1,       7/9 if (a1 , a2 , a3 , a4 ) = (1, 1, 1, 2),  1 > lct X > 4/5 if (a1 , a2 , a3 , a4 ) = (1, 1, 2, 2),    6/7 if (a1 , a2 , a3 , a4 ) = (1, 1, 2, 3),     1 in the remaining cases, 4

if X is general (see [27], [37], [29], [30]). The global log canonical threshold of the hypersurface w2 = t3 + z 9 + y 18 + x18 ⊂ P 1, 1, 2, 6, 9 ∼ = Proj C x, y, z, t, w is equal to 17/18 (see [27]), where wt(x) = wt(y) = 1, wt(z) = 2, wt(t) = 6, wt(w) = 9. Example 1.16. It follows from Lemma 6.1 that a0 lct P a0 , a1 , . . . , an = Pn

i=0 ai

,

where P(a0 , a1 , . . . , an ) is well-formed (see [89]), and a0 6 a1 6 . . . 6 an . Example 1.17. Let X be a smooth hypersurface in P(1n+1 , d) of degree 2d. Then 1 lct X = n+1−d in the case when the inequalities 2 6 d 6 n − 1 hold (see Proposition 20 in [28]). Example 1.18. Let X be smooth surface del Pezzo. It follows from [31] that  2 1 if KX = 1 and | − KX | contains no cuspidal curves,     2   5/6 if KX = 1 and | − KX | contains a cuspidal curve,     2   5/6 if KX = 2 and | − KX | contains no tacnodal curves,    2  3/4 if KX = 2 and | − KX | contains a tacnodal curve, lct X =  3/4 if X is a cubic in P3 with no Eckardt points,      2  2/3 if either X is a cubic in P3 with an Eckardt point, or KX = 4,     1 1 2   1/2 if X ∼ = P × P or KX ∈ {5, 6},    1/3 in the remaining cases. It would be interesting to compute global log canonical thresholds of del Pezzo surfaces with at most canonical singularities that are of Picard rank one, which has been classified in [62]. Example 1.19. Let X be a singular cubic surface in P3 such that X has at most canonical singularities. The singularities of the surface X are classified in [16]. It follows from [32] that  2/3 if Sing X = A1 ,       1/3 if Sing X ⊇ A4 ,      1/3 if Sing X = D4 ,      1/3 if Sing X ⊇ A2 , A2 , lct X =  1/4 if Sing X ⊇ A5 ,       1/4 if Sing X = D5 ,      1/6 if Sing X = E6 ,     1/2 in the remaining cases. It is unknown whether lct(X) ∈ Q or not2 (cf. Question 1 in [181]). Conjecture 1.20. There is an effective Q-divisor D ∼Q −KX on the variety X such that lct X = lct X, D ∈ Q. Let G ⊂ Aut(X) be an arbitrary subgroup. Definition 1.21. Global   lct X, G = sup ε ∈ Q 

G-invariant log canonical threshold of the Fano variety X is  ε the log pair X, D has log canonical singularities for every  n . G-invariant linear system D ⊂ − nKX and every n ∈ Z>0 

2It is even unknown whether lct(X) ∈ Q or not if X is a del Pezzo surfaces with log terminal singularities. 5

Remark 1.22. To define the threshold lct(X, G) ∈ R ∪ {+∞}, we only need to assume that − nKX 6= ∅ for some n 0. But all known applications require −KX to be ample, and G to be compact. In the case when the Fano variety X is smooth and G is compact, the equality lct X, G = αG X , holds (see Appendix A), where ( lct X, G = sup λ ∈ Q

αG (X) is the α-invariant introduced in [179]. It is clear that ) the log pair X, λD has log canonical singularities for every G-invariant effective Q-divisor D ∼Q −KX

in the case when |G| < +∞. Note that 0 6 lct(X) 6 lct(X, G) ∈ R ∪ {+∞}. Example 1.23. The simple group PGL(2, F7 ) is a group of automorphisms of the quartic x3 y + y 3 z + z 3 x = 0 ⊂ P2 ∼ = Proj C[x, y, z] , which induces an embedding PGL(2, F7 ) ⊂ Aut(P2 ). Then lct(P2 , PGL(2, F7 )) = 4/3 (see [31]). 2 = 5. Then Example 1.24. Let X be a smooth del Pezzo surface such that KX • the isomorphism Aut(X) ∼ = S5 holds (see [161]), • the equalities lct(X, S5 ) = lct(X, A5 ) = 2 hold (see [31]).

Example 1.25. Let X be the cubic surface in P3 given by the equation x3 + y 3 + z 3 + t 3 = 0 ⊂ P3 ∼ = Proj C x, y, z, t , and let G = Aut(X) ∼ = Z33 o S4 . Then lct(X, G) = 4 by [31]. The following result was proved in [179], [132], [49] (see Appendix A). Theorem 1.26. Suppose that X has at most quotient singularities, and the inequality dim X lct X, G > dim X + 1 holds. Then X admits an orbifold K¨ ahler–Einstein metric. Let us show how to apply Theorem 1.26 (cf. Examples 1.13, 1.14, 1.15). P Example 1.27. Let X be a quasismooth hypersurface in P(a0 , a1 , a2 , a3 ) of degree 3i=0 ai − 1, where a0 6 a1 6 a2 6 a3 . Then it follows from [49], [81], [13], [2] (cf. [14], [15]) that • either the surface X is smooth, which implies that n o a0 , a1 , a2 , a3 ∈ 1, 1, 1, 1 , 1, 1, 1, 2 , 1, 1, 2, 3 , and all possible values of lct(X) are contained in Example 1.18, • or (a0 , a1 , a2 , a3 ) = (2, 2n + 1, 2n + 1, 4n + 1) and lct(X) = 1, where n ∈ Z>2 , • or we have the following sporadic possibilities: – (a0 , a1 , a2 , a3 ) = (2, 3, 3, 5) and lct(X) > 33/38; – (a0 , a1 , a2 , a3 ) = (1, 2, 3, 5) and lct(X) > 2/3; – (a0 , a1 , a2 , a3 ) = (1, 3, 5, 7) and lct(X) > 2/3 if X ⊂ P(1, 3, 5, 7) is general; – (a0 , a1 , a2 , a3 ) = (1, 3, 5, 8) and lct(X) > 11/16 if X ⊂ P(1, 3, 5, 8) is general; – the inequality lct(X) > 1 holds and   2, 3, 5, 9 , 3, 3, 5, 5 , 3, 5, 7, 11 , 3, 5, 7, 14 , 3, 5, 11, 18          5, 14, 17, 21 , 5, 19, 27, 31 , 5, 19, 27, 50 , 7, 11, 27, 37  . a0 , a1 , a2 , a3 ∈   7, 11, 27, 44 , 9, 15, 17, 20 , 9, 15, 23, 23 , 11, 29, 39, 49         11, 49, 69, 128 , 13, 23, 35, 57 , 13, 35, 81, 128 6

P Example 1.28. Let X be a quasismooth hypersurface in P(a0 , . . . , a4 ) of degree 4i=0 ai − 1, where a0 6 a1 6 a2 6 a3 6 a4 . Then it follows from [82] that • the inequality lct(X) > 3/4 holds for at least 1936 quintuples (a0 , a1 , a2 , a3 , a4 ), • the inequality lct(X) > 1 holds for at least 1605 quintuples (a0 , a1 , a2 , a3 , a4 ). Example 1.29. Let X be one of the following smooth Fano varieties: • a Fermat hypersurface in Pn of degree n/2 6 d 6 n (cf. Example 1.25); • a general complete intersection of three quadrics in P6 that is given by 6 X

x2i

=

i=0

6 X

λi x2i

=

i=0

6 X

µi x2i = 0 ⊆ P6 ∼ = Proj C x0 , . . . , x6 ,

i=0

where λi and µi are complex numbers; • a smooth complete intersection of two quadrics in P5 that is given by 5 X

x2i =

i=0

5 X

ζ i x2i = 0 ⊆ P5 ∼ = Proj C x0 , . . . , x5 ,

i=0

where ζ is a primitive sixth root of unity; • a smooth complete intersection of a quadric and a cubic in P5 that is given by 5 X

x3i

=

i=0

5 X

ζ i x2i = 0 ⊆ P5 ∼ = Proj C x0 , . . . , x5 ,

i=0

where ζ is a nontrivial cube root of unity; • a hypersurface in P(1n+1 , q) of degree pq that is given by the equation p

w =

5 X

n+1 xpq ,q ∼ = Proj C x0 , . . . , xn , w , i ⊆P 1

i=0

such that pq − q 6 n, where wt(x0 ) = . . . = wt(xn ) = 1, wt(w) = q ∈ Z>0 and p ∈ Z>0 ; and let G = Aut(X). Then G is finite, and the inequality lct(X, G) > 1 holds (see [179], [132]). Example 1.30. Let X be a blow up of P3 along a disjoint union of two lines, let G be a maximal compact subgroup in Aut(X). Then the inequality lct(X, G) > 1 holds by [132] (cf. Lemma 9.26). If a variety with quotient singularities admits an orbifold Kähler–Einstein metric, then • either its canonical divisor is numerically trivial; • or its canonical divisor is ample (variety of general type); • or its canonical divisor is antiample (Fano variety). Remark 1.31. Every variety with at most quotient singularities that has numerically trivial or ample canonical divisor always admits an orbifold Kähler–Einstein metric (see [5], [190], [191]). There are several known obstructions for the Fano variety X to admit a Kähler–Einstein metric. For example, if the variety X is smooth, then it does not admit a Kähler–Einstein metric if • either the group Aut(X) is not reductive (see [122]), • or the tangent bundle of X is not polystable with respect to −KX (see [114]), • or the Futaki character of holomorphic vector fields on X does not vanish (see [68]), • or the pair (X, −KX ) is not K-semistable (see [182], [53], [54], [162], [163]). Example 1.32. The following varieties admit no Kähler–Einstein metrics: • a blow up of P2 in one or two points (see [122]), • a smooth Fano threefold P(OP2 ⊕ OP2 (1)) (see [176]), • a smooth Fano fourfold P α∗ OP1 (1) ⊕ β ∗ OP2 (1) , where α : P1 × P2 → P1 and β : P1 × P2 → P2 are natural projections (see [68]). 7

Example 1.33. Let X be a smooth Fano threefold such that Pic X = Z − KX 3 = 22. Then lct(X) 6 2/3 by Lemma 11.3 (see Section 3). But and −KX • the tangent bundle of the threefold X is stable (see [176]), • the group Aut(X) is trivial if the threefold X is general (see [167]), • there is X such that Aut(X) = {1} and X admits no Kähler–Einstein metrics (see [182]); • there is X, whose group of automorphisms Aut(X) is not non-reductive (see [142]); • if Aut(X) ∼ = PSL(2, C), then X has a Kähler–Einstein metric (see [55] and Remark 3.2).

Recently new obstruction for the existence of orbifold Kähler–Einstein metrics on Fano varieties with at most quotient singularities have been found (see [69], [175], [11]). P Example 1.34. Let X be a quasismooth hypersurface in P(a0 , . . . , an ) of degree d < ni=0 ai , where a0 6 . . . , 6 an and P(a0 , . . . , an ) is well-formed (see [89]). Then X is a Fano variety. If n X ai > d + na0 , i=0

then X admits no orbifold K¨ ahler–Einstein metric (see [69], [175]). The problem of existence of K¨ ahler–Einstein metrics on smooth toric Fano varieties is completely solved. Namely, the following result holds (see [115], [7], [187], [133]). Theorem 1.35. If X is smooth and toric, then the following conditions are equivalent: • the variety X admits a K¨ ahler–Einstein metric; • the Futaki character of holomorphic vector field of X vanishes; • the baricenter of the reflexive polytope of X is zero. It should be pointed out that the assertion of Theorem 1.26 gives only a sufficient condition for the existence of a K¨ ahler–Einstein metric on X (cf. [119], [182]). Example 1.36. Let X be a general cubic surface in P3 that has an Eckardt point (see Definition 4.1). Then Aut(X) ∼ = Z2 (see [52]) and 2 lct X, Aut X = lct X = 3 by [31]. But every smooth del Pezzo surface that has a reductive automorphism groups admits a Kähler–Einstein metric (see [183], [180]). Example 1.37. Let X be a general hypersurface in P(15 , 3) of degree 6. Then Aut(X) ∼ = Z2 (see [123]) and 1 lct X, Aut X = lct X = 2 by [28]. But X admit a K¨ ahler–Einstein metric (see [3]). The problem of existence of K¨ ahler–Einstein metrics on singular Fano varities that have quotient singularities is not well studied even for del Pezzo surfaces with canonical singularities. Example 1.38. Let X be a cubic surface in P3 . Then • the surface X admits a K¨ ahler–Einstein metric if Sing(X) = ∅ (see [180]), • the surface X does not admit an orbifold Kähler–Einstein metric in the case when it has at least one singular point that is not a singular point of type A1 or A2 (see [51]), • the surface that is given by the equation xyz + xyt + xzt + yzt = 0 ⊂ P3 ∼ = Proj C x, y, z, t admits an orbifold K¨ ahler–Einstein metric and has 4 singular points of type A1 (see [32]), • the surface that is given by the equation xyz = t3 ⊂ P3 ∼ = Proj C x, y, z, t admits an orbifold K¨ ahler–Einstein metric and has 3 singular points of type A2 (see [32]). 8

Example 1.39. Let X be a complete intersection of two quadrics in P4 . Then • the surface X admits a K¨ ahler–Einstein metric if Sing(X) = ∅ (see [180]), • the surface X does not admit an orbifold Kähler–Einstein metric in the case when it has at least one singular point that is not an ordinary double point (see [88]), • if the surface X ⊂ P4 can be given by the equations 4 4 X X x2i = λi x2i = 0 ⊂ P4 ∼ = Proj C x0 , . . . , x4 i=0

i=0

and X has at most ordinary double points, then lct(X, Z42 ) = 1 (see [116]). 2 6 2. Then Remark 1.40. Let X be a del Pezzo surface with canonical singularities such that KX • the surface X admits a K¨ ahler–Einstein metric in the following cases: 2 = 2 and X has at most singular points of type A or A (see [70]); – if KX 1 2 2 = 1 and X has at most ordinary double points (see [31]); – if KX • we expect X to admit no Kähler–Einstein metrics if X has relatively bad singularities (cf. [51], [113]).

The numbers lct(X) and lct(X, G) play an important role in birational geometry. Example 1.41. Let V and V¯ be varieties with at most terminal and Q-factorial singularities, and let Z be a smooth curve. Suppose that there is a commutative diagram ρ V _ _ _ _ _ _/ V¯ π

π ¯

Z Z such that π and π ¯ are flat morphisms, and ρ is a birational map that induces an isomorphism ¯ V \X ∼ = V¯ \ X, ¯ are scheme fibers of π and π where X and X ¯ over a point O ∈ Z, respectively. Suppose that ¯ • the fibers X and X are irreducible and reduced, • the divisors −KV and −KV¯ are π-ample and π ¯ -ample, respectively, ¯ have at most log terminal singularities, • the varieties X and X and ρ is not an isomorphism. Then it follows from [136] and [32] that ¯ 6 1, (1.42) lct X + lct X ¯ are Fano varieties by the adjunction formula. where X and X In general, the inequality 1.42 is sharp (see [152], [72], [73], [137]). Example 1.43. Let π : V → Z be a surjective flat morphism such that • the variety V is a smooth threefold, • the variety Z is a smooth curve, • the divisor −KV is π-ample, let X be a scheme fiber of the morphism π over a point O ∈ Z such that X is a smooth cubic surface in P3 that has an Eckardt poin P ∈ X (cf. Definition 4.1), let L1 , L2 , L3 ⊂ X be the lines that pass through the point P . Then it follows from [41] that there is a commutative diagram ψ ¯ U _ _ _ _ _ _/ U @@

@@ β @@ @@ ρ _ _ _ _ _ _ _ _ _ _ _ _ _ _/ V¯ V ? ?? } ?? }} } ? } ? π ? ~}} π¯ α

Z Z such that α is a blow up of the point P , the map ψ is an antiflip in the proper transforms of the curves L1 , L2 , L3 , and β is a contraction of the proper transform of the fiber X. Then 9

• • • •

the the the the

birational map ρ is not an isomorphism, threefold V¯ has terminal and Q-factorial singularities, divisor −KV¯ is a Cartier π ¯ -ample divisor, map ρ induces an isomorphism ¯ V \X ∼ = V¯ \ X,

¯ is a scheme fiber of π where X ¯ over the point O, ¯ • the surface X is a cubic surface with a singular point of type D4 . ¯ = 1 (see Examples 1.18 and 1.19). The latter assertion implies that lct(X) + lct(X) Global log canonical thresholds can be used to prove that some Fano varieties are non-rational. Definition 1.44. The variety X is said to be birationally superrigid if the following conditions hold: • rk Pic(X) = 1; • the variety X has terminal Q-factorial singularities; • there is no rational dominant map ρ : X 99K Y such that – general fiber of the map ρ is rationally connected, – the inequality dim(Y ) > 1 holds; • there is no non-biregular birational map ρ : X 99K Y such that – the variety Y has terminal Q-factorial singularities; – the equality rk Pic(Y ) = 1 holds. The following result is known as the Noether–Fano inequality (see [40], [96], [22], [159]). Theorem 1.45. The following conditions are equivalent: • the variety X is birationally superrigid; • the following conditions hold: – the equality rk Pic(X) = 1 holds; – the variety X has terminal Q-factorial singularities; – for every linear system M on the variety X that does not have fixed components, the log pair (X, λM) has canonical singularities, where KX + λM ∼Q 0. Proof. Because one part of the required assertion is well-known (see [40], [22], [159]), we prove only another part of the required assertion. Suppose that • the variety X is birationally superrigid, • but there is a linear system M on the variety X such that M has no fixed components, the singularities of (X, λM) are not canonical, where KX + λM ∼Q 0. It follows from [75] that there is birational morphism π : V → X such that • the variety V is smooth, • the proper transform of M on the variety V has no base points, and let B be the proper transform of the linear system M on the variety V . Then r r X X KV + λB ∼Q π ∗ KX + λM + ai Ei ∼Q ai Ei , i=1

i=1

where Ei is an exceptional divisor of π, and ai ∈ Q. It follows from [9] that there is a commutative diagram ρ

U

~~

~

~

~

V A A φ

AAπ AA A /X

such that ρ is a birational map, the morphism φ is birational, the divisor r r X X KU + λρ B ∼Q φ∗ KX + λM + ai ρ Ei ∼Q ai ρ Ei i=1 10

i=1

is φ-nef, the variety U is Q-factorial, the log pair (U, λρ(B) has terminal singularities. The morphism φ is not an isomorphism. It follows from [169, 1.1] that ai > 0 =⇒ dim ρ Ei 6 dim X − 2, but it follows from the construction of the map ρ that there is k ∈ {1, . . . , r} such that • the inequality ak < 0 holds, • the subvariety ρ(Ek ) ⊂ U is a divisor, because the singularities of the log pair (X, λM) are not canonical. The divisor KU +λρ(B) is not pseudo-effective. Then it follows from [9] that there is a diagram ψ U _ _ _ _ _ _/ Y φ

X

τ

Z

such that ψ is a birational map, the morphism τ is a Mori fibred space (see [102]), and the divisor − KY + λ ψ ◦ ρ B is τ -ample. The variety Y has terminal Q-factorial singularities, and rk Pic(Y /Z) = 1. Then • the birational map ψ ◦ ρ ◦ π −1 is not an isomorphism, because KX + λM ∼Q 0, • general fiber of the morphism τ is rationally connected (see [193]), which contradicts the assumption that X is birationally superrigid. Birationally superrigid Fano varieties are non-rational. In particular, if the variety X is birationally superrigid, then dim(X) 6= 2 (cf. [117], [118], [95]). Example 1.46. It follows from [97] that smooth quartic hypersurface x4 + xw3 + y 4 − 6y 2 z 2 + z 4 + t4 + t3 w = 0 ⊂ P4 ∼ = Proj C x, y, z, t, w is smooth and unirational (cf. [120]) and birationally superrigid (cf. [38]). Example 1.47. The following smooth Fano varieties are birationally superrigid: • a smooth hypersurface in Pn of degree n for 4 6 n 6 12 (see [97], [147], [19], [155], [59]); • a general hypersurface in Pn of degree n > 4 (see [151]); • a smooth hypersurface in P(1n+1 , n) of degree 2n > 6 (see [94], [148]). • a general complete intersection of hypersurfaces F1 , . . . , Fk ⊂ Pn such that k X

deg Fi = n > 3k + 1 > 7,

i=1

where deg(Fk ) > . . . > deg(F2 ) > deg(F1 ) > 2 (see [154]); • a smooth fourfold complete intersection in P6 of degree 8 containing no planes (see [21]); • a smooth Fano variety X such that there is a double cover τ : X −→ V ⊂ Pn where V is a hypersurface, τ is ramified in a divisor R ∈ |OPn (2n − 2deg(V ))|V |, and – either deg(V ) = 2 and n > 5 (see [148]), – or V and R ∈ |OPn (2n − 2deg(V ))|V | are general and n > 5 (see [153]), – or 3 6 deg(V ) 6 4 and n > 8 (see [24]). • a sextic hypersurface in P6 with at most ordinary double points (see [25]). Example 1.48. Let π : X → P3 be a double cover branched along a surface S ⊂ P3 of degree 6 such that the sextic surface S has at most ordinary double points. Then • the inequality |Sing(S)| 6 65 holds (see [84], [188]), • for any 65 > k ∈ Z>0 , there exists S ⊂ P3 such that |Sing(S)| = k (see [18], [6]), • the variety X is birationally superrigid in the case when rk Cl(X) = 1 (see [97], [34]), • the equality rk Cl(X) = 1 holds if |Sing(S)| 6 14 (see [34]), 11

• suppose that the surface S is a Barth sextic (see [6]) that is given by 2 4 τ 2 x2 −y 2 τ 2 y 2 −z 2 τ 2 z 2 −x2 = t2 1+2τ x2 +y 2 +z 2 −t2 ⊂ P3 ∼ = Proj C x, y, z, t , √ where τ = (1 + 5)/2; then rk Cl(X) = 14 (see [56]) and the diagram

Y ρ

/ P4 ψ

X

π

/ P3

commutes (see [56], [140]), where – the variety Y is a determinantal quartic threefold in P4 such that |Sing(Y )| = 42, – the map ψ is the projection from a singular point of the quartic Y , – the map ρ is a birational map, which implies that the threefold X is rational. The following result is proved in [157]. Theorem 1.49. Let X1 , X2 , . . . , Xr be birationally superrigid Fano varieties such that lct(X1 ) > 1, lct(X2 ) > 1, . . . , lct(Xr ) > 1. Then • the variety X1 × . . . × Xr is non-rational and Bir X1 × . . . × Xr = Aut X1 × . . . × Xr , • for every rational dominant map ρ : X1 × . . . × Xr 99K Y, whose general fiber is rationally connected, there is a commutative diagram X1 × . . . × XrW

W W W W ρW W W W W W W Xi1 × . . . × Xik _ _ _ _ _ _ _ _ _ _ _ _W+/ Y ξ π

for some {i1 , . . . , ik } ⊆ {1, . . . , r}, where ξ is a birational map, and π is a projection. Varieties satisfying all hypotheses of Theorem 1.49 exist (see Examples 1.11, 1.13, 1.14, 1.47). Example 1.50. Let X be a hypersurface that is given by w2 = x6 + y 6 + z 6 + t6 + x2 y 2 zt ⊂ P 1, 1, 1, 1, 3 ∼ Proj C x, y, z, t, w , = where wt(x) = wt(y) = wt(z) = wt(t) = 1 and wt(w) = 3. Then • the threefold X is smooth and birationally superrigid (see [94]), • it follows from [37] that the equality lct(X) = 1 holds. Example 1.51. Let X be a hypersurface that is given by  2n n n−4 n X X X  w2 = x2n aij xi  ⊂ P 1n+1 , n ∼ = Proj C x0 , x1 , . . . , xn , w i +ε i=0

i=1

j=0

for some ε ∈ C 3 aij , where wt(xi ) = 1 and wt(w) = n. Suppose that any n + 1 forms among x0 , x1 , . . . , xn ,

n X i=0

a1i xi ,

n X i=0

a2i xi , . . . ,

n X

an−4 i xi

i=0

are linearly independent. Put ∆ = max{|aij |}. Suppose that the inequalities n > 8 and 2m 1 > |ε| 2n∆ + 2∆ 12

hold. Then X is birationally superrigid and lct(X) = 1 (see [148], [160]). The hypersurface  2n n n−4 n X X X 2 i  w2 = n2n x2n j + 1 xi  i + i=0

i=1

j=0

is birationally superrigid, and its global log canonical threshold is 1 (see [160]). Suppose, in addition, that the subgroup G ⊂ Aut(X) is finite (cf. [71]). Definition 1.52. The Fano variety X is G-birationally superrigid if • the G-invariant subgroup of the group Cl(X) is isomorphic to Z, • the variety X has terminal singularities, • there is no G-equivariant rational dominant map ρ : X 99K Y such that – general fiber of the map ρ is rationally connected, – the inequality dim(Y ) > 1 holds, • there is no G-equivariant non-biregular birational map ρ : X 99K Y such that – the G-invariant subgroup of the group Cl(Y ) is isomorphic to Z, – the variety Y has terminal singularities. Arguing as in the proof of Theorem 1.45, we obtain the following result. Theorem 1.53. The following conditions are equivalent: • the variety X is G-birationally superrigid; • the following conditions hold: – the G-invariant subgroup of the group Cl(X) is isomorphic to Z; – the variety X has terminal singularities; – for every G-invariant linear system M on variety X that has no fixed components, the log pair (X, λM) is canonical, where KX + λM ∼Q 0. If X is birationally superrigid, then X is G-birationally superrigid for any G ⊂ Aut(X). Example 1.54. Let X be a smooth surface in P(1, 1, 2, 3) of degree 6 such that the G-invariant subgroup of the group Pic(X) is Z. Then X is G-birationally superrigid (see [117], [118],[95]). The proof of Theorem 1.49 implies the following result (see [31]). Theorem 1.55. Let Xi be a Fano variety, and let Gi ⊂ Aut(Xi ) be a finite subgroup such that • the variety Xi is Gi -birationally superrigid, • the inequality lct(Xi , Gi ) > 1 holds for any i = 1, . . . , r. Then the following assertions hold: • there is no G1 × . . . × Gr -equivariant birational map ρ : X1 × . . . × Xr 99K Pn ; • every G1 × . . . × Gr -equivariant birational automorphism of X1 × . . . × Xr is biregular; • for every G1 × . . . × Gr -equivariant rational dominant map ρ : X1 × . . . × Xr 99K Y, whose general fiber is rationally connected, there a commutative diagram X1 × . . . × XrW

W W W W ρW W W W W W W Xi1 × . . . × Xik _ _ _ _ _ _ _ _ _ _ _ _W+/ Y ξ π

where ξ is a birational map, π is a natural projection, and {i1 , . . . , ik } ⊆ {1, . . . , r}. Varieties satisfying all hypotheses of Theorem 1.55 do exist (see Example 1.25). Example 1.56. The simple group A6 is a group of automorphisms of the sextic 10x3 y 3 + 9zx5 + 9zy 5 + 27z 6 = 45x2 y 2 z 2 + 135xyz 4 ⊂ P2 ∼ = Proj C x, y, z , 13

which induces an embedding A6 ⊂ Aut(P2 ). It follows from [44] that P2 is A6 -birationally superrigid. But the equality lct(P2 , A6 ) = 2 holds (see [31]). Thus, there is an induced embedding A6 × A6 ∼ = Ω ⊂ Bir P4 such that Ω is not conjugate to any subgroup in Aut(P4 ) by Theorem 1.55. Example 1.57. Suppose that X be a smooth cubic surface in P3 that is given by x2 y + xz 2 + zt2 + tx2 = 0 ⊂ P3 ∼ = Proj C x, y, z, t . Then Aut(X) ∼ = S5 (see [52]). Hence, by [31] lct X, S5 = lct X, A5 = 2, and the surface X is A5 -birationally superrigid (see Example 1.54). Let us consider Fano varieties that are close to being birationally superrigid. Definition 1.58. The Fano variety X is birationally rigid3 if • the equality rk Pic(X) = 1 holds, • the variety X has Q-factorial and terminal singularities, • there is no rational dominant map ρ : X 99K Y such that – a general fiber of the map ρ is rationally connected, – the inequality dim(Y ) > 1 holds, • there is no birational map ρ : X 99K Y such that – the varieties Y and X are not biregular, – the variety Y has terminal Q-factorial singularities, – the equality rk Pic(Y ) = 1 holds. Arguing as in the proof of Theorem 1.45, we obtain the following result. Theorem 1.59. The following conditions are equivalent: • the variety X is birationally rigid; • the following conditions hold: – the equality rk Pic(X) = 1 holds; – the variety X has Q-factorial and terminal singularities; – for every linear system M on the Fano variety X that does not have fixed components, there is birational automorphism ξ ∈ Bir(X) such that the log pair X, λ ξ M has canonical singularities, where KX + λ ξ(M) ∼Q 0. Remark 1.60. For every n > 5, there exists a smooth Fano variety X of dimension n such that Pic X = Z − KX and the variety X is not birationally rigid (see [17]). Birationally rigid Fano varieties are non-rational (see [40], [96], [22], [159]). Example 1.61. The following varieties are birationally rigid but not birationally superrigid: • a general complete intersection of a quadric and a cubic in P5 (see [99]); • a smooth double cover of a quadric in P4 branched over a surface of degree 8 (see [94]). One usually seeks for the birational automorphism in Definition 1.58 in a given set of birational automorphisms. This leads to the following definition. 3There are several definitions of birational rigidity and birational superrigidity (see [40], [42], [96], [22], [159]). 14

Definition 1.62. Suppose that X is birationally rigid. A subset Γ ⊂ Bir(X) untwists all maximal singularities if for every linear system M on the variety X that has no fixed components, there is a birational automorphism ξ ∈ Γ such that the log pair X, λ ξ M has canonical singularities, where λ is a rational number such that KX + λ ξ(M) ∼Q 0. If X is birationally rigid and there is Γ ⊂ Bir(X) that untwists all maximal singularities, then the group Bir(X) is generated by Γ and Aut(X). Example 1.63. Let X be a sufficiently general hypersurface in Pn of degree n > 5 that has one ordinary singular point O = Sing(X) of multiplicity n − 2. Then the projection ψ : X 99K Pn−1 from the point O induces an involution that untwists all maximal singularities (see [156]). If X is defined over a perfect field, then Definition 1.58 still makes sense (see [117], [118], [95]). Definition 1.64. The variety X is universally birationally rigid if for any variety U , the variety X ⊗ Spec C U is birationally rigid over a field of rational functions C(U ) of the variety U . Example 1.65. Let X be a smooth Fano threefold such that there is a double cover π : X −→ Q ⊂ P3 , where Q is a quadric threefold, and π is branched in a surface S ⊂ Q of degree 8. Put n o C = C ⊂ X C is a smooth curve such that − KX · C = 1 , then C is a one-dimensional family. For every curve C ∈ C there is a commutative diagram X ψC

π

P2

/Q φ C

P2

where φC is a projection from the line π(C). General fiber of the map ψC is an elliptic curve, the map ψC induces an elliptic fibration with a section and an involution τC ∈ Bir(X). Then ψC ∈ Aut X ⇐⇒ C ⊂ S, and S contains no curves in C if X is general. It follows from [94] that there is an exact sequence 1 −→ Γ −→ Bir X −→ Aut X −→ 1, where Γ is a free product of subgroups that are generated by non-biregular birational involutions constructed above. Hence the Fano variety X is universally birationally rigid (see [94]). Example 1.66. Let X be a quartic threefold in P4 that has at most ordinary double points. Then • the inequality |Sing(X)| 6 45 holds (see [186]), • in general, the variety X is not birationally superrigid if Sing(X) 6= ∅, • the variety X is universally birationally rigid if rk Cl(X) = 1 (see [97], [149], [125], [172]), • the inequality rk Cl(X) 6 16 holds (see [100], [101]), • the equality rk Cl(X) = 1 holds if |Sing(X)| 6 8 (see [23]), • the equality rk Cl(X) = 1 holds if the following conditions hold: – the inequality |Sing(X)| 6 12 holds; – the quartic X contains neither planes or quadric surfaces (see [171]); • in the case when |Sing(X)| = 45, it follows from [83] that X can be given by the equation w4 − w x3 + y 3 + z 3 + t3 + 3xyzt = 0 ⊂ P4 ∼ = Proj C x, y, z, t, w , the quartic X is determinantal and rational, and rk Cl(X) = 16 (see [76], [100], [101]). 15

Birationally superrigid Fano varieties are universally birationally rigid. Definition 1.67. Suppose that X is universally birationally rigid. A subset Γ ⊂ Bir(X) universally untwists all maximal singularities if for every variety U the induced subset Γ ⊂ Bir X ⊆ Bir X ⊗ Spec C U untwists all maximal singularities on X ⊗ Spec(C(U )). An identity map universally untwists all maximal singularities if X is birationally superrigid. Remark 1.68. Suppose that X is birationally rigid, and dim(X) 6= 1. Let Γ ⊆ Bir(X) be a subset. It follows from [107] that the following conditions are equivalent: • the subset Γ universally untwists all maximal singularities; • the subset Γ untwists all maximal singularities, and Bir(X) is countable. Example 1.69. In the assumptions of Example 1.15, suppose that X is general. Then • the hypersurface X is universally birationally rigid (see [43]), • there are involutions τ1 , . . . , τk ∈ Bir(X) such that that the sequence of groups

1 −→ τ1 , . . . , τk −→ Bir X −→ Aut X −→ 1 is exact (see [43], [35]), where hτ1 , . . . , τk i is a subgroup generated by τ1 , . . . , τk , • the subgroup hτ1 , . . . , τk i universally untwists all maximal singularities (see [43]). All relations between the involutions τ1 , . . . , τk are found in [35]. The papers [164], [165], [35], [26] classify all maps ψ : X 99K P2 such that the diagram V @ @@ ~~ @@ω ~ ~ @@ ~ ~ @ ~ ~ _ _ _ _ _ _ _ / P2 X α

ψ

commutes, where α is a birational morphism, and ω is an elliptic fibration. The papers [164], [165], [36] classify all maps φ : X 99K P1 such that the diagram V ~ @@@ η @@ ~~ ~ @@ ~~ @ ~~ _ _ _ _ _ _ _ / P1 X φ β

commutes, where β is a birational morphism, and η is a fibration into surfaces of Kodaira dimension zero. Let X1 , . . . , Xr be Fano varieties that have at most Q-factorial and terminal singularities, let ci × Xi+1 × . . . × Xr πi : X1 × . . . × Xi−1 × Xi × Xi+1 × . . . × Xr −→ X1 × . . . × Xi−1 × X be a natural projection, and let Xi be a scheme general fiber of the projection πi , which is defined ci × Xi+1 × . . . × Xr ). Suppose that rk Pic(X1 ) = . . . = rk Pic(Xr ) = 1. over C(X1 × . . . × Xi−1 × X Remark 1.70. There are natural embeddings of groups r Y D E Bir Xi ⊆ Bir X1 , . . . , Bir Xr ⊆ Bir X1 × . . . × Xr . i=1

The following generalization of Theorem 1.49 holds (see [27]). Theorem 1.71. Suppose that X1 , X2 , . . . , Xr are universally birationally rigid. Then • the variety X1 × . . . × Xr is non-rational and D E Bir X1 × . . . × Xr = Bir X1 , . . . , Bir Xr , Aut X1 × . . . × Xr , 16

• for every rational dominant map ρ : X1 × . . . × Xr 99K Y , whose general fiber is rationally connected, there is a subset {i1 , . . . , ik } ⊆ {1, . . . , r} and a commutative diagram σ X1 × . . . × Xr _ _ _ _ _ _/ X1 × . . . × RXr

R R ρ R R R R R _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ /( Y Xi1 × . . . × Xik ξ π

where ξ and σ are birational maps, and π is a projection, in the case when the inequalities lct(X1 ) > 1, lct(X2 ) > 1, . . . , lct(Xr ) > 1 hold. Corollary 1.72. Suppose that there are subgroups Γ1 ⊆ Bir(X1 ), . . . , Γr ⊆ Bir(Xr ) that universally untwists all maximal singularities, and lct(X1 ) > 1, lct(X2 ) > 1, . . . , lct(Xr ) > 1. Then r DY E Bir X1 × . . . × Xr = Γi , Aut X1 × . . . × Xr . i=1

The following four examples are implied by Examples 1.14, 1.15, 1.47, 1.69 and [123]. Example 1.73. Let X be a general hypersurface in P(1, 1, 4, 5, 10) of degree 20. The sequence m Y 1 −→ Z2 ∗ Z2 −→ Bir |X × .{z . . × X} −→ Sm −→ 1 i=1

m times

is exact, where Sm is a permutation group, and Z2 ∗ Z2 is the infinite dihedral group. Example 1.74. Let X be a general hypersurface in P(1, 1, 3, 4, 5) of degree 13. Then Bir X × V ∼ = Z2 ∗ Z2 ∗ Z2 , where V is a general hypersurface in Pn of degree n > 6. Example 1.75. Let X be a general hypersurface in P(1, 1, 2, 3, 3) of degree 9. Then E D 2 Bir X × V ∼ = a, b, c a2 = b2 = c2 = abc = 1 , where V is a general hypersurface in Pn of degree n > 6. Example 1.76. Let X be a general hypersurface in P(1, 1, 2, 2, 3) of degree 8. Then Bir X × V ∼ = Z2 ∗ Z2 ∗ Z2 ∗ Z2 ∗ Z2 , where V is a general hypersurface in Pn of degree n > 6. Suppose now that X is a smooth Fano threefold (see [98]). Let o n ‫ ג‬X ∈ 1.1, 1.2, . . . , 1.17, 2.1, . . . , 2.36, 3.1, . . . , 3.31, 4.1, . . . , 4.13, 5.1, . . . , 5.7, 5.8 be the ordinal number of the deformation type of the threefold X in the notation of Table 1. Remark 1.77. The threefold X lies in 105 deformation families (see [92], [93], [126], [128], [129], [127]). The main purpose of this paper is to prove the following result. Theorem 1.78. The following assertions hold: • lct(X) = 1/5 for ‫(ג‬X) ∈ {2.36, 3.29}; • lct(X) = 1/4 for o n ‫ ג‬X ∈ 1.17, 2.28, 2.30, 2.33, 2.35, 3.23, 3.26, 3.30, 4.12 ; • lct(X) = 1/3 for o n ‫ ג‬X ∈ 1.16, 2.29, 2.31, 2.34, 3.9, 3.18, . . . , 3.22, 3.24, 3.25, 3.28, 3.31, 4.4, 4.8, . . . 4.11, 5.1, 5.2 ; • lct(X) = 3/7 for ‫(ג‬X) = 4.5; 17

• lct(X) = 1/2 for 1.11, 1.12, 1.13, 1.14, 1.15, 2.1, 2.3, 2.18, 2.25, 2.27, 2.32, 3.4, 3.10, 3.11, 3.12, ‫ ג‬X ∈ ; 3.14, 3.15, 3, 16, 3.17, 3.24, 3.27, 4.1, 4.2, 4.3, 4.6, 4.7, 5.3, 5.4, 5.5, 5.6, 5.7, 5.8 • if the threefold X is general in moduli, then  1/3 if ‫(ג‬X) = 2.23,       1/2 if ‫(ג‬X) ∈ {2.5, 2.8, 2.10, 2.11, 2.14, 2.15, 2.19, 2.24, 2.26, 3.2, 3.5, . . . , 3.8, 4.13},  lct X = 2/3 if ‫(ג‬X) = 3.3,    3/4 if ‫(ג‬X) ∈ {2.4, 3.1},     1 if ‫(ג‬X) = 1.1. Hence, if the threefold X is general in moduli, then we do not know lct(X) only when o n ‫ ג‬X ∈ 1.2, 1.3, 1.4, 1.5, . . . , 1.10, 2.2, 2.6, 2.7, 2.9, 2.12, 2.13, 2.16, 2.17, 2.20, 2.21, 2.22, 3.13 , and the generality condition in Theorem 1.78 can not be omitted in many cases. Example 1.79. Suppose that ‫(ג‬X) = 4.13. Note that this deformation type was omitted in [126], and it has been discovered only twenty years later (see [127]). There is a birational morphism α : X −→ P1 × P1 × P1 that contracts a surface E ⊂ X to a curve C such that C · F1 = C · F2 = 1 and C · F3 = 3, where F1 ∼ = F2 ∼ = F3 ∼ = P1 × P1 are fibers of three different projections P1 × P1 × P1 → P1 , respectively. Then lct X = 1/2 by Theorem 1.78 if X is general. There is a surface G ∈ |F1 + F2 | such that C ⊂ G. Then ¯ + E + F¯3 , −KX ∼ 2G ¯ are proper transforms of F3 and G, respectively. Then lct(X) 6 1/2. But where F¯3 ⊂ X ⊃ G ¯ + E + F¯3 6 4/9 < 1/2 lct X 6 lct X, 2G in the case when the intersection F3 ∩ C consists of a single point. We hope that the proof of Theorem 1.78 can be used to • study the slope stability of the threefold X in the sense of [162] and [163], • study the problem of existence of a Kähler–Einstein metric on the threefold X, • compute lct(X, G) for various subgroups G ⊂ Aut(X). Remark 1.80. The stability of the tangent bundle of X was studied in [176]. It is known that • the tangent bundle of X is unstable with respect to −KX when o n ‫ ג‬X ∈ 2.35, 2.36, 3.29, 3.30, 3.31, 4.11, 4.12 , • the tangent bundle of X is semistable with respect to −KX when o n ‫ ג‬X ∈ 2.33, 2.34, 3.27, 3.28, 4.10, 5.2, . . . , 5.8 , • the tangent bundle of X is stable with respect to −KX when o n ‫ ג‬X 6∈ 2.33, 2.34, 2.35, 2.36, 3.27, . . . , 3.31, 4.10, 4.11, 4.12, 5.2, . . . , 5.8 . We organize the paper in the following way: • in Section 2, we consider auxiliary results that are used in the proof of Theorem 1.78; • in Section 3, we find the global log canonical threshold of the Mukai–Umemura threefold; • in Section 4, we prove Theorem 4.2 that is required for Example 5.4 and Lemma 8.2; • in Section 5, we consider facts on surfaces that are used in the proof of Theorem 1.78; • in Section 6, we compute global log canonical thresholds of toric Fano varieties; 18

• in Section 7, we prove Theorem 1.78 for smooth Fano threefolds of index 2, i.e., for o n ‫ ג‬X ∈ 1.11, 1.12, 1.13, 1.14, 1.15, 2.32, 2.35, 3.27 ; • • • •

in in in in

Section Section Section Section

8, we prove Theorem 1.78 in the case when rk Pic(X) = 2; 9, we prove Theorem 1.78 in the case when rk Pic(X) = 3; 10, we prove Theorem 1.78 in the case when rk Pic(X) > 4; 11, we find upper bounds for lct(X) in the case when o n ‫ ג‬X ∈ 1.8, 1.9, 1.10, 2.9, 2.12, 2.13, 2.16, 2.17, 2.20, 2.21, 2.22, 3.13 ;

• in Appendix A, written by J.-P. Demailly, the relation between global log canonical thresholds of smooth Fano varieties and the α-invariants of smooth Fano varieties introduced by G. Tian in [179] for the study of the existence of Kähler–Einstein metrics has been studied; • in Appendix B, we put Table 1 that contains the list of all smooth Fano threefolds together with the known values and bounds for their global log canonical thresholds. We use a standard notation D1 ∼ D2 (resp., D1 ∼Q D2 ) for the linearly equivalent (resp., Q-linearly equivalent) divisors (resp., Q-divisors). If a divisor (resp., a Q-divisor) D is linearly equivalent to a line bundle L (resp., is Q-linearly equivalent to a divisor that is linearly equivalent to L), we write D ∼ L (resp., D ∼Q L). Recall that Q-linear equivalence coincides with numerical equivalence in the case of Fano varieties. A divisor D on Pn1 × Pn2 × . . . × Pnm is said to be of multidegree (a1 , a2 , . . . , am ) ∈ Zm if m P D∼ ai Hi , where i=1 Hi ∼ πi∗ OPni ai , is a pull-back of a hyperplane section of Pni under the projection πi : Pn1 × Pn2 × . . . × Pnm → Pni is a natural projection. A curve C ⊂ Pn1 × Pn2 × . . . × Pnm is said to be of multidegree (a1 , a2 , . . . , am ) ∈ Zm if πi∗ OPni 1 · C = ai for i = 1, . . . , m. The projectivisation PY (E) of a vector bundle E on a variety Y is the variety of hyperplanes in the fibers of E. The symbol Fn denotes the Hirzebruch surface P(OP1 ⊕ OP1 (n)). We always refer to a smooth Fano threefold X using the ordinal number ‫(ג‬X) introduced in Table 1. We are very grateful to J.-P. Demailly for writing Appendix A, and to A. Iliev, A. G. Kuznetsov and Yu. G. Prokhorov for useful discussions. The first author would like to express his gratitude to IHES (Bures-sur-Yvette, France) and MPIM (Bonn, Germany) for hospitality. 2. Preliminaries Let X be a variety with log terminal singularities. Let us consider a Q-divisor r X BX = ai Bi , i=1

where Bi is a prime Weil divisor on the variety X, and ai is an arbitrary non-negative rational number. Suppose that BX is a Q-Cartier divisor such that Bi 6= Bj for i 6= j. ¯ → X be a birational morphism such that X ¯ is smooth (see [75]). Put Let π : X BX¯ =

r X

¯i , ai B

i=1

¯i is a proper transform of the divisor Bi on the variety X. ¯ Then where B n X ci Ei , KX¯ + BX¯ ∼Q π ∗ KX + BX + i=1 19

where ci ∈ Q, and Ei is an exceptional divisor of the morphism π. Suppose that ! ! n r [ [ [ ¯i B Ei i=1

i=1

is a divisor with simple normal crossing. Put B

¯ X

= BX¯ −

n X

ci Ei .

i=1

Definition 2.1. The singularities of (X, BX ) are log canonical (resp., log terminal) if • the inequality ai 6 1 holds (resp., the inequality ai < 1 holds), • the inequality cj > −1 holds (resp., the inequality cj > −1 holds), for every i = 1, . . . , r and j = 1, . . . , n. One can show that Definition 2.1 does not depend on the choice of the morphism π. Put     [ [ [ LCS X, BX =  Bi   π Ei  ( X, ai >1

ci 6−1

then LCS(X, BX ) is called the locus of log canonical singularities of the log pair (X, BX ). Definition 2.2. A proper irreducible subvariety Y ( X is said to be a center of log canonical singularities of the log pair (X, BX ) if one of the following conditions is satisfied: • either the inequality ai > 1 holds and Y = Bi , • or the inequality ci 6 −1 holds and Y = π(Ei ), ¯ → X. for some choice of the birational morphism π : X Let LCS(X, BX ) be the set of all centers of log canonical singularities of (X, BX ). Then Y ∈ LCS X, BX =⇒ Y ⊆ LCS X, BX and LCS(X, BX ) = ∅ ⇐⇒ LCS(X, BX ) = ∅ ⇐⇒ the log pair (X, BX ) is log terminal. Remark 2.3. Let H be a linear system on X that has no base points, let H be a sufficiently general divisor in the linear system H, and let Y ( X be an irreducible subvariety. Put m X Y = Zi , H

i=1

where Zi ⊂ H is an irreducible subvariety. It follows from Definition 2.2 (cf. Theorem 2.20) that n o Y ∈ LCS X, BX ⇐⇒ Z1 , . . . , Zm ⊆ LCS H, BX . H

Example 2.4. Let α : V → X be a blow up of a smooth point O ∈ X. Then BV ∼Q α∗ BX − multO BX E, where multO (BX ) ∈ Q, and E is the exceptional divisor of the blow up α. Then multO BX > 1 if the log pair (X, BX ) is not log canonical at the point O. Put B V = BV + multO BX − dim X + 1 E, and suppose that multO (BX ) > dim(X) − 1. Then O ∈ LCS(X, BX ) if and only if • either E ∈ LCS(V, B V ), i. e. multO (BX ) > dim(X), • or there is a subvariety Z ( E such that Z ∈ LCS(V, B V ). 20

The locus LCS(X, BX ) ⊂ X can be equipped with a scheme structure (see [132], [169]). Put n r X X ¯i , I X, BX = π∗ dci eEi − bai cB i=1

i=1

and let L(X, BX ) be a subscheme that corresponds to the ideal sheaf I(X, BX ). Definition 2.5. For the log pair (X, BX ), we say that • the subscheme L(X, BX ) is the subscheme of log canonical singularities of (X, BX ), • the ideal sheaf I(X, BX ) is the multiplier ideal sheaf of (X, BX ). It follows from the construction of the subscheme L(X, BX ) that Supp L X, BX = LCS X, BX ⊂ X. The following result is the Nadel–Shokurov vanishing theorem (see [169], [111, Theorem 9.4.8]). Theorem 2.6. Let H be a nef and big Q-divisor on X such that KX + BX + H ∼Q D for some Cartier divisor D on the variety X. Then for every i > 1 H i X, I X, BX ⊗ D = 0. Proof. It follows from the Kawamata–Viehweg vanishing theorem (see [105]) that n r X X ¯i = 0 Ri π∗ π ∗ KX + BX + H + dci eEi − bai cB i=1

i=1

for every i > 0. It follows from the equality of sheaves n r X X ¯i = I X, BX ⊗ D π ∗ π ∗ K X + BX + H + dci eEi − bai cB i=1

i=1

and from the degeneration of a local-to-global spectral sequence that n r X X i ∗ i ¯ ¯i , dci eEi − H X, I X, BX ⊗ D = H X, π KX + BX + H + bai cB i=1

i=1

for every i > 0. But for i > 0, the cohomology group n r X X ¯ π ∗ KX + BX + H + ¯i H i X, dci eEi − bai cB i=1

i=1

is trivial by the Kawamata–Viehweg vanishing theorem (see [105]).

For every Cartier divisor D on the variety X, let us consider the exact sequence of sheaves 0 −→ I X, BX ⊗ D −→ OX D −→ OL(X, BX ) D −→ 0, and let us consider the corresponding exact sequence of cohomology groups H 0 OX D −→ H 0 OL(X, BX ) D −→ H 1 I X, BX ⊗ D . Theorem 2.7. Suppose that −(KX + BX ) is nef and big. Then LCS(X, BX ) is connected. Proof. Put D = 0. Then it follows from Theorem 2.6 that the sequence C = H 0 OX −→ H 0 OL(X, BX ) −→ H 1 I X, BX = 0 is exact. Thus, the locus LCS X, BX = Supp L X, BX is connected.

Let us consider few elementary applications of Theorem 2.7 (cf. Example 1.18). 21

Lemma 2.8. Suppose that LCS(X, BX ) 6= ∅, where X ∼ = Pn , and BX ∼Q −λKX for some rational number 0 < λ < n/(n + 1). Then • the inequality dim(LCS(X, BX )) > 1 holds, • the subscheme L(X, BX ) does not contain isolated zero-dimensional components. Proof. Suppose that there is a point O ∈ X such that LCS X, λBX = O ∪ Σ, where Σ ⊂ X is a possibly empty subset such that O 6∈ X. Let H be a general line in X ∼ = P2 . Then the locus LCS X, λBX + H = O ∪ H ∪ Σ is disconnected. But the divisor −(KX +λBX +H) is ample, which contradicts Theorem 2.7. Lemma 2.9. Suppose that LCS(X, BX ) 6= ∅, where X ∼ = P3 , and

BX ∼Q −λKX for some rational number 0 < λ < 1/2. Then LCS(X, BX ) contains a surface. Proof. Suppose that LCS(X, BX ) contains no surfaces. Let S ⊂ P3 be a general plane. The locus LCS P3 , BX + S is connected by Theorem 2.7. Then (S, BX |S ) is not log terminal by Remark 2.3. But the locus LCS S, BX S

consists of finitely many points, which is impossible by Lemma 2.8.

Lemma 2.10. Suppose that LCS(X, BX ) 6= ∅, where X is a smooth quadric threefold in P4 , and BX ∼Q −λKX for some rational number 0 < λ < 1/2. Then LCS(X, BX ) contains a surface. Proof. Let L ⊂ X be a general line, let P1 ∈ L 3 P2 be two general points, let H1 and H2 be the hyperplane sections of X that are tangent to X at the points P1 and P2 , respectively. Then 3 H1 + H2 = LCS X, λBX ∪ L LCS X, λBX + 4 is disconnected, which is impossible by Theorem 2.7. Remark 2.11. One can prove Lemmas 2.9, 2.10 and 2.29 using the following trick. Suppose that BX ∼Q −λKX for some λ ∈ Q such that 0 < λ < 1/2, where X is either P3 , or P1 × P2 , or a smooth quadric threefold, and the set LCS(X, BX ) contains no surfaces. Then LCS X, BX ⊆ Σ, where Σ ⊂ X is a (possibly reducible) curve. For a general φ ∈ Aut(X) we have φ Σ ∩ Σ = ∅, which implies that LCS(X, φ(BX )) ∩ LCS(X, BX ) = ∅. But [ LCS X, φ BX + BX = LCS X, φ BX LCS X, BX whenever φ is sufficiently general. The latter contradicts Theorem 2.7 since λ < 1/2. Lemma 2.12. Suppose that LCS(X, BX ) 6= ∅, where X is a blow up of P3 in one point, and BX ∼Q −λKX for some rational number 0 < λ < 1/2. Then LCS(X, BX ) contains a surface. 22

Proof. Suppose that the set LCS(X, BX ) contains no surfaces. Let α : X −→ P3 be the blow up of a point, and let E be the exceptional divisor of α. In the case when LCS X, λBX 6⊆ E, we can apply Lemma 2.9 to the pair (P3 , α(BX )) to get a contradiction. Hence LCS(X, BX ) ⊆ E. Let H ⊂ P3 a general hyperplane, and let H1 ⊂ P3 ⊃ H2 be general hyperplanes that pass ¯ H ¯ 1 and H ¯ 2 the proper transforms of these planes on X. Then through α(E). Denote by H, 1 ¯ ¯ ¯ H1 + H2 + 2H LCS X, BX + 2 is disconnected, which is impossible by Theorem 2.7.

Lemma 2.13. Suppose that X is a cone in P4 over a smooth quadric surface, and BX ∼Q −λKX for some rational number 0 < λ < 1/3. Then LCS(X, BX ) = ∅. Proof. Suppose that LCS(X, BX ) 6= ∅. Let S ⊂ X be a general hyperplane section. Then LCS S, BX = ∅, S

because S ∼ = P1 × P1 and lct(P1 × P1 ) = 1/2 (see Example 1.18). One has |LCS(X, BX )| < +∞ by Remark 2.3. Then the locus LCS X, BX + S is disconnected, which contradicts Theorem 2.7.

The following result is a corollary Theorem 2.6 (see [132, Theorem 4.1]). Lemma 2.14. Suppose that −(KX + BX ) is nef and big and dim(LCS(X, BX )) = 1. Then • the locus LCS(X, BX ) is a connected union of smooth rational curves, • every two irreducible components of the locus LCS(X, BX ) meet in at most one point, • every intersecting irreducible components of the locus LCS(X, BX ) meet transversally, • no three irreducible components of the locus LCS(X, BX ) meet in one point, • the locus LCS(X, BX ) does not contain a cycle of smooth rational curves. Proof. Arguing as in the proof of Theorem 2.7, we see that the locus LCS(X, BX ) is a connected tree of smooth rational curves with simple normal crossings. To consider another application of Theorem 2.7, we need the following result (see [150], [21]). Lemma 2.15. Suppose that X is a smooth complete intersection ∩ki=1 Gi ⊂ Pm , and BX ∼Q OPm (1) , X

where Gi is a hypersurface. Let S ( X be an irreducible subvariety such that dim(S) > k. Then multS BX 6 1. Proof. We may assume dim(S) = k 6 (m − 1)/2. Let P be a sufficiently general point in Pm , and let C ⊂ Pm be a cone over the subvariety S with vertex in the point P . Then C ∩ X = S ∪ R, where R is a curve on X. Let us calculate |R ∩ S|. Let π : X −→ Pm−1 , π1 : G1 −→ Pm−1 , . . . , πk : Gk −→ Pm−1 be projections from the point P , let D ⊂ X and Di ⊂ Gi be the ramification subvarieties of the projections π and πi , respectively. Put C ∩ Gi = S ∪ Ri . Then Ri ∩ S = Di ∩ S 23

by [155, Lemma 3]. Hence, it follows from R = ∩ki=1 Ri and D = ∩ki=1 Di that R ∩ S = D ∩ S. Let (z0 , . . . , zm ) be homogeneous coordinates on Pm such that P = p0 , . . . , pm , and Gi ⊂ Pm is given by the equation Fi (z0 , . . . , zm ) = 0. Then D ⊂ X is cut out by m m m X X X ∂F1 z0 , . . . , zm ∂F2 z0 , . . . , zm ∂Fk z0 , . . . , zm pi = pi = . . . = pi = 0. ∂zi ∂zi ∂zi i=0

i=0 on Pm

Let Fr be a linear system m X i=0

i=0

that contains divisors ∂Fr z0 , . . . , zm = 0, r = 1, . . . , k. λi ∂zi

The variety X is smooth. Hence k Y R ∩ S = D ∩ S = deg S deg Fi − 1 , i=1

because F1 , . . . , Fk do not base points on X. Therefore, we have the inequality deg S

k Y

X deg Fi − 1 = BX · R > multS BX = multS BX R ∩ S ,

i=1

O∈R∩S

which implies multS (BX ) 6 1.

Using Remark 2.3, Theorem 2.7 and Lemma 2.15, we obtain the following result. Corollary 2.16. Let X is a smooth complete intersection ∩ki=1 Gi ⊂ Pm such that m+1−

k X

deg Gi > k + 1,

i=1

where Gi is a hypersurface in

Pm .

Then X is a Fano variety and 1 lct X = . Pk m + 1 − i=1 deg Gi

Let us consider another simple application of Theorem 2.7 and Lemma 2.15. Lemma 2.17. Let X be a cubic hypersurface in P4 such that |Sing(X)| < +∞. Suppose that BX ∼Q −KX , and there is a positive rational number λ < 1/2 such that LCS(X, λBX ) 6= ∅. Then LCS X, λBX = L, where L is a line in X ⊂ P4 such that L ∩ Sing(X) 6= ∅. Proof. Let S be a general hyperplane section of X. Then S ∪ LCS X, λBX ⊆ LCS X, λBX + S , which implies that dim(LCS(X, λBX )) > 1 by Theorem 2.7. Then LCS S, λBX 6= ∅ S

by Remark 2.3. But |LCS(S, λBX |S )| < +∞ by Lemma 2.15. There is a point O ∈ S such that LCS S, λBX = O S

by Theorem 2.7. Therefore, there is a line L ⊂ X such that LCS(X, λBX ) = L by Remark 2.3. Arguing as in the proof of Lemma 2.15, we see that L ∩ Sing(X) 6= ∅. Similar to Lemma 2.17, one can prove the following result. 24

Lemma 2.18. Suppose that there is a double cover τ : X → P3 branched over an irreducible reduced quartic surface R ⊂ P3 that has at most ordinary double points, the equivalence BX ∼Q −λKX holds and LCS(X, BX ) 6= ∅, where λ < 1/2. Then Sing(X) 6= ∅ and LCS X, BX = L, where L is an irreducible curve on X such that −KX · L = 2 and L ∩ Sing(X) 6= ∅. Proof. We have −KX ∼ 2H, where H is a Cartier divisor on X such that H ∼ τ ∗ OP 3 1 . The variety X is a Fano threefold, and H 3 = 2. Then LCS X, BX + H must be connected by Theorem 2.7. Thus, there is a curve C ∈ LCS X, BX , which implies that multC (BX ) > 1/λ > 2. Let S be a general surface in |H|. Put BS = BX |S . Then 1 −KS ∼ H S ∼Q BS , λ but the log pair (S, BS ) is not log canonical in every point of the intersection S ∩ LCS(X, BX ). The surface H is a smooth surface in P(1, 1, 1, 2) of degree 4. Let P be any point in S ∩ LCS(X, BX ). Then there is a birational morphism ρ : S −→ S¯ such that S¯ is a cubic surface in P3 and ρ is an isomorphism in a neighborhood of P . Then ¯ ρ BS S, ¯ ρ(BS )) 6= ∅. But is not log terminal at the point ρ(P ). Thus, we have LCS(S, 1 ρ(BS ) ∼Q −KS¯ ∼ OP3 1 , λ S¯ ¯ ρ(BS )) consists of one point by Lemma 2.15 and Theorem 2.7. Then which implies that LCS(S, P = S ∩ C = S ∩ LCS X, BX if the point P is sufficiently general. Therefore, we see that LCS X, BX = C, the curve C is irreducible and −KX · C = 2. Then τ (C) ⊂ P3 is a line. Suppose that C ∩ Sing(X) = ∅. Let us derive a contradiction. Suppose that τ (C) ⊂ R. Take a general point O ∈ C. Let τ O ∈ Π ⊂ P3 be a plane that is tangent to R at the point τ (O). Arguing as in the proof of Lemma 2.15, we see that R|Π is reduced along τ (C), because τ (C) ∩ Sing(R) = ∅. Fix a general line Γ ⊂ Π ⊂ P3 ¯ ⊂ X be an irreducible curve such that τ (Γ) ¯ = Γ. Then such that τ (O) ∈ Γ. Let Γ ¯ 6⊆ Supp BX , Γ because Γ spans a dense subset in P3 when we vary the point O ∈ C and the line Γ ⊂ Π. Note ¯ equals either 1 or 2, and multO (Γ) ¯ = 2 in the case when H · Gamma ¯ that H · Γ = 2. Hence ¯ ¯ ¯ ¯ ¯ H · Γ > 2λH · Γ = Γ · BX > multO Γ multC BV > H · Γ, 25

which is a contradiction. Thus, we see that τ (C) 6⊂ R. There is an irreducible reduced curve C¯ ⊂ X such that τ C¯ = τ C ⊂ P3 and C¯ 6= C. Let Y be a general surface in |H| that passes through the curves C¯ and C. Then Y is smooth, because C ∩ Sing(X) = ∅, and C¯ · C¯ = C · C = −2 on the surface Y . By construction, we have Y 6⊂ Supp(BX ). Put BY = BX |Y . Then BY = multC¯ BX C¯ + multC BX C + ∆ where ∆ is an effective Q-divisor on the surface Y such that C¯ 6⊂ Supp(∆) 6⊃ C. But BY ∼Q 2λ C¯ + C , which implies, in particular, that 2λ − multC BX C · C = multC¯ BX − 2λ C¯ · C + ∆ · C > multC¯ BX − 2λ C¯ · C > 0, because ∆ · C > 0 and C¯ · C > 0. Then multC¯ (BX ) > 2λ, because C · C < 0. Thus, we have −∆ ∼Q multC¯ BX − 2λ C¯ + multC BX − 2λ C which is impossible, because multC (BX ) > 2λ and Y is projective.

One can generalize Theorem 2.7 in the following way (see [169, Lemma 5.7]). Theorem 2.19. Let ψ : X → Z be a morphism. Then the set ¯ B X¯ LCS X, is connected in a neighborhood of every fiber of the morphism ψ ◦ π : X → Z in the case when • the morphism ψ is surjective and has connected fibers, • the divisor −(KX + BX ) is nef and big with respect to ψ. Let us consider one important application of Theorem 2.19 (see [108, Theorem 5.50]). Theorem 2.20. Suppose that B1 is a Cartier divisor, a1 = 1, and B1 has at most log terminal singularities. Then the following assertions are equivalent: • the log pair (X, BX ) is log canonicalP in a neighborhood of the divisor B1 ; • the singularities of the log pair (B1 , ri=2 ai Bi |B1 ) are log canonical. Proof. Suppose that the singularities of the log pair (X, PBX ) are not log canonical in a neighborhood of the divisor B1 ⊂ X. Let us show that (B1 , ri=2 ai Bi |B1 ) is not log canonical. In the case when am > 1 and Bm ∩ B1 6= ∅ for some m > 2, the log pair r X B1 , ai Bi B1

i=2

is not log canonical by Definition 2.1. Thus, we may assume that ai 6 1 for every i. Then r X X, B1 + λai Bi i=2

is not log canonical as well for some rational number λ < 1. Then r r n X X X ¯1 + ¯i ∼Q π ∗ KX + B1 + K¯ +B λai B λai Bi + di Ei X

i=2

i=2

i=1

for some rational numbers d1 , . . . , dn . It follows from Theorem 2.19 that ¯1 ∩ Ek 6= ∅ B 26

and the inequality dk 6 −1 holds for some k. But r r X X ¯i ∼Q φ∗ KB + λa B KB¯1 + λai B i i 1 B1

i=2

B1

i=2

+

n X

di Ei

i=1

B1

,

¯1 → B1 is a birational morphism that is induced by π. where φ : B P Thus, the log pair (B1 , ri=2 λai Bi |B1 ) is not log terminal. Then the log pair r X B1 , ai Bi B1

i=2

is not log canonical. The rest of the proof is similar (see the proof of [105, Theorem 7.5]).

The simplest application of Theorem 2.20 is a non-obvious result (see [108, Corollary 5.57]). Lemma 2.21. Suppose that dim(X) = 2 and a1 6 1. Then r X ai Bi · B1 > 1 i=2

whenever (X, BX ) is not log canonical at some point O ∈ B1 such that O 6∈ Sing(X) ∪ Sing(B1 ). Proof. Suppose that (X, BX ) is not log canonical in a point O ∈ B1 . By Theorem 2.20, we have r r X X ai Bi · B1 > multO ai Bi >1 i=2

B1

i=2

if O 6∈ Sing(X) ∪ Sing(B1 ), because (X, B1 +

Pr

i=2 ai Bi )

is not log canonical at the point O.

Let us consider another application of Theorem 2.20 (cf. Lemma 2.30). Lemma 2.22. Suppose that X is a Fano variety with log terminal singularities. Then 1 1 lct P × X = min , lct X . 2 Proof. The inequalities 1/2 > lct(V × U ) 6 lct(X) are obvious. Suppose that 1 1 , lct X , lct P × X < min 2 and let us show that this assumption leads to a contradiction. There is an effective Q-divisor D ∼Q −KP1 ×X such that the log pair P1 × X, λD is not log canonical in some point P ∈ P1 × X, where λ < min(1/2, lct(X)). Let F be a fiber of the projection P1 × X → P1 such that P ∈ F . Then D = µF + Ω, where Ω is an effective Q-divisor on P1 × X such that F 6⊂ Supp(Ω). Let L be a general fiber of the projection P1 × X → X. Then 2 = D · L = µ + Ω · L > µ, which implies that the log pair (P1 × X, F + λΩ) is not log canonical at the point P . Then F, λΩ F

is not log canonical at the point P by Theorem 2.20. But Ω ∼Q D ∼Q −KF , F

F

which is impossible, because X ∼ = F and λ < lct(X). 27

Let P be a point in X. Let us consider an effective divisor ∆=

r X

εi Bi ∼Q BX ,

i=1

where εi is a non-negative rational number. Suppose that • the divisor ∆ is a Q-Cartier divisor, • the equivalence ∆ ∼Q BX holds, • the log pair (X, ∆) is log canonical in the point P ∈ X. Remark 2.23. Suppose that (X, BX ) is not log canonical in the point P ∈ X. Put ai α = min εi 6= 0 , εi where α is well defined, because there is εi 6= 0. Then α < 1, the log pair ! r X ai − αεi Bi X, 1−α i=1

is not log canonical in the point P ∈ X, the equivalence r X ai − αεi i=1

1−α

Bi ∼Q BX ∼Q ∆

holds, and at least one irreducible component of the divisor Supp(∆) is not contained in ! r X ai − αεi Bi . Supp 1−α i=1

The assertion of Remark 2.23 is obvious. Nevertheless it is very useful. Lemma 2.24. Suppose that X ∼ = C1 × C2 , where C1 and C2 are smooth curves, suppose that BX ∼Q λE + µF where E ∼ = C1 and F ∼ = C2 are curves on the surface X such that E·E =F ·F =0 and E · F = 1, and λ and µ are non-negative rational numbers. Then • the pair (X, BX ) is log terminal if λ < 1 and µ < 1, • the pair (X, BX ) is log canonical if λ 6 1 and µ 6 1. Proof. Suppose that λ, µ < 1, but (X, BX ) is not log terminal at some point P ∈ X. Then multP BX > 1, and we may assume that E 6⊂ Supp(BX ) or F 6⊂ Supp(BX ) by Remark 2.23. But E · BX = µ, F · BX = λ, which immediately leads to a contradiction, because multP (BX ) > 1.

Let [BX ] be a class of Q-rational equivalence of the divisor BX . Put n o lct X, BX = inf lct X, D D is an effective Q-divisor on X such that D ∼Q BX > 0, and put lct(X, [BX ]) = +∞ if BX = 0. Note that BX is an effective by assumption. Remark 2.25. The equality lct(X, [−KX ]) = lct(X) holds (see Definition 1.7). Arguing as in the proof of Lemma 2.22, we obtain the following result. 28

Lemma 2.26. Suppose that there is a surjective morphism with connected fibers φ : X −→ Z such that dim(Z) = 1. Let F be a fiber of φ that has log terminal singularities. Then either h i lctF X, BX > lct F, BX F , or there is a positive rational number ε < lct(F, [BX |F ]) such that F ⊆ LCS(X, εBX ). Proof. Suppose that lctF (X, BX ) < lct(F, [BX |F ]). Then there is a rational number h i ε < lct F, BX F such that the log pair (X, εBX ) is not log canonical at some point P ∈ F . Put BX = µF + Ω, where Ω is an effective Q-divisor on X such that F 6⊂ Supp(Ω). We may assume that εµ 6 1. Then (X, F + εΩ) is not canonical at the point P . Then F, εΩ F

is not log canonical at P by Theorem 2.20. But Ω|F ∼Q BX |F , which is a contradiction.

Let us show how to apply Lemma 2.26. Lemma 2.27. Let Q ⊂ P4 be a cone over a smooth quadric surface, and let α : X → Q be a blow up along a smooth conic C ⊂ Q \ Sing(Q). Then lct(X) = 1/3. ¯ be a proper Proof. Let H be a general hyperplane section of Q ⊂ P4 that contains C, and let H transform of the surface H on the threefold X. Then ¯ + 2E, −KX ∼ 3H where E is the exceptional divisor of α. In particular, the inequality lct(X) 6 1/3 holds. We suppose that lct(X) < 1/3. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some positive rational number λ < 1/3. There is a commutative diagram X ~ AAA β ~ AA ~ AA ~~ A ~ ~ _ _ _ _ _ _ _ / P1 Q ψ α

¯ and ψ is a projection from the twowhere β is a morphism given by the linear system |H|, dimensional linear subspace that contains the conic C. Suppose that LCS(X, λD) contains a surface M ⊂ X. Then D = µM + Ω, where µ > 1/λ, and Ω is an effective Q-divisor such that M 6⊂ Supp(Ω). Let F be a general fiber of β. Then F ∼ = P1 × P1 and D = µM + Ω ∼Q −KF , F

F

F

which immediately implies that M is a fiber of the morphism β. But α D = µα M + α Ω ∼Q −KQ ∼ 3α M , which is impossible, because µ > 1/λ > 3. Thus, the set LCS(X, λD) contains no surfaces. There is a fiber S of the morphism β such that S 6= S ∩ LCS X, λD 6= ∅, which implies that S is singular by Lemma 2.26, because lct(P1 × P1 ) = 1/2. 29

Thus, the surface S is an irreducible quadric cone in P3 . Then LCS X, λD ⊆ S by Theorem 2.7. We may assume that either S 6⊂ Supp(D) or E 6⊂ Supp(D) by Remark 2.23, because 2 X, S + E 3 has log canonical singularities, and the equivalence 3S + 2E ∼Q D holds. Put Γ = E ∩ S. The curve Γ is an irreducible conic in S. Then LCS X, λD ⊆ Γ by Lemma 2.13. Intersecting D with a general ruling of the cone S ⊂ P3 , and intersecting D with a general fiber of the projection E → C, we see that Γ 6⊆ LCS X, λD , which implies that LCS(X, λD) consists of a single point O ∈ Γ by Theorem 2.7. Let R be a general (not passing through O) surface in |α∗ (H)|. Then 1 ¯ = R ∪ O, H + 2R LCS X, λD + 2 ¯ + 2R ∼Q D and λ < 1/3. which is impossible by Theorem 2.7, since −KX ∼ H

The following generalization of Lemma 2.26 is proved in [79]. Theorem 2.28. Suppose that there is a surjective morphism with connected fibers φ : X −→ Z such that φ is smooth in a neighborhood of a fiber F of the morphism φ. Then either h i lctF X, BX > lct F, BX F , or the equality lctO (X, BX ) = lctQ (X, BX ) holds for every two points O ∈ F 3 Q. Let us consider two elementary applications of Theorem 2.28. Lemma 2.29. Suppose that LCS(X, BX ) 6= ∅, where X ∼ = P1 × P2 and BX ∼Q −λKX for some rational number 0 < λ < 1/2. Then LCS(X, BX ) contains a surface. Proof. Suppose that LCS(X, BX ) contains no surfaces. By Theorems 2.7 and 2.28, we have LCS X, BX = F, where F is a fiber of the natural projection π2 : X → P2 . Let S be a general surface in ∗ π1 OP2 1 , let M1 and M2 be general fibers of the natural projection π1 : X → P1 . Then the locus 1 LCS X, λD + M1 + M2 + 3S =F ∪S 2 is disconnected, which is impossible by Theorem 2.7. Lemma 2.30. Let V and U be smooth Fano varieties. Then lct V × U = min lct V , lct U . 30

Proof. The inequalities lct(U ) > lct(V × U ) 6 lct(U ) are obvious. Suppose that lct V × U < min lct V , lct U , and let us show that this assumption leads to a contradiction. There is an effective Q-divisor D ∼Q −KV ×U such that the log pair V × U, λD is not log canonical in some point P ∈ V × U , where λ < min(lct(V ), lct(U )). Let us identify V with a fiber of the projection V × U → U that contains the point P . Then lctO V × U, D = lctQ V × U, D for every points O ∈ V 3 Q by Theorem 2.28, because the inequalities h i lct V > λ > lctV V × U, D > lct V, D V = lct V, − KV = lct V are inconsistent. So, the log pair (V × U, λD) is not log canonical in every point of V ⊂ V × U . Let us identify U with a general fiber of the projection V × U → V . Then D ∼Q −KU , U

and (U, λD|U ) is not log canonical in U ∩ V by Remark 2.3 (applied dim(V ) times here), which contradicts the inequality λ < lct(U ). We believe that the assertion of Lemma 2.30 holds for log terminal varieties (cf. Lemma 2.22). 3. The Mukai–Umemura threefold The main purpose of this section is to compute the global log canonical threshold of one remarkable smooth Fano threefold (cf. [55]) to illustrate the proof of Theorem 1.78. Lemma 3.1. Let X be the smooth Fano threefold such that4 Pic X = Z − KX , 3 = 22 holds, and Aut(X) ∼ PSL(2, C). Then lct(X) = 1/2. the equality −KX = Proof. Let U ⊂ C[x, y] be a subspace of forms of degree 12. Consider U ∼ = C13 as the affine part of P U ⊕C ∼ = P13 , and let us identify P(U ) with the hyperplane at infinity. The natural action of SL(2, C) on C[x, y] induces an action on P(U ⊕ C). Put φ = xy x10 − 11x5 y 5 − y 10 ∈ U and consider the closure SL(2, C) · [φ + 1] ⊂ P(U ⊕ C). It follows from [130] that X∼ = SL 2, C · φ + 1 , and the embedding X ⊂ P(U ⊕ C) ∼ = P13 is induced by | − KX |. The action of SL(2, C) on X has the following orbits (see [98, Theorem 5.2.13]): • the three-dimensional orbit Σ3 = SL(2, C) · [φ + 1]; • the two-dimensional orbit Σ2 = SL(2, C) · [xy 11 ]; • the one-dimensional orbit Σ1 = SL(2, C) · [y 12 ]. The orbit Σ3 is open, the orbit Σ1 ∼ = P1 is closed, and Σ2 = Σ1 ∪ Σ2 , so that the orbit Σ2 is neither open nor closed. One has X ∩ P(U ) = Σ1 ∪ Σ2 and X = Σ1 ∪ Σ2 ∪ Σ3 . Put R = X ∩ P(U ). It follows from [130] that 4The threefold X satisfying these assumptions is unique (see [130] and [142]). 31

• • • • • •

the surface R is swept out by lines on X ⊂ P13 , the surface R contains all lines on X ⊂ P13 , for any lines L1 ⊂ R ⊃ L2 such that L1 6= L2 , one has L1 ∩ L2 = ∅, the surface R is singular along the orbit Σ1 ∼ = P1 , the normalization of the surface R is isomorphic to P1 × P1 , for every point P ∈ Σ1 , the surface R is locally isomorphic to x2 = y 3 ⊂ C3 ∼ = Spec C x, y, z ,

which implies that lct(X, R) = 5/6. The structure of the surface R can be seen as follows. We see that nh o 11 i Σ2 = ax + by cx + dy ad − bc = 1 ⊂ P U , which implies that there is a birational morphism ν : P1 × P1 → R that is defined by h 11 i ν : a : b × c : d 7→ ax + by cx + dy ∈ R, so that ν is a normalization of the surface R. Let V5 be a smooth Fano threefold such that −KV5 ∼ 2H and H 3 = 5, where H is a Cartier divisor on V5 . Then |H| induces an embedding X ⊂ P6 . Let L ∼ = P1 be a line on X. Then NL/X ∼ = OP1 (−2) ⊕ OP1 (1). Let αL : UL −→ X be a blow up of the line L, and let EL be the exceptional divisor of αL . Then EL ∼ = F3 , and it follows from Theorem 4.3.3 in [98] (see [45], [178]) that there is a commutative diagram ρL UL _ _ _ _ _ _/ WL αL

βL

X _ _ _ψ _ _ _/ V5 L

where ρL is a flop in the exceptional section of E ∼ = F3 , the morphism βL contracts a surface DL ⊂ WL to a smooth rational curve of degree 5, and ψL is a double projection from the line L. ¯ L ⊂ X be the proper transform of the surface DL . Then Let D ¯L = 3 multL D ¯ L ∼ −KX . It follows from [63] that X \ D ¯L ∼ and D = C3 (cf. [138], [139], [65]). ˘L ⊂ D ¯ L that is given by It follows from [64] that there is an open subset D µ0 x4 + µ1 yz + µ2 z 3 x3 + µ3 y 3 + µ4 y 2 z 2 + µ5 yz 4 x2 + µ6 y 4 z + µ7 y 3 z 3 x + µ8 y 6 + µ9 y 5 z 2 = 0 ˘ L is given by the equations x = y = z = 0, and in C3 ∼ = Spec(C[x, y, z]), where L ∩ Σ1 ∈ D µ0 = −28 52 , µ1 = 29 33 5, µ2 = −26 34 5, µ3 = −28 33 7, µ4 = −24 34 127, µ5 = 29 35 , µ6 = 22 36 89, µ7 = −28 36 , µ8 = −36 53 , µ9 = 25 37 . ¯ L ) = 4, and it follows from Proposition 8.14 in [105] that Put OL = Σ1 ∩ L. Then multOL (D 1¯ LCS X, D L = OL 2 ¯ L ) = 1/2. Thus, we see that lct(X) 6 1/2. and lct(X, D Suppose that lct(X) < 1/2. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some positive rational number λ < 1/2. 32

By Remark 2.23, we may assume that R 6⊂ Supp(D), because lct(X, R) = 5/6. Let C be a line in X such that C 6⊂ Supp(D). Then 1 = D · C > multOC D multOC C = multOC D , which implies that OC 6∈ LCS(X, λD). In particular, we see that Σ1 6∈ LCS(X, λD). Let Γ be an irreducible curve in Supp(D) such that OC ∈ Γ. Then ¯C ¯C multΓ D multΓ D 1¯ DC + λD = + λmultΓ D 6 + λmultOC D < 1, multΓ 2 2 2 ¯ C ) = C, because D ¯ C 6= R. Thus, we see that because λ < 1/2 and Sing(D 1¯ Γ 6⊆ LCS X, DC + λD ⊇ LCS X, λD ∪ OC , 2 which is impossible by Theorem 2.7, because OC 6∈ LCS(X, λD) and λ < 1/2.

The threefold satisfying all hypotheses of Lemma 3.1 is called the Mukai–Mumemura threefold. Remark 3.2. Let X be the Mukai–Mumemura threefold. Then it follows from [55] that 5 lct X, SO 3 = . 6 Remark 3.3. Let X be a smooth Fano threefold such that Pic(X) = Z[−KX ]. Then it follows from the papers [143], [91], [74] that the following conditions are equivalent: 3 = 22 and the threefold X is the Mukai–Mumemura threefold; • −KX 3 • −KX > 16 and for any curve P1 ∼ = L ⊂ X such that −KX · L = 1, we have NL/X ∼ = OP1 1 ⊕ OP1 − 2 , 3 > 6 and for any two curves L ⊂ X ⊃ L such that • −KX 1 2

−KX · L1 = −KX · L2 = 1, either L1 = L2 , or L1 ∩ L2 = ∅. Remark 3.4. Let Z ⊂ P2 be a plane quartic curve. Then Z is given by an equation ζ x, y, z = 0 ⊂ P2 ∼ Proj C[x, y, z] , = where ζ(x, y, z) is a form of degree 4. A polar hexagon of the curve Z ⊂ P2 is the union Γ=

6 [

ξi x, y, z = 0 ⊂ P2

i=1

such that ζ(x, y, z) =

P6

4 i=1 ξi (x, y, z),

where ξi (x, y, z) is a non-zero linear form. Put n o Xζ = Γ ∈ Hilb6 P2 Γ is polar to Z ⊂ P2 ,

where we identify the polar hexagon Γ with a point in Hilb6 (P2 ). Then it follows from [167] that the variety Xζ is a smooth Fano threefold such that Pic Xζ = Z − KXζ 3 = 22 holds in the case when the homogeneous form ζ(x, y, z) is sufficiently and the equality −KX ζ general5. It follows from [130] that Xζ is the Mukai–Mumemura threefold if 2 ζ x, y, z = x2 + y 2 + z 2 .

5Varieties X that are smooth compactifications of C3 were studied in [138], [139], [63], [144], [64], [65]. ζ 33

4. Cubic surfaces Let X be a cubic surface in

P3

that has at most one ordinary double point.

Definition 4.1. A point O ∈ X is said to be an Eckardt point if O 6∈ Sing(X) and O = L1 ∩ L2 ∩ L3 , where L1 , L2 , L3 are different lines on the surface X ⊂ P3 . General cubic surfaces have no Eckardt points. It follows from Example 1.18 and 1.19 that ( 3/4 when X has no Eckardt points and Sing X = ∅ lct X = 2/3 when X has an Eckardt point or Sing X 6= ∅, Let D be an effective Q-divisor on X such that D ∼Q −KX , and let ω be a positive rational number such that ω < 3/4. In this section we prove the following result (cf. [20], [33], [31]). Theorem 4.2. Suppose that (X, ωD) is not log canonical. Then LCS X, ωD = O, where O ∈ X is either a singular point or an Eckardt point. Suppose that (X, ωD) is not log canonical. Let P be a point in LCS(X, ωD). Suppose that • neither P = Sing(X), • nor P is an Eckardt point. Lemma 4.3. One has LCS(X, ωD) = P . Proof. Suppose that LCS(X, ωD) 6= P . Then there is a curve C ⊂ X such that P ∈ C ⊆ LCS X, ωD by Theorem 2.7. Then there is an effective Q-divisor Ω on X such that C 6⊂ Supp(Ω) and D = µC + Ω, where µ > 1/ω. Let H be a general hyperplane section of X. Then 3 = H · D = µH · C + H · Ω > µdeg C , which implies that either deg(C) = 1, or deg(C) = 2. Suppose that deg(C) = 1. Let Z be a general conic on X such that −KX ∼ C + Z. Then ( 2µ if C ∩ Sing X = ∅ 2 = Z · D = µZ · C + Z · Ω > µZ · C = 3µ/2 if C ∩ Sing X 6= ∅, which implies that µ 6 4/3. But µ > 1/ω > 4/3, which gives a contradiction. We see that deg(C) = 2. Let L be a line on X such that −KX ∼ C + L. Then D = µC + λL + Υ, where λ ∈ Q such that λ > 0, and Υ is an effective Q-divisor such that C 6⊂ Supp(Υ) 6⊃ L. Then ( 2µ − λ if C ∩ Sing X = ∅ 1 = L · D = µL · C + λL · L + L · Υ > µL · C + λL · L = 3µ/2 − λ/2 if C ∩ Sing X 6= ∅, which implies that µ 6 7/6 < 4/3, because λ 6 4/3 (see the case when deg(C) = 1). But µ > 4/3, which gives a contradiction. Let π : U → X be a blow up of P , and let E be the π-exceptional curve. Then ¯ ∼Q π ∗ D + multP D E, D ¯ is a proper transform of D on the surface U . The log pair where multP (D) > 1/ω and D ¯ + ωmultP D − 1 E U, ω D 34

is not log canonical at some point Q ∈ E. Then either multP (D) > 2/ω, or ¯ + multP D > 2/ω > 8/3, (4.4) multQ D ¯ + (ωmultP (D) − 1)E is effective. because the divisor ω D Let T be the unique hyperplane section of X that is singular at P . We may assume that Supp T 6⊆ Supp D by Remark 2.23, because (X, ωT ) is log canonical. The following cases are possible: • the curve T is irreducible; • the curve T is a union of a line and an irreducible conic; • the curve T consists of 3 lines. Hence T is reduced. Note that multP (T ) = 2 since P is not an Eckardt point. We exclude these cases one by one. Lemma 4.5. The curve T is reducible. Proof. Suppose that T is irreducible. Then there is a commutative diagram U ~ @@@ ψ ~ π ~~ @@ @@ ~~ ~~ _ _ _ _ _ _ _ / P2 X ρ where ψ is a double cover branched over a quartic curve, and ρ is the projection from P ∈ X. Let T¯ be the proper transform of T on the surface U . Suppose that Q ∈ T¯. Then ¯ > multQ T¯ multQ D ¯ > multQ T¯ 8/3−multP D > 8/3−multP D , 3−2multP D = T¯ ·D which implies that multP (D) 6 1/3. But multP (D) > 4/3. Thus, we see that Q 6∈ T¯. Let τ ∈ Aut(U ) be an involution6 induced by ψ. It follows from [118] that ∗ ∗ τ π − KX ∼ π ∗ − 2KX − 3E, ˘ = π ◦ τ (Q). Then Q ˘ 6= P , because Q 6∈ T¯. and τ (T¯) = E. Put Q ˘ Then T 6= H, because P 6= Q ˘ Let H be the hyperplane section of X that is singular at Q. and T is smooth outside of the point P . Hence P 6∈ H, because otherwise 3 = H · T > multP H multP T + multQ˘ H multQ˘ T > 4. ¯ be the proper transform of H on the surface U . Put R ¯ = τ (H) ¯ and R = π(R). ¯ Then Let H ¯ ∼ π ∗ − 2KX − 3E, R ¯ must be singular at the point Q. ant the curve R ¯ we see that Suppose that R is irreducible. Taking into account the possible singularities of R, 3 X, R 8 is log canonical. Thus, we may assume that R 6⊆ Supp(D) by Remark 2.23. Then ¯·D ¯ > multQ (R)mult ¯ ¯ 6 − 3multP D = R Q D > 2 8/3 − multP D , which implies that multP (D) < 2/3. But multP (D) > 4/3. The curve R must be reducible. ˘ The curves R and H are reducible. So, there is a line L ⊂ X such that P 6∈ L 3 Q. ¯ ¯ ¯ ¯ Let L be the proper transform of L on the surface U . Put Z = τ (L). Then L · E = 0 and ¯ · T¯ = L ¯ · π ∗ (−KX ) = 1, L ¯ Then which implies that Z¯ · E = 1 and Z¯ · π ∗ − KX = 2. We have Q ∈ Z. ¯ > multQ D ¯ > 8/3 − multP D > 2 − multP D 2 − multP D = Z¯ · D ¯ Hence, we see that Z¯ ⊆ Supp(D). ¯ in the case when Z¯ ⊆ 6 Supp(D). 6The involution τ induces an involution in Bir(X) that is called a Geiser involution. 35

¯ Then Z is a conic such that P ∈ Z and Put Z = π(Z). −KX ∼ L + Z, which means that L ∪ Z is cut out by the plane in P3 that passes through Z. Put D = εZ + Υ, where ε ∈ Q such that ε > 0, and Υ is an effective Q-divisor such that Z 6⊂ Supp(Υ). We may assume that L 6⊆ Supp(Υ) by Remark 2.23. Then ( 2ε if Z ∩ Sing X = ∅, 1 = L · D = εZ · L + L · Υ > εZ · L = 3ε/2 if Z ∩ Sing X 6= ∅, which implies that ε 6 2/3. ¯ be the proper transform of Υ on the surface U . Then the log pair Let Υ ¯ + ωmultP D − 1 E U, εω Z¯ + ω Υ ¯ Then is not log canonical at Q ∈ Z. ¯ · Z¯ + ωmultP D − 1 = ω Υ ¯ + ωmultP D − 1 E · Z¯ > 1 ωΥ by Lemma 2.21, because ε 6 2/3. In particular, we see that ( 2 − multP D + ε if Z ∩ Sing X = ∅, ¯ = 2 − multP D − εZ¯ · Z¯ = 8/3 − multP D < Z¯ · Υ 2 − multP D + ε/2 if Z ∩ Sing X 6= ∅, which implies that ε > 2/3. But ε 6 2/3.

Therefore, there is a line L1 ⊂ X such that P ∈ L1 . Put D = m1 L1 + Ω, where m1 ∈ Q such that Ω is an effective Q-divisor such that L1 6⊂∈ Supp(Ω). Then ( 1 + m1 if L1 ∩ Sing X = ∅, 4/3 < 1/ω < Ω · L1 = 1 − m1 L1 · L1 = 1 + m1 /2 if L1 ∩ Sing X 6= ∅. Corollary 4.6. The following inequality holds: ( 1/3 if L1 ∩ Sing X = ∅, m1 > 2/3 if L1 ∩ Sing X = 6 ∅. Remark 4.7. Suppose that X is singular. Put O = Sing(X). It follows from [16] that O = Γ1 ∩ Γ2 ∩ Γ3 ∩ Γ4 ∩ Γ5 ∩ Γ6 , where Γ1 , . . . , Γ6 are different lines on the surface X ⊂ P3 . The equivalence −2KX ∼

6 X

Γi

i=1

holds. Suppose that L1 = Γ1 . Let Π2 , . . . , Π6 ⊂ P3 be planes such that L1 ⊂ Πi ⊃ Γi , and let Λ2 , . . . , Λ6 be lines on the surface X such that L1 ∪ Γi ∪ Λi = Πi ∩ X ⊂ X ⊂ P3 , which implies that −KX ∼ L1 + Γi + Λi . Then −5KX ∼ 4L1 +

6 X i=2

6 6 X X Λi + L1 + Γi ∼ 4L1 + Λi − 2KX , i=2 36

i=2

which implies that −3KX ∼ 4L1 +

P6

i=2 Λi .

But the log pair ! P6 Λ i X, L1 + i=2 3

is log canonical at the point P . Thus, we may assume that ! 6 X Supp Λi 6⊆ Supp D i=2

thanks to Remark 2.23, because L1 ⊆ Supp(D). Then there is Λk such that 1 = D · Λk = m1 L1 + Ω · Λk = m1 + Ω · Λk > m1 , because O 6∈ Λk . Thus, we may assume that m1 6 1 if L1 ∩ Sing(X) 6= ∅. Arguing as in the proof of Lemma 2.15, we see that m1 6 1 if L1 ∩ Sing(X) = ∅. Lemma 4.8. There is a line L2 ⊂ X such that L1 6= L2 and P ∈ L2 . Proof. Suppose that there is no line L2 ⊂ X such that L1 6= L2 and P ∈ L2 . Then T = L1 + C, where C is an irreducible conic on the surface X ⊂ P3 such that P ∈ C. It follows from Remark 2.23 that we may assume that C 6⊆ Supp(Ω), because m1 6= 0. ¯ 1 and C¯ be the proper transforms of L1 and C on the surface U , respectively. Then Let L ¯ ∼Q m1 L ¯1 + Ω ¯ ∼Q π ∗ m1 L1 + Ω − m1 + multP Ω E ∼Q π ∗ D − multP D E, D ¯ is the proper transform of the divisor Ω on the surface U . We have where Ω ( 2/3 − m1 , if L1 ∩ Sing X = ∅, ¯ = 2 − multP D + m1 C¯ · L ¯ < 2/3 − m1 C¯ · L ¯1 = 0 6 C¯ · Ω 2/3 − m1 /2, if L1 ∩ Sing X 6= ∅, which implies that m1 < 2/3 if L1 ∩ Sing(X) = ∅. It follows from inequality 4.4 that ¯ > 8/3 − multP Ω − m1 1 + multQ L ¯1 . multQ Ω ¯ 1 . Then it follows from Lemma 2.21 that Suppose that Q ∈ L ( 1 + 2m1 , if L1 ∩ Sing X = ∅, ¯1 · Ω ¯ + multP Ω + m1 E = 1 − m1 L ¯1 · L ¯1 = 8/3 < L 1 + 3m1 /2, if L1 ∩ Sing X 6= ∅, which is impossible, because m1 6 1 if L1 ∩ Sing(X) 6= ∅ by Remark 4.7. ¯ 1 . Suppose that Q ∈ C. ¯ Then We see that Q 6∈ L ¯ = C¯ · Ω ¯ > 8/3 − multP Ω − m1 , 2 − multP Ω − m1 − m1 C¯ · L1 ¯ 1 > 0. Hence, we see that Q 6∈ C. ¯ which is impossible, because m1 C¯ · L There is a commutative diagram U π

ζ

/W

ψ

X _ _ _ ρ_ _ _/ P2 ¯ 1 , the morphism ψ is a double cover where ζ is a birational morphism that contracts the curve L branched over a quartic curve, and ρ is a linear projection from the point P ∈ X. Let τ be the birational involution of U induced by ψ. Then • the involution τ is biregular ⇐⇒ L1 ∩ Sing(X) = ∅, ¯ 1 if L1 ∩ Sing(X) 6= ∅, • the involution τ acts biregularly on U \ L ¯ • it follows from the construction of τ that τ (E) = C, 37

• if L1 ∩ Sing(X) = ∅, then ¯1 ∼ L ¯ 1 , τ ∗ E ∼ C, ¯ τ ∗ π ∗ − KX ∼ π ∗ − 2KX − 3E − L ¯ 1. τ∗ L Let H be the hyperplane section of X that is singular at π ◦ τ (Q) ∈ C. Then P 6∈ H, because ¯ be the proper transform of H on the surface U . Then C is smooth. Let H ¯ 1 6⊆ Supp H ¯ 6⊇ C, ¯ L ¯ = τ (H) ¯ and R = π(R). ¯ Then R ¯ is singular at the point Q, and and we can put R ¯ ∼ π ∗ − 2KX − 3E − L ¯ 1, R because R does not pass through a singular point of X if Sing(X) 6= ∅. Suppose that R is irreducible. Then R + L1 ∼ −2KX , but the log pair 3 X, R + L1 8 is log canonical. Thus, we may assume that R 6⊆ Supp(D) by Remark 2.23. Then ¯ > 2 8/3 − m1 − multP Ω , ¯1 · L ¯1 = R ¯·Ω ¯ > 2multQ Ω 5 − 2 m1 + multP Ω + m1 1 + L which implies that m1 < 0. The curve R must be reducible. There is a line L ⊂ X such that P 6∈ L and π ◦ τ (Q) ∈ L. Then L ∩ L1 = ∅, because π ◦ τ (Q) ∈ C and (C + L1 ) · L = T · L = 1. Thus, there is unique conic Z ⊂ X such that −KX ∼ L + Z and P ∈ Z. Then Z is irreducible and P = Z ∩ L1 , because (L + Z) · L1 = 1. ¯ and Z¯ be the proper transform of L and Z on the surface U , respectively. Then Let L ( 2 if L ∩ Sing X = ∅, ¯ C¯ = Z¯ ·E = 1, L ¯ 1 · Z¯ = L·E ¯ ¯ L ¯ 1 = 0, Z¯ · Z¯ = 1− L· ¯ Z, ¯ L· ¯ Z¯ = L· = L· 3/2 if L ∩ Sing X 6= ∅. ¯ = L. ¯ Then Q ∈ Z. ¯ Suppose that Z¯ 6⊆ Supp(Ω). ¯ Then We have τ (Z) ¯ > 8/3 − m1 − multP Ω , 2 − m1 − multP Ω = Z¯ · Ω ¯ But the log pair which is impossible. Thus, we see that Z¯ ⊆ Supp(Ω). X, ω(L + Z) ¯⊆ ¯ by Remark 2.23. Put is log canonical at the point P . Hence, we may assume that L 6 Supp(Ω) D = εZ + m1 L1 + Υ, where Υ is an effective Q-divisor such that Z 6⊂ Supp(Υ) 6⊃ L1 . Then ( 2ε if L ∩ Sing X = ∅, 1 = L · D = εL · Z + m1 L · L1 + L · Υ = εL · Z + L · Υ > εL · Z = 3ε/2 if L ∩ Sing X 6= ∅, ¯ 1 = ∅. Then it follows from Lemma 2.21 that which implies that ε 6 2/3. But Z¯ ∩ L ¯ > 8/3 − multP D , 2 − multP D − εZ¯ · Z¯ = Z¯ · Υ ¯ is a proper transform of Υ on the surface U . We deduce that ε > 2/3. But ε 6 2/3. where Υ Therefore, we see that T = L1 + L2 + L3 , where L3 is a line such that P 6∈ L3 . Put D = m1 L1 + m2 L2 + ∆, where ∆ is an effective Q-divisor such that L2 6⊆ Supp(∆) 6⊇ L2 . The inequalities m1 > 1/3 and m2 > 1/3 hold by Corollary 4.6. We may assume that L3 6⊆ Supp ∆ by Remark 2.23. If the singular point of X (provided that there exists one) is contained in either L1 or L2 , we may assume without loss of generality that it is contained in L1 . Then L3 · L2 = 1 and L3 · L1 = 1/2 in the case when L1 ∩ Sing(X) 6= ∅, and L3 · L2 = L3 · L1 = 1 in the case when L1 ∩ Sing(X) = ∅. Then 1 − m1 L1 · L3 − m2 = L3 · ∆ > 0. 38

¯ 1 and L ¯ 3 be the proper transforms of L1 and L2 on the surface U , respectively. Then Let L ¯ 1 + m2 L ¯2 + ∆ ¯ ∼Q π ∗ m1 L1 + m2 L2 + ∆ − m1 + m2 + multP ∆ E, m1 L ¯ is the proper transform of ∆ on the surface U . The inequality 4.4 implies that where ∆ ¯2 . ¯ 1 − m1 1 + multQ L ¯ > 8/3 − multP ∆ − m1 1 + multQ L (4.9) multQ ∆ ¯ 2 does not contain the point Q. Lemma 4.10. The curve L ¯ 2 . Then Proof. Suppose that Q ∈ L ¯2 · ∆ ¯ > 8/3 − multP ∆ − m1 − m2 1 − multP ∆ − m1 + m2 = L by Lemma 2.21. Thus, we have m2 > 5/6. It follows from Lemma 2.21 that 1 − m2 − m1 L1 · L1 = ∆ · L1 > 4/3 − m2 , but L1 · L1 = −1 if L1 ∩ Sing(X) = ∅, and L1 · L1 = −1/2 if L1 ∩ Sing(X) 6= ∅. Then ( 1/3 if L1 ∩ Sing X = ∅, m1 > 2/3 if L1 ∩ Sing X 6= ∅, by Corollary 4.6, which is impossible because m2 > 5/6 and 1 > m1 L1 · L3 + m2 . ¯ 1 does not contain the point Q. Lemma 4.11. The curve L ¯ 1 . Arguing as in the proof of Lemma 4.10, we see that Proof. Suppose that Q ∈ L L1 ∩ Sing X 6= ∅, ¯1 · L ¯ 1 = −1/2. Then m1 > 10/9, because which implies that L ¯2 · ∆ ¯ + multP ∆ − m1 − m2 E > 8/3 1 + 3m1 /2 = L by Lemma 2.21. But m1 6 1 by Remark 4.7.

¯1 3 ¯ 2 . There is a commutative diagram Therefore, we see that L 6 Q 6∈ L U π

ζ

/W

ψ

X _ _ _ ρ_ _ _/ P2 ¯ 1 and L ¯ 2 , the morphism ψ is a where ζ is a birational morphism that contracts the curves L double cover branched over a quartic curve, and ρ is the projection from the point P . Let τ be the birational involution of U induced by ψ. Then • the involution τ is biregular ⇐⇒ L1 ∩ Sing(X) = ∅, ¯ 1 if L1 ∩ Sing(X) 6= ∅, • the involution τ acts biregularly on U \ L ¯ 2) = L ¯ 2 holds, • the equality τ (L ¯ 1) = L ¯ 1 and • if L1 ∩ Sing(X) = ∅, then τ (L ¯1 − L ¯ 2. τ ∗ π ∗ − KX ∼ π ∗ − 2KX − 3E − L ¯ 3 be a proper transform of L3 on the surface U . Then τ (E) = L ¯ 3 and Let L L1 ∪ L2 63 π ◦ τ Q ∈ L3 . Lemma 4.12. The line L3 is the only line on X that passes through the point π ◦ τ (Q). 39

Proof. Suppose that there is a line L ⊂ X such that L 6= L3 and π ◦ τ (Q) ∈ L. Then L ∩ L1 = L ∩ L2 = ∅, because π ◦ τ (Q) ∈ L3 and (L1 + L2 + L3 ) · L = 1. Thus, there is unique conic Z ⊂ X such that −KX ∼ L + Z and P ∈ Z. Then Z is irreducible, because P 6∈ L and P is not an Eckardt point. ¯ and Z¯ be the proper transform of L and Z on the surface U , respectively. Then Let L ( 2 if L ∩ Sing X = ∅, ¯·L ¯ 3 = Z¯ · E = 1, Z¯ · Z¯ = 1 − L ¯ · Z, ¯ L ¯ · Z¯ = L 3/2 if L ∩ Sing X 6= ∅, ¯ 1 · Z¯ = L ¯ 2 · Z¯ = L ¯·E =L ¯·L ¯1 = L ¯·L ¯ 2 = 0. and L ¯ ¯ ¯ ¯ because We have τ (Z) = L. Then Q ∈ Z, which implies that Z¯ ⊆ Supp(∆), ¯ > 8/3 − multP ∆ − m1 − m2 2 − multP ∆ − m1 − m2 = Z¯ · Ω ¯ On the other hand, the log pair in the case when Z¯ 6⊆ Supp(∆). X, ω(L + Z) ¯⊆ ¯ by Remark 2.23. Put is log canonical at the point P . Hence, we may assume that L 6 Supp(∆) D = εZ + m1 L1 + m2 L2 + Υ, where Υ is an effective Q-divisor such that Z 6⊆ Supp(Υ). Then ( 1 = L · D = εL · Z + m1 L · L1 + L · Υ = εL · Z + L · Υ > εL · Z =

2ε if L ∩ Sing X = ∅, 3ε/2 if L ∩ Sing X 6= ∅,

¯ 1 = ∅. Then it follows from Lemma 2.21 that which implies that ε 6 2/3. But Z¯ ∩ L ¯ > 8/3 − multP D , 2 − multP D − εZ¯ · Z¯ = Z¯ · Υ ¯ is a proper transform of Υ on the surface U . We deduce that ε > 2/3. But ε 6 2/3. where Υ Therefore, there is an unique irreducible conic C ⊂ X such that −KX ∼ L3 + C and π ◦ τ (Q) ∈ C. Then C + L3 is a hyperplane section of X that is singular at π ◦ τ (Q). ¯ and Z = π(Z). ¯ Let C¯ be the proper transform of C on the surface U . Put Z¯ = τ (C) Lemma 4.13. One has L1 ∩ Sing(X) 6= ∅. Proof. Suppose that L1 ∩ Sing(X) = ∅. Then C ∩ L1 = C ∩ L2 = ∅, because (L1 + L2 + L3 ) · C = L3 · C = 2. One can easily check that ¯1 − L ¯ 2, Z¯ ∼ π ∗ − 2KX − 4E − L and Z is singular at P . Then −2KX ∼ Z + L1 + L2 . But the log pair 1 U, Z + L1 + L2 2 is log canonical at P . Thus, we may assume that Z 6⊆ Supp(D) by Remark 2.23. We have Q ∈ Z¯ and Z¯ · E = 2. Then it follows from the inequality 4.4 that ¯ > multQ D ¯ > 8/3 − multP D , 4 − 2multP D = Z¯ · D which implies that multP (D) < 4/3. But multP (D) > 4/3. Thus, we see that L1 ∩ L3 = Sing(X) 6= ∅. Then L1 ∩ L2 ∈ C, which implies that ¯ 2, ¯1 − L Z¯ ∼ π ∗ − 2KX − 4E − 2L and Z is smooth rational cubic curve. Then −2KX ∼ Z + 2L1 + L2 . But the log pair 1 Z + 2L1 + L2 U, 2 40

is log canonical at P . Thus, we may assume that Z 6⊆ Supp(D) by Remark 2.23. ¯ 1 = 1. Then it follows from the inequality 4.4 that We have Q ∈ Z¯ and Z¯ · E = L ¯ > 8/3 − multP ∆ − m1 − m2 , ¯ > multQ ∆ 3 − multP ∆ − 2m1 − m2 = Z¯ · ∆ which implies that m1 < 1/3. But m1 > 2/3 by Corollary 4.6. The obtained contradiction completes the proof Theorem 4.2. 5. Del Pezzo surfaces Let X be a del Pezzo surface that has at most canonical singularities, let O be a point of the surface X, and let BX be an effective Q-divisor on the surface X. Suppose that • the point O is either smooth or an ordinary double point of X, • the surface X is smooth outside the point O ∈ X. 2 = 2 and the equivalence Lemma 5.1. Suppose that Sing (X) = O, KX

BX ∼Q −µKX holds, where 0 < µ < 2/3. Then LCS(X, µBX ) = ∅. Proof. Suppose that LCS(X, µBX ) 6= ∅. There is a curve P1 ∼ = L ⊂ X such that LCS X, µBX 6⊆ L the equality L · L = −1 holds, and L ∩ Sing(X) = ∅. Thus, there is a birational morphism π : X → S that contracts the curve L. Then LCS S, µπ BX 6= ∅ due to the choice of the curve L ⊂ X. But −KS ∼Q π(BX ), and S ⊂ P3 is a cubic surface that has at most one ordinary double point, which is impossible (see Examples 1.19 and 1.18). 2 = 5, the equivalence Lemma 5.2. Suppose that Sing(X) = ∅, KX

BX ∼Q −µKX holds, where µ ∈ Q is such that 0 < µ < 2/3. Assume that LCS(X, BX ) 6= ∅. Then • either the set LCS(X, BX ) contains a curve, • or there are a curve P1 ∼ = L ⊂ X and a point P ∈ L such that L · L = −1 and LCS X, BX = P. Proof. Suppose that LCS(X, BX ) contains no curves. Then it follows from Theorem 2.7 that LCS X, BX = P, where P ∈ X is a point. We may assume that P does not lie on any curve P1 ∼ = L ⊂ X such that the equality L · L = −1 holds. Then there is a birational morphism φ : X −→ P2 such that φ is an isomorphism in a neighborhood of the point P . Note that φ P ∈ LCS P2 , φ BX 6= ∅, the set LCS(P2 , φ(BX )) contains no curves, and φ BX ∼Q −µKP2 . Since µ < 2/3, the latter is impossible by Lemma 2.8.

2 = 5. Let α : V → X be a blow up of O, and Example 5.3. Suppose that O = Sing(X) and KX let E be the exceptional divisor of α. Then there is a birational morphism ω : V → P2 such that • the morphism ω contracts the curves E1 , E2 , E3 , E4 , • the curve ω(E) is a line in P2 that contains ω(E1 ), ω(E2 ), ω(E3 ), but ω(E) 63 ω(E4 ). 41

Let Z be a line in P2 such that ω(E1 ) ∈ Z 3 ω(E4 ). Then 2E + Z¯ + 2E1 + E2 + E3 ∼ −KV , where Z¯ ⊂ V is a proper transform of Z. One has 1 lct X, α Z¯ + 2α E1 + α E2 + α E3 = , 2 which implies lct(X) 6 1/2. Suppose that −KX ∼Q 2BX , but (X, BX ) is not log canonical. Then KV + BV + mE ∼Q α∗ KX + BX , for some m > 0, and BV is the proper transform of BX on the surface V . Then V, BV + mE is not log canonical at some point P ∈ V . There is a birational morphism π : V → U such that • the surface U is a smooth del Pezzo surface with KU2 = 6, • the morphism π is an isomorphism in a neighborhood of P ∈ X, which implies that (U, π(BV ) + mπ(E)) is not log canonical at π(P ). But 1 π BV + mπ E ∼Q − KU , 2 which is impossible, because lct(U ) = 1/2 (see Example 1.18). We see that lct(X) = 1/2. 2 = 4. Arguing as in Example 5.3, we see that the equality Example 5.4. Suppose that KX ( 1/2 when O = Sing X , lct X = 2/3 when Sing X = ∅,

holds (cf. [189]). Take λ 6 1. Suppose that BX ∼Q −KX , and (X, λBX ) is not log canonical at some point P ∈ X \ O. There is a commutative diagram V } @@@ β } } @@ }} @@ }~ } X _ _ _ _ _ _ _/ U α

ψ

where U is a cubic surface in P3 that has canonical singularities, the morphism α is a blow up of the point P , the morphism β is birational, and ψ is a projection from the point P ∈ X. Then KV + λBV + λmultP BX − 1 E ∼Q α∗ KX + λBX , where E is the exceptional divisor of α, and BV is the proper transform of BX . Note that V, λBV + λmultP BX − 1 E is not log canonical at some point Q ∈ E and multP (BX ) > 1/λ. Then the log pair V, λBV + λmultP BX − λ E is not log canonical at the point Q ∈ E as well. But the equivalences BV + multP BX − 1 E ∼Q −KV + α∗ KX + BX ∼Q −KV , hold. Suppose that P is not contained in any line on the surface X. Then • the morphism β : V → U is an isomorphism, • the cubic surface U is smooth outside of the point ψ(O), • the point ψ(O) is at most ordinary double point of the surface U , 42

which implies that λ > 2/3 (see Example 1.19). Suppose that λ = 3/4. Then the point ψ Q ∈ U ⊂ P3 must be an Eckardt point (see Definition 4.1) of the surface U by Theorem 4.2. But β E ⊂ U ⊂ P3 is a line. So, there are two conics C1 6= C2 contained in X such that P = C1 ∩ C2 and C1 + C2 ∼ −KX . 2 = 6 such that there is a diagram Lemma 5.5. Suppose that O = Sing(X) and KX

~~ ~~ ~ ~ ~ ~ α

X

V @ @

@@ β @@ @@

P2

where β is a blow up of three points P1 , P2 , P3 ∈ P2 lying on a line L ⊂ P2 , and α is a birational ¯ to the point O such that β(L) ¯ = L. Then morphism that contracts an irreducible curve L LCS X, λBX = O in the case when LCS(X, λBX ) 6= ∅, BX ∼Q −KX and λ < 1/2. Proof. Suppose that BX ∼Q −KX and ∅ 6= LCS X, λBX 6= O, ¯ ⊂ V be its proper transform. Then let M ⊂ P2 be a general line, and let M ¯ −KX ∼ 2α M ¯ ). Thus, the set LCS(X, λBX ) contains a curve, because otherwise the locus and O ∈ α(M ¯ LCS X, λBX + α M would be disconnected, which is impossible by Theorem 2.7. Let C ⊂ X be an irreducible curve such that C ⊆ LCS(X, λBX ). Then BX = εC + Ω, where ε > 2, and Ω is an effective Q-divisor such that C 6⊂ Supp(Ω). Let Γi ⊂ X be a proper transform of a general line in P2 that passes through Pi . Then O 6∈ Γ1 ∪ Γ2 ∪ Γ3 and −KX · Γ1 = −KX · Γ2 = −KX · Γ3 = 2. But −KX ∼Q Γ1 + Γ2 + Γ3 , which implies that there is m ∈ {1, 2, 3} such that C · Γm 6= 0. Then 2 = BX · Γm = εC + Ω · Γm > εC · Γm > ε > 2, because Γm 6⊂ Supp(BX ). The obtained contradiction completes the proof.

2 = 6. Let α : V → X be a blow up of the point Remark 5.6. Suppose that O = Sing(X) and KX O ∈ X, and let E be the exceptional divisor of α. Then ∗ KV + BV + mE ∼Q α KX + BX

for some m > 0, and BV is the proper transform of BX on V . Note that lct(X) 6 1/3. Suppose that lct(X) < 1/3, i. e. there exists an effective Q-divisor BX ∼Q −KX , such that the log pair (X, 1/3BX ) is not log canonical. Then the log pair 1 V, BV + mE 3 43

is not log canonical at some point P ∈ V . There is a birational morphism π : V −→ U such that U is either F1 or P1 × P1 and π is an isomorphism in a neighborhood of P ∈ X. Then the log pair 1 U, π(BV ) + mπ(E) 3 is not log canonical at the point π(P ). But we know that −KU ∼Q π BV + mπ E , so the latter contradicts Example 1.18. Hence lct(X) = 1/3. Lemma 5.7. Suppose that X ∼ = P(1, 1, 2) and BX ∼Q −KX , but there is a point P ∈ X such that O 6= P ∈ LCS X, λBX for some λ < 1/2. Take L ∈ |OP(1,1,2) (1)| such that P ∈ L. Then L ⊆ LCS(X, λBX ). Proof. Suppose that there is a curve Γ ∈ LCS(X, λBX ) such that P ∈ Γ 6= L. Then BX = µΓ + Ω, where µ > 2, and Ω is an effective Q-divisor such that Γ 6⊂ Supp(Ω). Hence µΓ + Ω ∼Q 4L and Γ ∼ mL, where m ∈ Z>0 . But m > 2, because P ∈ Γ 6= L, which is a contradiction. Suppose that L 6⊆ LCS(X, λBX ). Then it follows from Theorem 2.7 that LCS X, λBX = P, because we proved that LCS(X, λBX ) contains no curves passing through P ∈ X. Let C ∈ |OP(1,1,2) (1)| be a general curve. Then LCS X, λBX + C = P ∪ C, which is impossible by Theorem 2.7.

Lemma 5.8. Suppose that X ∼ = F1 . Then there are 0 6 µ ∈ Q 3 λ > 0 such that BX ∼Q µC + λL, where C and L are irreducible curves on the surface X such that C · C = −1, C · L = 1 and L · L = 0. Suppose that µ < 1 and λ < 1. Then LCS(X, BX ) = ∅. Proof. The set LCS(X, BX ) contains no curves, because L and C generate the cone of effective divisors of the surface X. Suppose that LCS(X, BX ) contains a point O ∈ X. Then KX + BX + 1 − µ C + 2 − λ L ∼Q − L + C , because −KX ∼Q 2C + 3L. But it follows from Theorem 2.6 that the map 0 = H 0 OX − L − C −→ H 0 OL(X, BX ) 6= 0 is surjective, because the divisor (1 − µ)C + (2 − λ)L is ample. 2 = 7. Then Lemma 5.9. Suppose that Sing(X) = ∅ and KX

L1 · L1 = L2 · L2 = L3 · L3 = −1, L1 · L2 = L2 · L3 = 1, L1 · L3 = 0 where L1 , L2 , L3 are exceptional curves on X. Suppose that LCS(X, BX ) 6= ∅ and BX ∼Q −µKX , where µ < 1/2. Then LCS(X, BX ) = L2 . 44

Proof. Let P be a point in LCS(X, BX ). Then P ∈ L2 , because lct(P1 × P1 ) = 1/2, and there is a birational morphism X → P1 × P1 that contracts only the curve L2 . Suppose that LCS(X, BX ) 6= L2 . Then LCS(X, BX ) = P by Theorem 2.7. We may assume that P 6∈ L3 . There is a birational morphism φ : X → P2 that contracts the curves L1 and L3 . Let C1 and C3 be proper transforms on X of sufficiently general lines in P2 that pass through the points φ(L1 ) and φ(L3 ), respectively. Then −KX ∼ C1 + 2C3 + L3 , and C1 63 P 6∈ C2 . Therefore, we see that 1 ⊆ C2 ∪ P ∪ L3 , C2 ∪ P ⊆ LCS X, λD + C1 + 2C2 + L3 2 which is impossible by Theorem 2.7, because P 6∈ L3 .

2 = 7 holds, the equivalence Lemma 5.10. Suppose that O = Sing(X), the equality KX

4 BX ∼Q C + L 3 1 ∼ ∼ holds, where L = P = C are curves on the surface X such that L · L = −1/2, C · C = −1, C · L = 1, but the log pair (X, BX ) is not log canonical at some point P ∈ C. Then P ∈ L. Proof. Let S be a quadratic cone in P3 . Then S ∼ = P(1, 1, 2) and there is a birational morphism φ : X −→ S ⊂ P3 that contracts the curve C to a smooth point Q ∈ S. Then Q ∈ φ(L) ∈ |OP(1,1,2) (1)|. Suppose that P 6∈ L. Then it follows from Remark 2.23 that to complete the proof we may assume that either C 6⊂ Supp(BX ) or L 6⊂ Supp(BX ), because the log pair 4 X, C + L 3 is log canonical in the point P ∈ X. Suppose that C 6⊂ Supp(BX ). Then 1 = BX · C > multP BX > 1, 3 which is impossible. Therefore, we see that C ⊂ Supp(BX ). Then L 6⊂ Supp(BX ). Put BX = εC + Ω, where Ω is an effective Q-divisor such that C 6⊂ Supp(Ω). Then 1 = BX · L = ε + Ω · L > ε, 3 which implies that ε 6 1/3. Then 1 < Ω · C = 1/3 + ε 6 2/3 by Lemma 2.21, which is a contradiction.

6. Toric varieties The purpose of this section is to prove Lemma 6.1 (cf. [7], [174], [39]). Let N = Zn be a lattice of rank n, and let M = Hom(N, Z) be the dual lattice. Put MR = M ⊗Z R and NR = N ⊗Z R. Let X be a toric variety defined by a complete fan Σ ⊂ NR , let ∆1 = v1 , . . . , v m be a set of generators of one-dimensional cones of the fan Σ. Put

n o ∆ = w ∈ M w, vi > −1 for all i = 1, . . . , m . Put T = (C∗ )n ⊂ Aut(X). Let N be the normalizer of T in Aut(X) and W = N /T . 45

Lemma 6.1. Let G ⊂ W be a subgroup. Suppose that X is Q-factorial. Then 1 o, n

lct X, G = 1 + max w, v w ∈ ∆G , v ∈ ∆1 where ∆G is the set of the points in ∆ that are fixed by the group G. Proof. Put µ = 1 + max{hw, vi | w ∈ ∆G , v ∈ ∆1 }. Then µ ∈ Q is the largest number such that −KX ∼Q λR + H, where R is an integral T o G-invariant effective divisor, and H is an effective Q-divisor. Hence 1 lct X, G 6 . µ Suppose that lct(X, G) < 1/µ. Then there is an effective G-invariant Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some rational λ < 1/µ. There exists a family {Dt | t ∈ C} of G-invariant effective Q-divisors such that • the equivalence Dt ∼Q D holds for every t ∈ C, • the equality D1 = D holds, • for every t 6= 0 there is φt ∈ Aut(X) such that ∼ D, Dt = φt D = • the divisor D0 is T -invariant, which implies that (X, λD0 ) is not log canonical (see [49]). On the other hand, the divisor D0 does not have components with multiplicity greater then µ, which implies that (X, λD0 ) is log canonical (see [61], [121]), which is a contradiction. Corollary 6.2. Let X = PPn (OPn ⊕ OPn (−a1 ) ⊕ . . . ⊕ OPn (−ak )), ai > 0 for i = 1, . . . , k. Then 1 o. n lct(X) = P 1 + max k, n + ki=1 ai Proof. Note that X is a toric variety, and ∆1 consists of the following vectors: k

n

z }| { z }| { (1, 0, . . . , 0, 0, 0, . . . , 0), ... (0, . . . , 0, 1, 0, 0, . . . , 0), (−1, . . . , −1, 0, 0, . . . , 0), (0, 0, . . . , 0, 1, 0, . . . , 0), ... (0, 0, . . . , 0, 0, . . . , 0, 1), (−a1 , . . . , −ak , −1, . . . , −1), which implies the required assertion by Lemma 6.1.

Applying Corollary 6.2, we obtain the following result. Corollary 6.3. In the notation of section 1 one has ( 1/4 whenever ‫(ג‬X) ∈ {2.33, 2.35}, lct X = 1/5 whenever ‫(ג‬X) = 2.36. On the other hand, straightforward computations using Lemma 6.1 imply the following result. Corollary 6.4. In the notation of section 1 one has   1/3 whenever ‫(ג‬X) ∈ {3.25, 3.31, 4.9, 4.11, 5.2},  lct X = 1/4 whenever ‫(ג‬X) ∈ {3.26, 3.30, 4.12},   1/5 whenever ‫(ג‬X) = 3.29. 46

Another application of Lemma 6.1 is an immediate proof of the following result, obtained in [77]. Example 6.5. Let X be a blow up of Pn , n > 2, at k + 1 points that are not contained in a linear subspace of dimension k − 1. Assume that k 6 n. Then the points are in general position (i. e. no r + 1 of them are contained in a linear subspace of dimension r − 1), so that X is toric, and 1 lct X = n+1 by Lemma 6.1. Note that X is a Fano variety only if k = 1, or n = 2 and k 6 3 (cf. [77, Chapter 5]). Remark 6.6. Let X be a blow up of Pn , n > 2, at n + 1 points that are not contained in a linear subspace of dimension n − 1. Then X is not Fano whether n > 3, but the points are still in general position, so that X is toric, and 1 lct X = n by Lemma 6.1. On the other hand, there is a natural action of the symmetric group Sn+1 on X such that lct(X, Sn+1 ) = 1. 7. Del Pezzo threefolds We use the assumptions and notation of Theorem 1.78. Suppose that −KX ∼ 2H, where H is a Cartier divisor that is indivisible in Pic(X). The purpose of this section is to prove the following result. Theorem 7.1. The equality lct(X) = 1/2 holds, unless ‫(ג‬X) = 2.35 when lct(X) = 1/4. It follows from Theorems 3.1.14 and 3.3.1 in [98] that n o ‫(ג‬X) ∈ 1.11, 1.12, 1.13, 1.14, 1.15, 1.17, 2.32, 2.35, 3.27 , and by [20] and [28] (see also Lemma 2.18) one has lct(X) = 1/2 if ‫(ג‬X) ∈ {1.12, 1.13}. It follows from Lemma 2.30 that lct(X) = 1/2 when ‫(ג‬X) = 3.27. Lemma 7.2. Suppose that ‫(ג‬X) = 2.35. Then lct(X) = 1/4. Proof. There is a birational morphism π : X → P3 that contracts a surface E ∼ = P2 to a point 3 P ∈ P . Hence lct(X) 6 1/4, because −KX ∼ 2E + 4T, where T is the proper transform of a plane in P3 that passes through P ∈ P3 . Suppose that lct(X) < 1/4. Then there is an effective Q-divisor D such that 1 D ∼Q E + T, 2 but the log pair (X, λD) is not log canonical for some rational number λ < 1. Let R be a proper transform on X of a general plane in P3 . Then LCS X, λD + R must be connected by Theorem 2.7, because −(KX + λD + R) is ample. Thus, we see that • the subscheme L(X, λD) is not zero-dimensional, • the locus LCS(X, λD) is not contained in E, which implies that (P3 , λπ(D)) is not log canonical. But lct(P3 ) = 1/4 and π(D) ∼Q OP3 (1), which is a contradiction. Remark 7.3. Actually, the assertion of Lemma 7.2 is contained in Corollary 6.3, but we still prefer to give a detailed proof that may have further applications. 47

We are left with the cases n o ‫(ג‬X) ∈ 1.11, 1.14, 1.15, 2.32 , while the inequality lct(X) 6 1/2 is obvious, because |H| is not empty. Lemma 7.4. Suppose that ‫(ג‬X) = 2.32. Then lct(X) = 1/2. Proof. We may suppose that lct(X) < 1/2. Then there is an effective Q-divisor D ∼Q H such that the log pair (X, λD) is not log canonical for some positive rational number λ < 1. The threefold X is a divisor on P2 × P2 of bi-degree (1, 1). There are two natural P1 -bundles π1 : X → P2 and π2 : X → P2 , and applying Theorem 2.28 to π1 and π2 we immediately obtain a contradiction. Remark 7.5. Suppose that Pic(X) = Z[H], and there is an effective Q-divisor D ∼Q H such that the log pair (X, λD) is not log canonical for some positive rational number λ < 1. Put D = εS + Ω ∼Q H, where S is an irreducible surface and Ω is an effective Q-divisor such that Supp Ω 6⊃ S. Then ε 6 1, because Pic(X) = Z[H], which implies that the set LCS(X, λD) contains no surfaces. Moreover, for any choice of H ∈ |H| the locus LCS X, λD + H is connected by Theorem 2.7. Let H be a general surface in |H|. Since LCS(X, λD + H) is connected, one obtains that the locus LCS(X, λD + H) has no isolated zero-dimensional components outside the base locus of the linear system |H|. Note that |H| has no base points at all, unless ‫(ג‬X) = 1.11 when Bs|H| consists of a single point O. Note that in the latter case O 6∈ LCS(X, λD), since X is covered by the curves of anticanonical degree 2 passing through O. Hence the locus LCS(X, λD) never has isolated zero-dimensional components; in particular, it ¯ Then contains an (irreducible) curve C. Put D|H = D. ¯ −KH ∼ H ∼Q D, H

¯ is not log canonical in every point of the intersection H ∩ C. The locus LCS(H, λD) ¯ but (H, λD) ¯ is zero-dimensional. We see that is connected by Theorem 2.7. But the scheme L(H, λD) H · C = H ∩ C = 1, and the locus LCS(X, λD) contains no curves besides the irreducible curve C. Lemma 7.6. Suppose that ‫(ג‬X) = 1.15. Then lct(X) = 1/2. Proof. We may suppose that lct(X) < 1/2. Then there is an effective Q-divisor D ∼Q H such that the log pair (X, λD) is not log canonical for some positive rational number λ < 1. The linear system |H| induces an embedding X ⊂ P6 . Thus, it follows from Remark 7.5 that the locus LCS(X, λD) consists of a single line C ⊂ X ⊂ P6 . It follows from [98, Proposition 3.4.1] (see [92] and [67]) that there is a commutative diagram V ? ?? β ~~ ~ ?? ~ ~ ?? ~ ~ ~ _ _ _ _ _ _ _/ Q X α

ψ

where Q is a quadric in P4 , the morphism α is a blow up of C, the morphism β is a blow up of a smooth rational cubic curve Z ⊂ Q, and the map ψ is a projection from the line C. Let S be the exceptional divisor of β, and let L be a fiber of the morphism β over a general ¯ = α(L). Then S¯ ∼ H, the curve L ¯ is a line, and the point of the curve Z. Put S¯ = α(S) and L surface S¯ is singular along C. Moreover, the singularity of S¯ at a general point of C is locally ¯ is log isomorphic to T × A1 , where T is a germ of a nodal curve. In particular, the pair (X, S) canonical. 48

We may assume that Supp(D) 6⊃ S¯ by Remark 2.23. Then ¯ · D > multC D > 1/λ > 1, 1=L which is a contradiction.

Lemma 7.7. Let ‫(ג‬X) = 1.14. Then lct(X) = 1/2. Proof. We may suppose that lct(X) < 1/2. Then there is an effective Q-divisor D ∼Q H such that the log pair (X, λD) is not log canonical for some positive rational number λ < 1. The linear system |H| induces an embedding X ⊂ P5 such that X is a complete intersection of two quadrics. Then the locus LCS(X, λD) consists of a single line C ⊂ X ⊂ P5 by Remark 7.5. It follows from [98, Proposition 3.4.1] that there is a commutative diagram V @ @@ β ~~ @@ ~ ~ @@ ~~ @ ~ ~ X _ _ _ _ _ _ _/ P3 α

ψ

where ψ is a projection from C, the morphism α is a blow up of the line C, and β is a blow up of a smooth curve Z ⊂ P3 of degree 5 and genus 2. Let S be the exceptional divisor of β, and let L be a fiber of the morphism β over a general ¯ = α(L). Then S¯ ∼ 2H, the curve L ¯ is a line, and point of the curve Z. Put S¯ = α(S) and L ¯ ¯ multC (S) = 3. But the log pair (X, 1/2S) is log canonical, which implies that we may assume that Supp(D) 6⊃ S¯ by Remark 2.23. Then ¯ · D > multC D > 1/λ > 1, 1=L which is a contradiction.

Remark 7.8. Let V ⊂ P5 be a complete intersection of two quadric hypersurfaces that has isolated singularities, and let BV be an effective Q-divisor on V such that BV ∼Q −KV and LCS V, µBV 6= ∅, where µ < 1/2. Arguing as in the proof of Lemma 7.7, we see that LCS V, µBV ⊆ L, where L ⊂ V is a line such that L ∩ Sing (V ) 6= ∅. Lemma 7.9. Suppose that ‫(ג‬X) = 1.11. Then lct(X) = 1/2. Proof. We may suppose that lct(X) < 1/2. Then there is an effective Q-divisor D ∼Q H such that the log pair (X, λD) is not log canonical for some positive rational number λ < 1. Recall that the threefold X can be given by an equation w2 = t3 + t2 f2 x, y, z + tf4 x, y, z + f6 x, y, z ⊂ P 1, 1, 1, 2, 3 ∼ = Proj C x, y, z, t, w , where wt(x) = wt(y) = wt(z) = 1, wt(t) = 2, wt(w) = 3, and fi is a polynomial of degree i. The locus LCS(X, λD) consists of a single curve C ⊂ X such that H · C = 1 by Remark 7.5. Let ψ : X 99K P2 be the natural projection. Then ψ is not defined in a point O that is cut out by x = y = z = 0. The curve C does not contain the point O, because otherwise we get 1 = Γ · D > multO D multO Γ > multC D > 1/λ > 1, where Γ is a general fiber of the projection ψ. Thus, we see that ψ(C) ⊂ P2 is a line. Let S be the unique surface in |H| such that C ⊂ S. Let L be a sufficiently general fiber of the rational map ψ that intersects the curve C. Then L ⊂ Supp (D), since otherwise 1 = D · L > multC (D) > 1/λ > 1. We may assume that D = S by Remark 2.23. Then S has a cuspidal singularity along C. 49

We may assume that the surface S is cut out on X by the equation x = 0, and we may assume that the curve C is given by w = t = x = 0. Then S is given by w2 = t3 + t2 f2 0, y, z + tf4 0, y, z ⊂ P 1, 1, 2, 3 ∼ = Proj C y, z, t, w , and f6 (x, y, z) = xf5 (x, y, z), where f5 (x, y, z) is a homogeneous polynomial of degree 5. Since the surface S is singular along the curve C, one has f4 x, y, z = xf3 x, y, z , where f3 (x, y, z) is a homogeneous polynomial of degree 3. Then every point of the set x = f5 x, y, z = t = w = 0 ⊂ P 1, 1, 1, 2, 3 must be singular on X, which is a contradiction, because X is smooth.

The assertion of Theorem 7.1 is completely proved. 8. Fano threefolds with ρ = 2 We use the assumptions and notation introduced in section 1. Lemma 8.1. Suppose that ‫(ג‬X) = 2.1 or ‫(ג‬X) = 2.3. Then lct(X) = 1/2. Proof. There is a birational morphism α : X → V that contracts a surface E ⊂ X to a smooth elliptic curve C ⊂ V , where V is one of the following Fano threefolds: • smooth hypersurface in P(1, 1, 1, 2, 3) of degree 6; • smooth hypersurface in P(1, 1, 1, 1, 2) of degree 4. The curve C is contained in a surface H ⊂ V such that Pic V = Z H and −KX ∼ 2H. Then C is a complete intersection of two surfaces in |H|, and ¯ + E, −KX ∼ 2H ¯ is a proper transform where E is the exceptional divisor of the birational morphism α, and H of the surface H on the threefold X. In particular, the inequality lct(X) 6 1/2 holds. We suppose that lct(X) < 1/2. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some positive rational number λ < 1/2. Then ∅ 6= LCS X, λD ⊆ E, since lct(V ) = 1/2 by Theorem 7.1 and α(D) ∼Q 2H ∼ −KV . Put k = H · C. Then k = H 3 ∈ {1, 2}. Note that NC/V ∼ = OC H C ⊕ O C H C , which implies that E ∼ = C × P1 . Let Z ∼ = C and L ∼ = P1 be curves on E such that Z ·Z =L·L=0 and Z · L = 1. Then

α∗ (H)|E

∼ kL, and since −2Z ∼ KE ∼ KX + E ∼ 2E − 2α∗ H ∼ −2kL + 2E ,

E

E

E

we see that E|E ∼ −Z + kL. Put D = µE + Ω, where Ω is an effective Q-divisor on X such that E 6⊂ Supp(Ω). The pair (X, E + λΩ) is not log canonical in the neighborhood of E. Hence, the log pair E, λΩ E

is also not log canonical by Theorem 2.20. But Ω ∼Q − KX − µE ∼Q 2α∗ H − 1 + µ E ∼Q 1 + µ Z + k 1 − µ L, E

E

E

and 0 6 λk(1 − µ) 6 1, which contradicts Lemma 2.24. 50

Lemma 8.2. Suppose that ‫(ג‬X) = 2.4 and X is general. Then lct(X) = 3/4. Proof. There is a commutative diagram X } AAA β } } AA }} AA }~ } P3 _ _ _ _ _ _ _/ P1 α

ψ

where ψ is a rational map, α is a blow up of a smooth curve C ⊂ P3 such that C = H1 · H2 for some H1 , H2 ∈ |OP3 (3)|, and β is a fibration into cubic surfaces. Let P be a pencil in |OP3 (3)| generated by H1 and H2 . Then ψ is given by P. We assume that X satisfies the following generality conditions: • every surface in P has at most one ordinary double point; • the curve C contains no Eckardt points7 (see Definition 4.1) of any surface in P. Let E be the exceptional divisor of the birational morphism α. Then 1 4¯ 1 4¯ H1 + E ∼Q H 2 + E ∼Q −KX , 3 3 3 3 ¯ i is a proper transform of Hi on the threefold X. We see that lct(X) 6 3/4. where H We suppose that lct(X) < 3/4. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some positive rational number λ < 3/4. Suppose that the set LCS(X, λD) contains a (irreducible) surface S ⊂ X. Then D = εS + ∆, where ε > 1/λ, and ∆ is an effective Q-divisor such that S 6⊂ Supp(∆). Then ¯ 1 , D H ¯ H1

¯ 1 6= ∅. But is not log canonical by Remark 2.3 if S ∩ H D ∼Q −KH¯ 1 . ¯ H1

¯ 1 to be a smooth cubic surface in P3 . Thus, it follows from Theorem 4.2 that We can choose H ¯ 1 = ∅, S∩H ¯ 1 . Thus, we see that α(S) is a cubic surface in P. Then which implies that S ∼ H εα S + α ∆ ∼Q OP3 4 , which is impossible, because ε > 1/λ > 4/3. Let F be a fiber of β such that F ∩ LCS(X, λD) 6= ∅. Put D = µF + Ω, where Ω is an effective Q-divisor such that F 6⊂ Supp(Ω). Then the log pair (F, λΩ|F ) is not log canonical by Theorem 2.20, because λµ < 1. It follows from Theorem 4.2 that LCS F, λΩ = O, F

where either O is an Eckardt point of the surface F , or O = Sing(F ). By Theorem 2.7 LCS X, λD = LCS X, λµF + λΩD = O, because it follows from Theorem 2.20 that (X, F + λΩD) is not log canonical at O and is log canonical in a punctured neighborhood of O. But O 6∈ E by our generality assumptions. Then α O ⊂ LCS P3 , λα D ⊆ α(O) ∪ C, 7Note that C does not contain singular points of the surfaces in P since C is a complete intersection of two

surfaces from P. 51

where α(O) 6∈ C. But λ < 3/4, which contradicts Lemma 2.8.

Lemma 8.3. Suppose that ‫(ג‬X) ∈ {2.5, 2.10, 2.14} and X is general. Then lct(X) = 1/2. Proof. There is a commutative diagram XA AA β ~~ ~ AA ~ ~ AA ~ ~ ~ V _ _ _ _ _ _ _/ P1 α

ψ

where V is a smooth Fano threefold such that −KV ∼ 2H for some H ∈ Pic(V ) and ‫(ג‬V ) ∈ 1.13, 1.14, 1.15 , the morphism α is a blow up of a smooth curve C ⊂ V such that C = H1 · H2 for some H1 , H2 ∈ |H|, H1 6= H2 , the morphism β is a del Pezzo fibration, and ψ is a linear projection. Let E be the exceptional divisor of the birational morphism α. Then ¯ 1 + E ∼ 2H ¯ 2 + E ∼ −KX , 2H ¯ i is a proper transform of Hi on the threefold X. We see that lct(X) 6 1/2. where H We suppose that lct(X) < 1/2. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some positive rational number λ < 1/2. Then ∅ 6= LCS X, λD ⊆ E, because α(D) ∼Q −KV and lct(V ) = 1/2 by Theorem 7.1. We assume that the threefold X satisfies the following generality condition: every fiber of the del Pezzo fibration β has at most one singular point that is an ordinary double point. Let F be a fiber of β such that F ∩ LCS(X, λD) 6= ∅. Put D = µF + Ω, where Ω is an effective Q-divisor on X such that F 6⊂ Supp(Ω). Then α D = µα F + α Ω ∼Q 2α F ∼Q −KV , which implies that µ 6 2. Then (F, λΩ|F ) is also not log canonical by Theorem 2.20. But Ω ∼Q −KF , F

which implies that lct(F ) 6 λ < 1/2. But F has at most one ordinary double point and KF2 = H 3 6 5, which implies that lct(F ) > 1/2 (see Examples 1.18, 1.19, 5.3 and 5.4), which is a contradiction. Lemma 8.4. Suppose that ‫(ג‬X) = 2.8 and X is general. Then lct(X) = 1/2. Proof. Let O ∈ P3 be a point, and let α : V7 → P3 be a blow up of the point O. Then V7 ∼ = P OP2 ⊕ OP2 1 , and there is a P1 -bundle π : V7 → P2 . Let E be the exceptional divisor of α. Then E is a section of π. 52

There is a quartic surface R ⊂ P3 such that Sing(R) = O, the point O is an isolated double point of the surface R, and there is a commutative diagram β

}} }} } } }~ }

X

η

z

V2 A

AA AA ω AA

V7 A } AA } α } AAπ } } AA ~}} ψ _ _ _ _ _ _ _ / 3 P P2

φ

where ω is a double cover branched in R, the morphism η is a double cover branched in the ¯ such proper transform of R, the morphism β is a birational morphism that contracts a surface E ¯ = E to the singular point of V2 and that η(E) ω Sing V2 = O, the map ψ is a projection from the point O, and φ is a conic bundle. We assume that X satisfies the following mild generality condition: the point O is an ordinary ¯∼ double point of the surface R. Then E = P1 × P1 . ¯ Let H be the proper transform on X of the general plane in P3 that passes through O. Then ¯ + E, ¯ −KX ∼ 2H which implies that lct(X) 6 1/2. We suppose that lct(X) < 1/2. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some positive rational number λ < 1/2. ¯ 6= ∅. Put It follows from Lemma 2.18 that LCS(X, D) ∩ E ¯ + Ω, D = µE ¯ 6⊂ Supp(Ω). Then where Ω is an effective Q-divisor on X such that E ¯ + Ω · Γ = 2µ + Ω · Γ > 2µ, 2 = D · Γ = µE where Γ is a general fiber of the conic bundle φ. Hence µ 6 1. Thus, the log pair ¯ λΩ E, ¯ E

¯ 6= ∅. But is not log canonical by Theorem 2.20, because LCS(X, D) ∩ E 1+µ Ω ∼Q − KE¯ , ¯ 2 E which is impossible by Lemma 2.24.

Lemma 8.5. Suppose that ‫(ג‬X) = 2.11 and X is general. Then lct(X) = 1/2. Proof. Let V be a cubic hypersurface in P4 . Then there is a commutative diagram V @ @@ β ~~ @@ ~ ~ @@ ~ ~ @ ~ ~ X _ _ _ _ _ _ _/ P2 α

ψ

such that α contracts a surface E ⊂ X to a line L ⊂ V , the map ψ is a linear projection from the line L, the morphism β is a conic bundle. We assume that X satisfies the following generality condition: the normal bundle NL/V to the line L on the variety V is isomorphic to OL ⊕ OL . Let H be a hyperplane section of V such that L ⊂ H. Then ¯ + E, −KX ∼ 2H ¯ ⊂ X is the proper transform of the surface H. In particular, lct(X) 6 1/2. where H 53

We suppose that lct(X) < 1/2. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some positive rational number λ < 1/2. Then ∅ 6= LCS X, λD ⊆ E, since lct(V ) = 1/2 and α(D) ∼Q −KV . Note that E ∼ = P1 × P1 by the generality condition. Let F ⊂ E be a fiber of the induced projection E → L, let Z ⊂ E be a section of this projection such that Z · Z = 0. Then α∗ (H)|E ∼ F and E|E ∼ −Z, because −2Z − 2F ∼ KE ∼ KX + E ∼ 2 E − α∗ H ∼ −2F + 2E . E

E

E

Put D = µE + Ω, where Ω is an effective Q-divisor on X such that E 6⊂ Supp(Ω). Then 2 = D · Γ = µE · Γ + Ω · Γ > µE · Γ = 2µ, where Γ is a general fiber of the conic bundle β. Thus, we see that µ 6 1. The log pair (E, λΩ|E ) is not log canonical by Theorem 2.20. But Ω ∼Q − KX − µE ∼Q 1 + µ Z + 2F, E

E

which contradicts Lemma 2.24, because µ 6 1 and λ < 1/2.

Lemma 8.6. Suppose that ‫(ג‬X) = 2.15 and X is general. Then lct(X) = 1/2. Proof. There is a birational morphism α : X → P3 that contracts a surface E ⊂ X to a smooth curve C ⊂ P3 that is complete intersection of an irreducible quadric Q ⊂ P3 and a cubic F ⊂ P3 . We assume that X satisfies the following generality condition: the quadric Q is smooth. ¯ be a proper transform of Q on the threefold X. Then there is a commutative diagram Let Q X@ @@ β }} @@ } } @@ } } @ }~ _ o _ _ _ _ _ _ 3 V P γ α

where V is a cubic in P4 that has one ordinary double point P ∈ V , the morphism β contracts ¯ to the point P , and γ is a linear projection from the point P . the surface Q Let E be the exceptional divisor of α. Then ¯+E −KX ∼ 2Q and β(E) ⊂ V is a surface that contains all lines on V that pass through P . We see that lct(X) 6 1/2. We suppose that lct(X) < 1/2. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some positive rational number λ < 1/2. It follows from Lemma 2.17 that either ¯ ∅ 6= LCS X, λD ⊆ Q, or the set LCS(X, λD) contains a fiber of the natural projection E → C. In both cases ¯ 6= ∅. LCS X, λD ∩ Q ¯∼ We have Q = P1 × P1 . Put

¯ + Ω, D = µQ ¯ 6⊂ Supp(Ω). Then where Ω is an effective Q-divisor on X such that Q α D ∼Q µQ + α Ω ∼Q −KP3 , ¯ λΩ| ¯ ) is not log canonical by Theorem 2.20. But which gives µ 6 2. The log pair (Q, Q 1+µ Ω ∼Q − KQ¯ , ¯ 2 Q which implies that µ > 1 by Lemma 2.24. 54

It follows from Remark 2.23 that we may assume that E 6⊂ Supp(D). Then ¯ · F + Ω · F = µ + Ω · F > µ, 1 = D · F = µQ where F is a general fiber the natural projection E → C. But µ > 1, which is a contradiction. Lemma 8.7. Suppose that ‫(ג‬X) = 2.18. Then lct(X) = 1/2. Proof. There is a smooth divisor B ⊂ P1 × P2 of bidegree (2, 2) such that the diagram X PPP PPP φ nnn PPP2 π PPP PPP (/ 2 1 2 P ×P P

φ1 nnnn

nn nnn n n v n P1 o π1

π2

commutes, where π is a double cover that is branched in B, the morphisms π1 and π2 are natural projections, the morphism φ1 is a quadric fibration, and φ2 is a conic bundle. Let H1 be a general fiber of π1 , and let H2 be a general surface in |π2∗ (OP2 (1))|. Then B ∼ 2H1 + 2H2 . ¯ 2 be a general surface in |φ∗ (OP2 (1))|. Then ¯ 1 be a general fiber of φ1 , and let H Let H 2 ¯ ¯ −KX ∼ H1 + 2H2 , which implies that lct(X) 6 1/2. We suppose that lct(X) < 1/2. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some positive rational number λ < 1/2. Applying Lemma 2.26 to the fibration φ1 we see that ∅ 6= LCS X, λD ⊆ Q, where Q is a singular fiber of φ1 . Moreover, applying Theorem 2.28 to the fibration φ2 , we see that ∅ 6= LCS X, λD ⊆ Q ∩ R, where R ⊂ X be an irreducible surface that is swept out by singular fibers of φ2 . In particular, the set LCS(X, λD) contains no surfaces. Suppose that LCS(X, λD) is zero-dimensional. Then 1 ¯ ¯2 ¯ 2, H1 + 2H = LCS(X, λD) ∪ H LCS X, λD + 2 which is impossible by Theorem 2.7. We see that the set LCS(X, λD) contains a curve Γ ⊂ Q ∩ R. Put D = µQ + Ω, where Ω is an effective Q-divisor such that Q 6⊂ Supp(Ω). Then Q, λΩ Q

is not log canonical along Γ by Theorem 2.20. But Ω ∼Q − KX − µQ ∼Q −KQ , Q

Q

P3

which implies that Γ is a ruling of the cone Q ⊂ by Lemma 5.7. Then φ2 (Γ) ⊂ P2 is a line, and φ2 Γ ⊆ φ2 R . But φ2 (R) ⊂ P2 is a curve of degree 4. Thus, we see that φ2 R = φ2 Γ ∪ Z, where Z ⊂ P2 is a reduced cubic curve. Then φ2 induces a double cover of φ2 Γ \ φ2 Γ ∩ Z 55

that must be unramified (see [192]). But the quartic curve φ2 (R) has at most ordinary double points (see [192], [166]). Then φ2 Γ ∩ Z| = 3, ∼ P1 . which is impossible, because φ2 (Γ) = Lemma 8.8. Suppose that ‫(ג‬X) = 2.19 and X is general. Then lct(X) = 1/2. Proof. It follows from [98, Proposition 3.4.1] that there is a commutative diagram XA AA β ~~ ~ AA ~ ~ AA ~ ~ ~ V _ _ _ _ _ _ _/ P3 α

ψ

where V is a complete intersection of two quadric fourfolds in P5 , the morphism α is a blow up of a line L ⊂ V , the morphism β is a blow up of a smooth curve C ⊂ P3 of degree 5 and genus 2, and the map ψ is a linear projection from the line L. Let E and R be the exceptional divisors of α and β, respectively. Then • the surface β(E) ⊂ P3 is an irreducible quadric, • the surface α(R) ⊂ V is swept out by lines that intersect the line L. We assume that X satisfies the following generality condition: the surface β(E) is smooth. Let H be any hyperplane section of V ⊂ P5 such that L ⊂ H. Then ¯ + E ∼ R + 2E ∼ −KX , 2H ¯ is a proper transform of H on the threefold X. We see that lct(X) 6 1/2. where H We suppose that lct(X) < 1/2. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some positive rational number λ < 1/2. Then ∅ 6= LCS X, λD ⊆ E ∼ = P1 × P1 , because α(D) ∼Q −KV and lct(V ) = 1/2 by Theorem 7.1. Let F be a fiber of the projection E → L, and let Z be a section of this projection such that Z · Z = 0. Then α∗ (H)|E ∼ F and E|E ∼ −Z, because −2Z − 2F ∼ KE ∼ KX + E ∼ 2 E − α∗ H ∼ 2E − 2F. E

E

E

By Remark 2.23, we may assume that either E 6⊂ Supp(D), or R 6⊂ Supp(D), because the log pair X, λ R + 2E is log canonical and −KX ∼ R + 2E. Put D = µE + Ω, where Ω is an effective Q-divisor on X such that E 6⊂ Supp(Ω). Suppose that µ 6 1. Then (X, E + λΩ) is not log canonical, which implies that E, λΩ E

is also not log canonical by Theorem 2.20. But Ω ∼Q − KX − µE ∼Q 1 + µ Z + 2F, E

E

which contradicts Lemma 2.24, because µ 6 1 and λ < 1/2. Thus, we see that µ > 1. Then we may assume that R 6⊂ Supp(D). Let Γ be a general fiber of the projection R → C. Then Γ 6⊂ Supp(D) and 1 = −KX · Γ = µE · Γ + Ω · Γ = µ + Ω · Γ > µ, which is a contradiction.

Lemma 8.9. Suppose that ‫(ג‬X) = 2.23 and X is general. Then lct(X) = 1/3. 56

Proof. There is a birational morphism α : X → Q such that Q ⊂ P4 is a smooth quadric threefold, and α contracts a surface E ⊂ X to a smooth curve C ⊂ Q that is a complete intersection of a hyperplane section H ⊂ Q and a divisor F ∈ |OQ (2)|. We assume that X satisfies the following generality condition: the quadric surface H is smooth. ¯ be a proper transform of H on the threefold X. Then there is a commutative diagram Let H X@ @@ β ~~ @@ ~ ~ @@ ~ @ ~ ~ Q o_ _ _ _γ _ _ _ V α

where V is a complete intersection of two quadrics in P5 such that V has one ordinary double ¯ to the point P , and γ is a projection from P . point P ∈ V , the morphism β contracts H Let E be the exceptional divisor of α. Then ¯ + 2E −KX ∼ 3H and β(E) ⊂ V is a surface that contains all lines that pass through P . In particular, lct(X) 6 1/3. We suppose that lct(X) < 1/3. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some positive rational number λ < 1/3. It follows from Remark 7.8 that either ¯ ∅ 6= LCS X, λD ⊆ H, or the set LCS(X, λD) contains a fiber of the natural projection E → C. In both cases ¯ 6= ∅. LCS X, λD ∩ H ¯ ∼ We have H = P1 × P1 . There is a non-negative rational number µ such that ¯ + Ω, D = µH ¯ 6⊂ Supp(Ω). Then where Ω is an effective Q-divisor on X such that H α D ∼Q µH + α Ω ∼Q −KQ , ¯ λΩ| ¯ ) is not log canonical by Theorem 2.20. But which gives µ 6 3. The log pair (H, H 1+µ KH¯ , Ω ∼Q − ¯ 2 H which implies that µ > 1 by Lemma 2.24. It follows from Remark 2.23 that we may assume that E 6⊂ Supp(D), because the log pair ¯ + 2E X, λ 3H is log canonical. Let F be a general fiber the natural projection E → C. Then ¯ · F + Ω · F = µ + Ω · F > µ, 1 = D · F = µH which is a contradiction, because µ > 1.

Lemma 8.10. Suppose that ‫(ג‬X) = 2.24 and X is general. Then lct(X) = 1/2. Proof. The threefold X is a divisor on P2 × P2 of bidegree (1, 2). Let Hi be a surface in ∗ πi OP2 1 , where πi : X → P2 is a projection of X onto the i-th factor of P2 × P2 , i ∈ {1, 2}. Then −KX ∼ 2H1 + H2 , which implies that lct(X) 6 1/2. Note that π1 is a conic bundle, and π2 is a P1 -bundle. Let ∆ ⊂ P2 be the degeneration curve of the conic bundle π1 . Then ∆ is a cubic curve. We suppose that X satisfies the following generality condition: the curve ∆ is irreducible. 57

Assume that lct(X) < 1/2. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some positive rational number λ < 1/2. Suppose that the set LCS(X, λD) contains a surface S ⊂ X. Then D = µS + Ω, where Ω is an effective Q-divisor such that S 6⊂ Supp(Ω), and µ > 1/λ. Let Fi be a general fiber of πi , i ∈ {1, 2}. Then 2 = D · Fi = µS · Fi + Ω · Fi > µS · Fi , but either S · F1 > 1 or S · F2 > 1. Thus, we see that µ 6 2, which is a contradiction. By Theorem 2.28 and Theorem 2.7, there is a fiber Γ2 of the P1 -bundle π2 such that ∅ 6= LCS X, λD = Γ2 , because the set LCS(X, λD) contains no surfaces. Applying Theorem 2.28 to the conic bundle π1 , we see that π1 Γ2 ⊂ ∆, which is impossible, because ∆ ⊂ P2 is an irreducible cubic curve and π1 (Γ2 ) ⊂ P2 is a line.

Lemma 8.11. Suppose that ‫(ג‬X) = 2.25. Then lct(X) = 1/2. Proof. Recall that X is a blow up α : X → P3 along a normal elliptic curve C of degree 4. ¯ ⊂ X be a proper transform of Q. Let Q ⊂ P3 be a general quadric containing C, and let Q Then ¯ + E, −KX ∼ 2Q where E is the exceptional divisor of α. In particular, lct(X) 6 1/2. We suppose that lct(X) < 1/2. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some positive rational number λ < 1/2. ¯ defines a quadric fibration Note that the linear system |Q| φ : X −→ P1 , such that every fiber of φ is irreducible. Then the log pair (X, λD) is log canonical along every ˜ of the fibration φ by Theorem 2.28 since lct(Q) ˜ = 1/2 (see Example 1.18). nonsingular fiber Q The locus LCS(X, λD) does not contain any fiber of φ, because α(D) ∼Q 2Q and every fiber of φ is irreducible. Therefore, we see that dim(LCS(X, λD)) 6 1. ¯ 1 of the fibration φ such that Let Z be an element in LCS(X, λD). There is a singular fiber Q ¯ 1 . Note that φ has 4 singular fibers and each of them is a proper transform of a quadric Z⊂Q cone in P3 with vertex outside C. ¯ 2 be a singular fiber of φ such that Q ¯ 1 6= Q ¯ 2 , let H ¯ be a proper transform of a general Let Q 3 3 ¯ ¯ 2 ), and let plane in P that is tangent to the cone α(Q2 ) ⊂ P along one of its rulings L ⊂ α(Q 3 ¯ R be a proper transform of a very general plane in P . Put 1 ¯2 + 2 − ε H ¯ + 3εR 1+ε Q ∆ = λD + 2 for some positive rational number ε < 1 − 2λ. Then 1 1 + ε + 2λ KX , ∆ ∼Q − λ + 1 + ε KX ∼Q − 2 2 which implies that −(KX + ∆) is ample. ¯ be a proper transform on X of the line L. Then Let L ¯ ⊂ LCS X, ∆ ⊂ Q ¯1 ∪ Q ¯ 2, Z ∪L which is impossible by Theorem 2.7, because −(KX + ∆) is ample. Lemma 8.12. Suppose that ‫(ג‬X) = 2.26 and X is general. Then lct(X) = 1/2. 58

Proof. Let V be a smooth Fano threefold such that −KV ∼ 2H and Pic V = Z H , where H is a Cartier divisor such that H 3 = 5 (i. e. ‫(ג‬V ) = 1.15). Then |H| induces an embedding X ⊂ P6 . It follows from [98, Proposition 3.4.1] (see also [67]) that there is a line L ⊂ X ⊂ P6 such that there is a commutative diagram X@ @@ β ~~ ~ @@ ~ ~ @@ ~ ~ ~ _ _ _ _ _ _ _/ Q V α

ψ

where Q is a smooth quadric hypersurface in P4 , the morphism α is a blow up of the line L ⊂ V , the morphism β is a blow up of a twisted cubic curve P1 ∼ = C ⊂ Q, and ψ is a projection from the line L. Let S be the exceptional divisor of the morphism β. Put S¯ = α(S). Then S¯ ∼ H, and S¯ is singular along the line L. Let E be the exceptional divisor of the morphism α. Then β E ∼ OP4 1 , Q

which implies that β(E) is an irreducible quadric surface. We suppose that X satisfies the following generality condition: the surface β(E) is smooth. The equivalence −KX ∼ 2S + 3E holds. Moreover, the log pair 1 2S + 3E X, 3 is log canonical but not log terminal. Thus, we see that lct(X) 6 1/3. We suppose that lct(X) < 1/3. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some positive rational number λ < 1/3. Then ∅ 6= LCS X, λD ⊆ E, because α(D) ∼Q −KV and lct(V ) = 1/2 by Theorem 7.1. Note that E ∼ = P1 × P1 by our generality condition. Let F be a fiber of the projection E → L, and let Z be a section of this projection such that Z · Z = 0. Then α∗ (H)|E ∼ F and E|E ∼ −Z, because ∗ −2Z − 2F ∼ KE ∼ KX + E ∼ 2 E − α H ∼ 2E − 2F. E

E

E

By Remark 2.23, we may assume that either E 6⊂ Supp(D), or S 6⊂ Supp(D). Put D = µE + Ω, where Ω is an effective Q-divisor on X such that E 6⊂ Supp(Ω). Suppose that µ 6 2. Then (X, E + λΩ) is not log canonical, which implies that E, λΩ E

is also not log canonical by Theorem 2.20. But Ω ∼Q − KX − µE ∼Q 1 + µ Z + 2F, E

E

which contradicts Lemma 2.24, because µ 6 2 and λ < 1/3. Thus, we see that µ > 2. Then we may assume that S 6⊂ Supp(D). Let Γ be a general fiber of the projection S → C. Then Γ 6⊂ Supp(D) and 1 = −KX · Γ = µE · Γ + Ω · Γ = µ + Ω · Γ > µ, which is a contradiction.

Lemma 8.13. Suppose that ‫(ג‬X) = 2.27. Then lct(X) = 1/2. 59

Proof. There is a morphism α : X → P3 contracting a surface E to a twisted cubic curve C ⊂ P3 , and X ∼ = P(E), where E is a stable rank two vector bundle on P2 with c1 (E) = 0 and c1 (E) = 2 such that the sequence 0 −→ OP2 − 1 ⊕ OP2 − 1 −→ OP2 ⊕ OP2 ⊕ OP2 ⊕ OP2 −→ E ⊗ OP2 1 −→ 0 is exact (see [50] and [177, Application 1]). ¯ ⊂ X be a proper transform of Q. Let Q ⊂ P3 be a general quadric containing C, and let Q Then ¯ + E, −KX ∼ 2Q where E is the exceptional divisor of α. In particular, we see that lct(X) 6 1/2. We suppose that lct(X) < 1/2. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some positive rational number λ < 1/2. Suppose that the set LCS(X, λD) contains a surface S ⊂ X. Put D = µF + Ω, where µ > 1/λ and Ω is an effective Q-divisor such that F 6⊂ Supp(Ω). Let φ : X → P2 be the natural P1 -bundle. Then 2 = D · Γ = µF · Γ + Ω · Γ = µF · Γ + Ω · F > µF · Γ, where Γ is a general fiber of φ. Thus, we see that F is swept out by the fibers of φ. Then α F ∼ OP3 d and d > 2. But α(D) ∼Q µα(F ) + α(Ω) ∼Q OP3 (4), which is a contradiction. We wee that the locus LCS(X, λD) contains no surfaces. Applying Theorem 2.28 to (X, λD) and φ, we see that L ⊆ LCS X, λD , where L is a fiber of φ. Note that α(L) is a secant line of the curve C ⊂ P3 . One has α L ⊆ LCS P3 , λα D ⊆ α LCS X, λD ∪ C, which is impossible by Lemma 2.9.

Lemma 8.14. Suppose that ‫(ג‬X) = 2.28. Then lct(X) = 1/4. Proof. There is a blow up α : X → P3 along a plane cubic curve C ⊂ P3 . One has −KX ∼ 4G + 3E, where E is the exceptional divisor of α and G is a proper transform of the plane in P3 that contains the curve C. In particular, we see that the inequality lct(X) 6 1/4 holds. We suppose that lct(X) < 1/4. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some rational number λ < 1/4. One has ∅ 6= LCS X, λD ⊆ E, since lct(P4 ) = 1/4. Computing the intersections with a strict transform of a general line in P3 intersecting the curve C, one obtains that LCS(X, λD) does not contain the divisor E. Moreover, any curve Γ ∈ LCS(X, λD) must be a fiber of the natural projection ψ : E −→ C by Lemma 2.14. Therefore, we see that either the locus LCS(X, λD) consists of a single point, or the locus LCS(X, λD) consists of a single fiber of the projection ψ by Theorem 2.7. Let R be a sufficiently general cone in P3 over the curve C, and let H be a sufficiently general plane in P3 that passes through the point Sing (R). Then [ 3 ¯ ¯ ¯ , LCS X, λD + R+H = LCS X, λD Sing R 4 60

¯ and H ¯ are proper transforms of R and H on the threefold X. But the divisor where R 3 ¯ ¯ ∼Q λ − 1/4 KX − KX + λD + R+H 4 is ample, which contradicts Theorem 2.7.

Lemma 8.15. Suppose that ‫(ג‬X) = 2.29. Then lct(X) = 1/3. Proof. There is a birational morphism α : X → Q such that Q is a smooth quadric hypersurface, and α is a blow up along a smooth conic C ⊂ Q. ¯ be a proper Let H be a general hyperplane section of Q ⊂ P4 that contains C, and let H transform of the surface H on the threefold X. Then ¯ + 2E, −KX ∼ 3H where E is the exceptional divisor of α. In particular, the inequality lct(X) 6 1/3 holds. We suppose that lct(X) < 1/3. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some rational λ < 1/3. Then ∅ 6= LCS X, λD ⊆ E, since lct(Q) = 1/3 (see Example 1.9) and α(D) ∼Q −KQ . ¯ has no base points and defines a morphism β : X → P1 , whose general The linear system |H| fiber is a smooth quadric surface. Then the log pair (X, λD) is log canonical along the smooth fibers of β by Theorem 2.28 (see Example 1.18). ¯ of the morphism β such that It follows from Theorem 2.7 that there is a singular fiber S ∼ H ∅ 6= LCS X, λD ⊆ E ∩ S, and α(S) ⊂ P3 is a quadratic cone. Put Γ = E ∩ S. Then Γ is an irreducible conic, the log pair 2 X, S + E 3 has log canonical singularities, and 3S + 2E ∼Q D. Therefore, it follows from Remark 2.23 that to complete the proof we may assume that either S 6⊂ Supp(D) or E 6⊂ Supp(D). Intersecting the divisor D with a strict transform of a general ruling of the cone α(S) ⊂ P3 and with a general fiber of the projection E → C, we see that Γ 6⊆ LCS X, λD , which implies that LCS(X, λD) consists of a single point O ∈ Γ by Theorem 2.7. Let R be a general (not passing through O) surface in |α∗ (H)|. Then 1 ¯ LCS X, λD + H + 2R = R ∪ O, 2 ¯ + 2R ∼Q D and λ < 1/3. which is impossible by Theorem 2.7, since −KX ∼ H

Lemma 8.16. Suppose that ‫(ג‬X) = 2.30. Then lct(X) = 1/4. Proof. There is a commutative diagram X@ @@ β }} } @@ } } @@ } ~}} P3 o_ _ _ γ_ _ _ _ Q α

where Q is a smooth quadric threefold in P4 , the morphism α is a blow up of a smooth conic C ⊂ P3 , the morphism β is a blow up of a point, and γ is a projection from a point. Let G be a proper transform on the variety X of the unique plane in P3 that contains the conic C. Then the surface G is contracted by the morphism β, and −KX ∼ 4G + 3E, where E is the exceptional divisor of the blow up α. Thus, we see that lct(X) 6 1/4. 61

We suppose that lct(X) < 1/4. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some rational λ < 1/4. Then ∅ 6= LCS X, λD ⊆ E ∩ G, because lct(P4 ) = 1/4 and lct(Q) = 1/3. We may assume that either G 6⊂ Supp(X) or E 6⊂ Supp(X) by Remark 2.23. Intersecting D with lines in G ∼ = P2 and with fibers of the projection E → C, we see that LCS X, λD ( E ∩ G which implies that there is a point O ∈ E ∩ G such that LCS(X, λD) = O by Theorem 2.7. Let R be a general surface in |α∗ (H)| and F a general surface in |α∗ (2H) − E|. Then 1 = R ∪ O, F + 2R LCS X, λD + 2 which is impossible by Theorem 2.7 since −KX ∼ F + 2R ∼Q D and λ < 1/4.

Lemma 8.17. Suppose that ‫(ג‬X) = 2.31. Then lct(X) = 1/3. Proof. There is a birational morphism α : X → Q such that Q is a smooth quadric hypersurface, and α is a blow up of the quadric Q along a line L ⊂ Q. Let H be a sufficiently general hyperplane section of the quadric Q ⊂ P4 that passes through ¯ be a proper transform of the surface H on the threefold X. Then the line L, and let H ¯ + 2E, −KX ∼ 3H where E is the exceptional divisor of α. In particular, lct(X) 6 1/3. We suppose that lct(X) < 1/3. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some rational λ < 1/3. Then ∅ 6= LCS X, λD ⊆ E, since lct(Q) = 1/3 and α(D) ∼Q −KQ . ¯ defines a P1 -bundle φ : X → P2 such that the induced morphism The linear system |H| E∼ = F1 → P2 contracts an irreducible curve Z ⊂ E. One has LCS X, λD = Z ⊂ E by Theorem 2.28. Put D = µE + Ω, where Ω is an effective Q-divisor on X such that E 6⊂ Supp(Ω). Then 2 = D · F = µE · F + Ω · F = µ + Ω · F > µ, where F is a general fiber of φ. Note that the log pair X, E + λΩ is not log canonical, because λ < 1/3. Then (E, λΩ|E ) is not log canonical by Theorem 2.20. Let C be a fiber of the natural projection E → L. Then Ω ∼Q 3C + 1 + µ Z, E

which implies that (E, λΩ|E ) is log canonical by Lemma 5.8, which is a contradiction. 62

9. Fano threefolds with ρ = 3 We use the assumptions and notation introduced in section 1. Lemma 9.1. Suppose that ‫(ג‬X) = 3.1 and X is general. Then lct(X) = 3/4. Proof. There is a double cover ω : X −→ P1 × P1 × P1 branched over a divisor of tridegree (2, 2, 2). The projection P1 × P1 × P1 −→ P1 onto the i-th factor induces a morphism πi : X → P1 , whose fibers are del Pezzo surfaces of degree 4. Let R1 be a singular fiber of the fibration π1 , let Q be a singular point of the surface R1 , and let R2 and R3 be fibers of π2 and π3 such that R2 3 Q ∈ R3 , respectively. Then multQ (R1 + R2 + R3 ) = 4, which implies that the log pair 3 X, R1 + R2 + R3 4 is not log terminal at Q. But −KX ∼ R1 + R2 + R3 . Thus, we see that lct(X) 6 3/4. We suppose that the threefold X satisfies the following generality condition: for an arbitrary point O ∈ X, there is k ∈ {1, 2, 3} such that • the fiber Fk of the fibration πk that contains O is smooth at the point O, • the singularities of the surface Fk consist of at most one ordinary double point, • for every smooth curve Γ ⊂ Fk such that −KFk · Γ = 1, we have O 6∈ Γ, • for every smooth curves ∆1 ⊂ Sk ⊃ ∆2 such that −KFk · ∆1 = −KFk · ∆2 = 2 and ∆1 + ∆2 ∼ −KFk , we have O 6= ∆1 ∩ ∆2 . We suppose that lct(X) < 3/4. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical at some point P ∈ X for some rational number λ < 3/4. Let Si be a fiber of πi such that P ∈ Si . Without loss of generality, we may assume that • the surface S1 is smooth at the point P , • the singularities of the surface S1 consist of at most one ordinary double point, • for every smooth curve L ⊂ S1 such that −KS1 · L = 1, we have P 6∈ L, • for every smooth curves C1 ⊂ S1 ⊃ C2 such that −KS1 · C1 = −KS1 · C2 = 2 and C1 + C2 ∼ −KS1 , we have P 6= C1 ∩ C2 . The surface S1 is a del Pezzo surface of degree 4. One has D = µS1 + Ω, where Ω is an effective Q-divisor on X such that S1 6⊂ Supp(Ω). Let φ : X → P1 × P1 be a natural conic bundle induced by the linear system S2 + S3 , and let Γ be a general fiber of the conic bundle φ. Then 2 = D · Γ = µS1 · Γ + Ω · Γ = 2µ + Ω · Γ > 2µ, which implies that µ 6 1. Then (X, S1 + λΩ) is not canonical at the point P . Hence S1 , λΩ S1

is not log canonical at the point P by Theorem 2.20. But Ω ∼Q D ∼Q −KSk , S1

S1 63

which is impossible (see Example 5.4 and mind the generality assumption).

Lemma 9.2. Suppose that ‫(ג‬X) = 3.2 and X is general. Then lct(X) = 1/2. Proof. The threefold X is a primitive Fano threefold (see [128, Definition 1.3]). Put U = P OP1 ×P1 ⊕ OP1 ×P1 − 1, −1 ⊕ OP1 ×P1 − 1, −1 , let π : U → P1 × P1 be a natural projection, and let L be a tautological line bundle on U . Then X ∈ 2L + π ∗ OP1 ×P1 2, 3 . Let us show that lct(X) 6 1/2. Let E1 and E2 be surfaces in X such that π E1 ⊂ P1 × P1 ⊃ π E2 are divisors on P1 × P1 of bi-degree (1, 0) and (0, 1), respectively. Then −KX ∼ L + 2E1 + E2 , X

which implies that lct(X) 6 1/2. We suppose that lct(X) < 1/2. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical at some point P ∈ X for some rational number λ < 1/2. It follows from [86, Proposition 3.8] that there is a commutative diagram γ1

γ2

/V o U1 hPPP n6 U2 O PPP nnn n PPP n α P nnn β1 PPPP nnn β2 n P n n ψ1 ψ2 nn X PPP PPP nn PPφP2 nnn n PPP ω n n n PPP nn n n P( n v / P1 o 1 1 1 P ×P P φ1

π2

π1

where V is a Fano threefold that has one ordinary double point O ∈ V such that Pic V = Z − KV and −KV3 = 16, the morphism α contracts a unique surface P1 × P1 ∼ =S⊂X such that S ∼ L|X to the point O ∈ V , the morphism βi contracts S to a smooth rational curve, the morphism γi contracts the curve βi (S) to the point O ∈ V so that the rational map γ2 ◦ γ1−1 : U1 99K U2 is a flop in β1 (S) ∼ = P1 , the morphism ψ2 is a quadric fibration, and the morphisms ψ1 , φ1 , φ2 are fibrations whose fibers are del Pezzo surfaces of degree 4, 3 and 6, respectively. The morphisms π1 and π2 are natural projections, and ω = π|X . Then Cl V = Z α E1 ⊕ Z α E2 , and ω is a conic bundle. The curve β1 (S) is a section of ψ1 , and β2 (S) is a 2-section of ψ2 . We assume that the threefold X satisfies the following mild generality condition: every singular fiber of the del Pezzo fibration φ2 has at most A1 singularities. Applying Lemma 2.26 to the fibration φ1 , we see that ∅ 6= LCS X, λD ⊆ S1 where S1 is a singular fiber of the del Pezzo fibration φ1 , because the global log canonical threshold of a smooth del Pezzo surface of degree 6 is equal to 1/2 by Example 1.18. Applying Lemma 2.26 to φ2 , we obtain a contradiction by Example 1.38. Lemma 9.3. Suppose that ‫(ג‬X) = 3.3 and X is general. Then lct(X) = 2/3. 64

Proof. The threefold X is a divisor on P1 × P1 × P2 of tridegree (1, 1, 2). In particular, −KX ∼ π1∗ OP1 1 + π2∗ OP1 1 + φ∗ OP2 1 , where π1 : X → P1 and π1 : X → P1 are fibrations into del Pezzo surfaces of degree 4 that are induced by the projections of the variety P1 ×P1 ×P2 onto its first and second factor, respectively, and φ : X → P2 is conic bundle that is induced by the projection P1 × P1 × P2 → P2 . Let α2 : X → P1 × P2 be a birational morphism induced by the linear system ∗ π2 OP1 1 + φ∗ OP2 1 , let Hi ∈ |πi∗ (OP1 (1))| and R ∈ |φ∗ (OP2 (1))| be general surfaces. Then H1 ∼ H2 + 2R − E2 , where E2 is the exceptional divisor of the birational morphism α2 . Hence 3 1 1 −KX ∼ H1 + H2 + R ∼Q H1 + H2 + E2 , 2 2 2 which implies that lct(X) 6 2/3. We suppose that the threefold X satisfies the following generality conditions: for an arbitrary point O ∈ X, there is k ∈ {1, 2} such that • the fiber Fk of the fibration πk that contains the point O is smooth at the point O, • the singularities of the surface Fk consist of at most one ordinary double point, • for every smooth curve Γ ⊂ Fk such that −KFk · Γ = 1, we have O 6∈ Γ if Sing(Fk ) 6= ∅. We suppose that lct(X) < 2/3. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical at some point P ∈ X for some rational number λ < 2/3. Let Si be a fiber of πi such that P ∈ Si . Then we may assume that • the surface S1 is smooth at the point P , • the singularities of the surface S1 consist of at most one ordinary double point, • for every smooth curve L ⊂ S1 such that −KS1 · L = 1, we have P 6∈ L if Sing(S1 ) 6= ∅. Put D = µS1 + Ω, where Ω is an effective Q-divisor such that S1 6⊂ Supp(Ω). Then H2 , λµS1 + λΩ H2

H2

is log canonical because lct(H2 ) = 2/3. Thus, we see that µ 6 1/λ. Hence S1 , λΩ S1

is not log canonical at the point P by Theorem 2.20. But Ω ∼Q −KS1 , S1

which is impossible (see Example 5.4).

Lemma 9.4. Suppose that ‫(ג‬X) = 3.4. Then lct(X) = 1/2. Proof. Let O be a point in P2 . Then there is a commutative diagram X η1

α

φ η2

/ F1 V PPP PPP nnn n n γ1 nn PPPγ2 n PPP ω β nnn PPP n n n n P n w '/ 2 P1 × P2 P1 o P π1 π2

υ

& / P1

such that πi and υ are natural projections, ω is a double cover branched over a divisor B ⊂ P1 ×P2 of bi-degree (2, 2), the morphism γ1 is a fibration into quadrics, γ2 and η2 are conic bundles, the morphism β is a blow up of the point O, the morphism α is a blow up of a smooth curve that 65

is a fiber of γ2 over the point O, the morphism η1 is a fibration into del Pezzo surfaces of degree 6, and φ is a fibration into del Pezzo surfaces of degree 4. Let H be a general fiber of η1 , and let S be a general fiber of φ. Then −KX ∼ H + 2S + E, where E is the exceptional divisor of α. Thus, we see that lct(X) 6 1/2. We suppose that lct(X) < 1/2. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some positive rational number λ < 1/2. Then ∅ 6= LCS X, λD ⊆ E, because α(D) ∼Q −KV and lct(V ) = 1/2 by Lemma 8.7. Let Γ be a fiber of η2 such that Γ ∩ LCS(X, λD) 6= ∅. Then Γ ⊆ LCS X, λD ⊆ E, by Theorem 2.28. Hence (H, λD|H ) is not log canonical at the points H ∩ Γ. But D ∼Q −KX ∼ −KH H

H

and lct(H) = 1/2, because H is a del Pezzo surface of degree 6, which is a contradiction.

Lemma 9.5. Suppose that ‫(ג‬X) = 3.5 and X is general. Then lct(X) = 1/2. Proof. There is a birational morphism α : X → P1 ×P2 that contracts a surface E ⊂ X to a curve C ⊂ P1 ×P2 of bidegree (5, 2). Let π1 : P1 ×P2 → P1 and π2 : P1 ×P2 → P2 be natural projections. There is Q ∈ π2∗ OP1 2 such that C ⊂ Q. Let H1 be a general fiber of π1 , let H2 be a surface in |π2∗ (OP1 (1))|. We have ¯ −KX ∼ 2H¯1 + H¯2 + Q, ¯ ⊂ X are proper transforms of H1 , H2 , Q, respectively. In particular, lct(X) 6 where H¯1 , H¯2 , Q 1/2. We suppose that X satisfies the following generality condition: every fiber F of π1 ◦ α is singular at most at one ordinary double point. Assume that lct(X) < 1/2. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some positive rational number λ < 1/2. Let S be an irreducible surface on the threefold X. Put D = µS + Ω, where Ω is an effective Q-divisor such that S 6⊂ Supp(Ω). Then 1 ¯ H1 , µS + Ω ¯1 2 H is log canonical (see Example 1.18). Thus, either µ 6 2, or S is a fiber of π1 ◦ α. Let Γ ∼ = P1 be a general fiber of the conic bundle π2 ◦ α. Then 2 = D · Γ = µS · Γ + Ω · Γ > µS · Γ, which implies that µ 6 2 in the case when S is a fiber of π1 ◦ α. We see that the set LCS(X, λD) contains no surfaces. Now, applying Lemma 2.26 to π1 ◦ α, we obtain a contradiction with Example 5.4. Lemma 9.6. Suppose that ‫(ג‬X) = 3.6 and X is general. Then lct(X) = 1/2. Proof. Let ε : V → P3 be a blow up of a line L ⊂ P3 . Then V ∼ = P OP 1 ⊕ O P 1 ⊕ O P 1 1 66

and there is a natural P2 -bundle η : V → P1 . There is a smooth elliptic curve C ⊂ P3 of degree 4 such that L ∩ C = ∅ and there is a commutative diagram γ

Y o ε

/ 1 q8 P q q qq β qqηq q q qqq

P3 o

φ

X

V

δ

where δ is a blow up of C, the morphism β is a blow up of the proper transform of the line L, the morphism γ is a blow up of the proper transform of the curve C, and φ is a del Pezzo fibration. We suppose that X satisfies the following generality condition: for every fiber F of φ, the surface F has at most one singular point that is an ordinary double point of the surface F . Let E and G be the exceptional surfaces of β and γ, respectively, let H ⊂ P3 be a general plane that passes through L, and let Q ⊂ P3 a quadric surface that passes through C. Then ¯ +Q ¯ + E, −KX ∼ 2H ¯ ⊂X⊃Q ¯ are proper transforms of H and Q, respectively. We have lct(X) 6 1/2. where H We suppose that lct(X) < 1/2. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some positive rational number λ < 1/2. It follows from Lemma 8.11 that lct(V ) = 1/2. Therefore, we see that ∅ 6= LCS X, λD ⊆ G. Note that every fiber of the fibration φ is a del Pezzo surface of degree 5 that has at most one ordinary double point. Thus, applying Lemma 2.26 to φ, we obtain a contradiction with Example 5.3. Lemma 9.7. Suppose that ‫(ג‬X) = 3.7 and X is general. Then lct(X) = 1/2. Proof. Let W be a divisor on P2 × P2 of bi-degree (1, 1). Then −KW ∼ 2H, where H is a Cartier divisor on W . There is a commutative diagram vv vv v vv v{ v β

P1 × P2 φ

P2 o

HH γ HH HH H#

P1 × P2

α

ξ

ED

X HH

ζ

W ρ

P1

/ P2

o

ω

ψ

BC

where φ and ψ are natural projections, α is a blow up of a smooth curve C ⊂ W such that C = H1 ∩ H2 , where H1 6= H2 are surfaces in |H|, the map ρ is induced by the pencil generated by H1 and H2 , the morphism ω is a del Pezzo fibration of degree 6, the morphisms ζ and ξ are P1 -bundles, ¯1 ⊂ X ⊃ M ¯ 2 such that φ◦β(M ¯ 1 ) = ξ(C) and ψ◦γ(M ¯ 2 ) = ζ(C). while β and γ contract surfaces M Note that lct(X) 6 1/2, because ¯ 1 + E, −KX ∼ 2H ¯ 1 ⊂ X is the proper transform of H1 , and E is the exceptional surface of α. where H We suppose that X satisfies the following generality condition: all singular fibers of the fibration ω satisfy the hypotheses of Lemma 5.5. Assume that lct(X) < 1/2. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some positive rational number λ < 1/2. Then ∅ 6= LCS X, λD ⊆ E, 67

because lct(W ) = 1/2 by Theorem 7.1. Applying Lemma 2.26, we see that ∅ 6= LCS X, λD ⊆ E ∩ F, where F is a singular fiber of ω. Note that F is a del Pezzo surface of degree 6. Put D = µF + Ω, where Ω is an effective Q-divisor such that F 6⊂ Supp(Ω). Then Ω ∼Q D ∼Q −KF , F

F

and the surface F is smooth along the curve E ∩F . But the log pair (F, λΩ|F ) is not log canonical at some point P ∈ E ∩ F by Theorem 2.20, which is impossible by Lemma 5.5. Remark 9.8. Let us use the notation and assumptions of the proof of Lemma 9.7. Then ∅ 6= LCS X, λD ⊆ E ∩ F, where F is a singular fiber of the fibration ω. Applying Theorem 2.28 to φ and ψ, we see that ¯1 ∩ M ¯2 , ∅ 6= LCS X, λD ⊆ E ∩ F ∩ M by Lemma 2.29. Regardless to how singular F is, if the threefold X is sufficiently general, then ¯1 ∩ M ¯2 = ∅, E∩F ∩M which implies that an alternative generality condition can be used in Lemma 9.7. Lemma 9.9. Suppose that ‫(ג‬X) = 3.8 and X is general. Then lct(X) = 1/2. Proof. Let π1 : F1 × P2 → F1 and π2 : F1 × P2 → P2 be natural projections. Then ∗ X ∈ α ◦ π1 OP2 1 ⊗ π2∗ OP2 2 , where α : F1 → P2 is a blow up of a point. Let H be a surface in |π2∗ (OP2 (1))|. Then −KX ∼ E + 2L + H, where E ⊂ X ⊃ L are irreducible surfaces such that π1 (E) ⊂ F1 is the exceptional curve of α, and π1 (L) ⊂ F1 is a fiber of the natural projection F1 → P2 . We have lct(X) 6 1/2. The projection π1 induces a fibration φ : X → P1 into del Pezzo surfaces of degree 5. We suppose that X satisfies the following generality condition: for every fiber F of φ, the surface F has at most one singular point that is an ordinary double point of the surface F . Assume that lct(X) < 1/2. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some positive rational number λ < 1/2. Applying Lemma 2.26 to the morphism φ, we obtain a contradiction with Example 5.3. Lemma 9.10. Suppose that ‫(ג‬X) = 3.9. Then lct(X) = 1/3. Proof. Let O1 ∈ V1 ∼ = V2 3 O2 be singular points of V1 ∼ = V2 ∼ = P(1, 1, 1, 2), respectively, let O1 6∈ S1 ∈ OP(1,1,1,2) (2) be a smooth surface, and let C1 ⊂ S1 ∼ = P2 be a smooth quartic curve. Then the diagram p X NNNN NNNβ2 ppp p p NNN pp p NNN p p x pp & β1

U1

α1

γ1

γ2

U2

α2

V1 N V2 NN pp p NN p N N p p ψ2 ψ1 xp & P2 68

commutes, where ψi is a natural projection, αi is a blow up of the point Oi with weights (1, 1, 1), the morphism γi is a P1 -bundle, and βi is a birational morphism that contracts a surface P 1 × C1 ∼ = Gi ⊂ X to a smooth curve C1 ∼ = Ci ⊂ Ui . Let Ei ⊂ X be the proper transform of the exceptional divisor of αi . Then S1 = α1 ◦ β1 E2 ⊂ V1 ∼ = P 1, 1, 1, 2 ∼ = V2 ⊃ α2 ◦ β2 E1 are surfaces in |OP(1,1,1,2) (2)| that contain the curves C1 and C2 , respectively. On the other hand, α1 ◦ β1 G2 ⊂ V1 ∼ = P 1, 1, 1, 2 ∼ = V2 ⊃ α2 ◦ β2 G1 are surfaces in |OP(1,1,1,2) (4)| that contain O1 ∪ C1 and O2 ∪ C2 , respectively. ¯ ⊂ X be the proper transform of a general surface in |OP(1,1,1,2) (1)|. Then Let H ¯ + E2 + E1 , −KX ∼ 3H which gives lct(X) 6 1/3. Suppose that lct(X) < 1/3. Then there is an effective Q-divisor 5 G1 + G2 − 5 E1 + E2 D ∼Q −KX ∼Q 2 such that the log pair (X, λD) is not log canonical for some λ < 1/3. Put D = µ1 E1 + µ2 E2 + Ω, where Ω is an effective Q-divisor on X such that E1 6⊆ Supp Ω ⊇ 6 E2 . Let Γ be a general fiber of the conic bundle γ1 ◦ β1 . Then 2 = Γ · D = Γ · µ1 E1 + µ2 E2 + Ω = µ1 + µ2 + Γ · Ω > µ1 + µ2 , and without loss of generality we may assume that µ1 6 µ2 . Then µ1 6 1. Suppose that there is a surface S ∈ LCS(X, λD). Then S 6= E1 and S 6= G1 , because α2 ◦ β2 (G1 ) ∈ |OP(1,1,1,2) (4)| and α2 ◦ β2 (D) ∈ |OP(1,1,1,2) (5)|. Hence S ∩ E1 6= ∅. But 2µ1 1 KE1 + Ω − KE1 ∼Q D = − 3 3 E1 E1 2 ∼ and E1 = P , which is impossible by Theorem 2.20, because λ < 1/3 = lct(P2 ). We see that the set LCS(X, λD) contains no surfaces. Let P ∈ LCS(X, λD) be a point. Suppose that P 6∈ G1 . Let Z be a fiber of γ1 such that β1 (P ) ∈ Z. Then Z ⊆ LCS U1 , λβ1 D ¯1 = β1 (E1 ). Then we have by Theorem 2.28. Put E ¯1 , λΩ ¯1 ∈ LCS E Z ∩E ¯ E1

by Theorem 2.20, which is impossible by Lemma 2.8, because µ1 6 1. Hence LCS(X, λD) ( G1 . Suppose that LCS(X, λD) ⊆ G1 ∩ G2 . Then LCS X, λD = 1 by Lemma 2.14 and Theorem 2.7. One has 1 ¯ ¯ ¯ ∪ E1 ∪ E1 , LCS X, λD ∪ H ⊆ LCS X, λD + E2 + E2 + H ⊂ LCS X, λD ∪ H 3 ¯ is a general surface in |(β1 ◦ γ1 )∗ (OP2 (1))| and which contradicts Theorem 2.7, because H 1 ¯ ∼Q λ − 1/3 KX . E2 + E2 + H λD + 3 69

Thus, we see that G1 ) LCS(X, λD) 6⊆ G1 ∩ G2 . Then ∅ 6= LCS U2 , λβ2 D ( β2 G1 , and it follows from Theorems 2.7 and 2.28 that there is a fibre L of the fibration γ2 such that LCS U2 , λβ2 D = L. Let B be a general surface in |α2∗ (OP(1,1,1,2) (2))|. Then β2 (D)|B ∼Q OP2 (5) and B ∼ = P2 . But LCS B, λβ2 D =L∩B B

and |L ∩ B| = 1, which is impossible by Lemma 2.8, because λ < 1/3.

Lemma 9.11. Suppose that ‫(ג‬X) = 3.10. Then lct(X) = 1/2. Proof. Let Q ⊂ P4 be a smooth quadric hypersurface. Let C1 ⊂ Q ⊃ C2 be disjoint (irreducible) conics. Then there is a commutative diagram XP nn~n~ @@P@PPPP n n @@ PPP nn ~~ PPP @@ nnn ~~ φ1 nnn PPφP2 @@ ~ ~ n PPP @@ ~ nn n ~ n PPP β β @@ 1 nn ~~ 2 n PPP @ ~ n n @ ~ n PPP @ ~ n n ~ n P' n w / P1 /Qo Y Y1 2 P1 o α α 2 1 ψ ψ 1

2

where the morphism αi is a blow up along the conic Ci , the morphism βi is a blow up along the proper transform of the conic Ci , the morphism ψi is a natural fibration into quadric surfaces, and φi is fibration, whose general fiber is isomorphic to a smooth del Pezzo surfaces of degree 6. Let Ei be the exceptional divisor of the morphism βi , and let Hi be a sufficiently general hyperplane section of the quadric Q that passes through the conic Ci . Then −KX ∼ H¯1 + 2H¯2 + E2 , ¯ i ⊂ X is the proper transform of the surface Hi . We see that lct(X) 6 1/2. where H We suppose that lct(X) < 1/2. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some positive rational number λ < 1/2. Using Example 1.18 and applying Lemma 2.28, we see that ∅ 6= LCS X, λD ⊆ S1 ∩ S2 , where Si is a singular fiber of φi . Hence, the set LCS(X, λD) contains no surfaces. It follows from Theorem 2.7 that either LCS(X, λD) is a point in E1 ∪ E2 , or LCS X, λD ∩ X \ E1 ∪ E2 6= ∅, which implies that we may assume that LCS(X, λD) is a point in E1 by Lemma 2.10. Since β2 is an isomorphism on X \ E2 , we see that P ∈ LCS Y1 , λβ2 D ⊂ P ∪ β2 E2 for some point P ∈ E1 . Then LCS(Y1 , λβ2 (D)) = P by Theorem 2.7, because P 6∈ β2 (E2 ). Let H be a general hyperplane section of the quadric Q. Then ˜ 1 + 2H ˜ ∼Q β2 D , −KY ∼ H 1

˜ ⊂ Y1 ⊃ H ˜ 1 are proper transforms of H and H1 , respectively. But where H 1 ˜ ˜ ˜ LCS Y1 , λβ2 D + H1 + 2 H = P ∪ H, 2 which is impossible by Theorem 2.7, because λ < 1/2. Lemma 9.12. Suppose that ‫(ג‬X) = 3.11. Then lct(X) = 1/2. 70

Proof. Let O ∈ P3 be a point, let δ : V7 → P3 be a blow up of the point O, and let E be the exceptional divisor of δ. Then ∼ P OP 2 ⊕ O P 2 1 , V7 = there is a natural P1 -bundle η : V7 → P2 , and E is a section of η. There is a linearly normal elliptic curve O ∈ C|subsetP3 such that the diagram γ

/ P3 o U gPPP k5 V7 kkk PPP k k k PPP kk η kkk α PPPP kkk β k PP k k k k 2 X GG ω BP δ

~~ ~~ ~ ~ ~~~ o P1

GG GGυ GG G#

φ

π2

P1 × P2

π1

commutes, where π1 and π2 are natural projections, the morphism γ contracts a surface C × P1 ∼ =G⊂U to the curve C, the morphism α is a blow up of the fiber of the morphism γ over the point O ∈ P3 , the morphism β is a blow up of the proper transform of C, the morphism ω is a fibration into quadric surfaces, the morphism φ is a fibration into del Pezzo surfaces of degree 7, and υ contracts a surface C × P1 ∼ =F ⊂X to an elliptic curve Z ⊂ P1 × P2 such that −KP1 ×P2 · Z = 13 and Z ∼ = C. Let H1 be a general fiber of φ, and let H2 be a general surface in |(η ◦ β)∗ (OP2 (1))|. Then −KX ∼ H1 + 2H2 , which implies that lct(X) 6 1/2. We suppose that lct(X) < 1/2. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some positive rational number λ < 1/2. Note that ¯ ∅ 6= LCS X, λD ⊆ E, ¯ is the exceptional divisor of α, because lct(U ) = 1/2 by Lemma 8.11. where E Let Γ ∼ = P2 be the general fiber of π2 ◦ υ. Then ¯ · Γ, 2 = −KX · Γ = D · Γ = 2E ¯⊂ which implies that E 6 LCS(X, λD). Applying Lemma 2.26 to the log pair V7 , λβ D ¯ ∩ G. Applying Lemma 2.29 to the log pair we see that LCS(X, λD) ⊆ E P1 × P2 , λυ D ¯ ∩ F ∩ G, where |E ¯ ∩ F ∩ G| = 1. Hence we see that LCS(X, λD) = E LCS X, λD + H2 = LCS X, λD ∪ H2 , and H2 ∩ LCS(X, λD) = ∅. But the divisor 1 1 KX + H1 − KX + λD + H2 = λ − 2 2 is ample, which is impossible by Theorem 2.7.

Lemma 9.13. Suppose that ‫(ג‬X) = 3.12. Then lct(X) = 1/2. 71

Proof. Let ε : V → P3 be a blow up of a line L ⊂ P3 . There is a natural P2 -bundle η : V → P1 , there is a smooth rational cubic curve C ⊂ P3 such that L ∩ C = ∅, and the diagram π1 PO 1 ofMMM MMM MMM φ MMM M

1

2

P; × P P PPP ww PPπP2 w w w PPP w ω w PPP ww ' 2 η n6 P p X HHH n p n p HH nnn pp HH nnn ppp n H p n γ n p H ψ β n H$ xppp nnn Y V NNN NNN vv vv NNN v v ε NNN vv α N& zvv P3

commutes, where α and β are blow ups of the curve C and its proper transform, respectively, the morphism γ is a blow up of the proper transform of the line L, the morphism ψ is a P1 -bundle, the morphism ω is a birational contraction of a surface F ⊂ X to a curve such that C ∪ L ⊂ α ◦ γ F ⊂ P3 , and α ◦ γ(F ) consists of secant lines of C ⊂ P3 that intersect L, the morphism φ is a fibration into del Pezzo surfaces of degree 6, the morphisms π1 and π2 are natural projections. Let E and G be exceptional divisors of β and γ, respectively, let Q ⊂ P3 be a general quadric surface that passes through C, let H ⊂ P3 be a general plane that passes through L. Then ¯ + 2H ¯ + G, −KX ∼ Q ¯ ⊂ X ⊃ H ¯ are proper transforms of Q ⊂ P3 ⊃ H, respectively. In particular, where Q lct(X) 6 1/2. We suppose that lct(X) < 1/2. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some positive rational number λ < 1/2. Note that ∅ 6= LCS X, λD ⊂ G, since lct(Y ) = 1/2 by Lemma 8.13. Applying Theorem 2.28 to φ we see that ∅ 6= LCS X, λD ⊂ G ∩ Sφ , where Sφ is a singular fiber of the del Pezzo fibration φ (see Example 1.18). Then we see that ∅ 6= LCS X, λD ⊂ G ∩ Sφ ∩ F, by applying Theorem 2.28 to the log pair (P1 × P2 , λω(D)) and to the P1 -bundle π2 . Let Z1 ∼ = P1 be a section of the natural projection P1 × P1 ∼ = G −→ L ∼ = P1 such that Z1 · Z1 = 0, and let Z2 a fiber of this projection. Then F ∼ Z1 + 3Z2 G

and Sφ |G ∼ Z1 . The curve F ∩ G is irreducible. Thus, we see that G ∩ F ∩ Sφ < +∞, which implies that the set LCS(X, λD) consists of a single point P ∈ G by Theorem 2.7. The log pair (V, λβ(D)) is not log canonical. Since β is an isomorphism on X \ E, we see that β P ∈ LCS V, λβ D ⊆ β P ∪ β E , which implies that LCS(V, λβ(D)) = β(P ) by Theorem 2.7. Let H ⊂ P3 be a general plane. Then 1 ˜ 1 + 3H ˜ ˜ LCS V, λβ D + H = β(P ) ∪ H, 2 72

˜ ⊂V ⊃H ˜ 1 are proper transforms of H ⊂ P3 ⊃ H1 , respectively. But where H ˜ 1 + 3H ˜ ∼Q β D , −KV ∼ H which contradicts Theorem 2.7, because λ < 1/2.

Lemma 9.14. Suppose that ‫(ג‬X) = 3.14. Then lct(X) = 1/2. Proof. Let P ∈ P3 be a point, and let α : V7 → P3 be a blow up of the point P . Then there is a natural P1 -bundle π : V7 → P2 . Let ζ : Z → P(1, 1, 1, 2) be a blow up of the singular point of P(1, 1, 1, 2). Then Z∼ = P OP2 ⊕ OP2 2 , and there is a natural P1 -bundle φ : Z → P2 . There is a plane Π ⊂ P3 and a smooth cubic curve C ⊂ Π such that P 6∈ Π and the diagram ω /Z X MM 88 r M r M r MMMβ γ rr 88 r M r 88 MMM r r r 88 M& x r r 88 φ V Y LL 7A 88 r LLL r A r 88 ζ AAπ r α L r L AA 88 L rr r ε LLL r A 88 L% rr x r 88 η _ _ _ _ _ _ _ _ _ _ / 3 2 Q h PO P 88 Q Q ξ 8 Q Q Q 888 ν ψ Q 8 / P(1, 1, 1, 1, 2) _ _ _ _ _ _ _ _ _ _ _ _ _ _/ P(1, 1, 1, 2) U υ

commutes (see [176, Example 3.6]), where we have the following notation: • the morphism ε is a blow up of the curve C; • the threefold U is a cubic hypersurface in P(1, 1, 1, 1, 2); • the rational map ξ is a projection from the point P ; • the morphism γ is a blow up of the point that dominates P ; • the morphism β is a blow up of the proper transform of the curve C; • the morphism η contracts the proper transform of Π to the point Sing(U ), • the morphism ω contracts a surface R ⊂ X to a curve such that β ◦ α R ⊂ P3 is a cone over the curve C whose vertex is the point P ; • the rational maps ψ and ν are natural projections; • the rational map υ is a linear projection from a point. ¯ ⊂ X be a proper Let E and G be exceptional divisors of γ and β, respectively, and let H 3 transform of a general plane in P that passes through the point P . Then ¯ + 3H ¯ + G, −KX ∼ Π ¯ ⊂ X is the proper transform of the plane Π. Thus, we see that lct(X) 6 1/3. where Π We suppose that lct(X) < 1/3. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some positive rational number λ < 1/3. ¯ ⊂ X be a proper transform of a general line in P3 that intersects the curve C. Then Let L ¯=Π ¯ ·L ¯ + 3H ¯ ·L ¯+G·L ¯ = 3H ¯ ·L ¯ = 3, D·L ¯ and E. which implies that LCS(X, λD) contains no surfaces except possibly Π Let Γ be a general fiber of π ◦ β. Then ¯ · Γ + 3H ¯ ·Γ+G·Γ=Π ¯ · Γ + G · Γ = 2, D·Γ=Π ¯ and E. Thus, by Lemma 2.9, we have which implies that LCS(X, λD) does not contain Π ∅ 6= LCS X, λD ( E ∪ G. 73

Suppose that LCS(X, λD) ⊆ E. Then ∅ 6= LCS V7 , λβ D ⊆ β E , which contradicts Theorem 2.28, because β(E) is a section of π. We see that LCS(X, λD) ( G. Applying Theorem 2.28 to (Z, λω(D)) and φ and Theorem 2.7 to (X, λD), we see that ∅ 6= LCS X, λD ⊆ F, where F is a fiber of the natural projection G → β(G). Then ∅ 6= LCS Y, λγ D ⊆ γ(F ), where γ(F ) is a fiber of the blow up ε over a point in the curve C. Let S ⊂ P3 be a general cone over the curve C, and let O ∈ C be an inflection point such that ε ◦ γ F 6= O. Let L ⊂ S be a line that passes through the point O, and let H ⊂ P3 be a plane that is tangent to the cone S along the line L. Since O is an inflection point of the curve C, the equality multL S · H = 3 ˘ H ˘ and L ˘ be the proper transforms of S, H and L on the threefold Y . Then holds. Let S, 2 ˘ ˘ ˘ LCS Y, λγ(D) + S+H = LCS Y, λγ D ∪ L 3 ˘ which is impossible by Theorem 2.7. due to generality in the choice of S. But −KY ∼ S˘ + H, Lemma 9.15. Suppose that ‫(ג‬X) = 3.15. Then lct(X) = 1/2. Proof. Let Q ⊂ P4 be a smooth quadric hypersurface, let C ⊂ Q be a smooth conic, and let ε : V → Q be a blow up of the conic C ⊂ Q. Then there is a natural morphism η : V → P1 induced by the projection Q 99K P1 from the two-dimensional linear subspace in P4 that contains the conic C ⊂ Q. Then a general fiber of η is a smooth quadric surface in P3 . Take a line L ⊂ Q such that L ∩ C = ∅; then there is a commutative diagram π1 PO 1 fMo MM MMM MMM φ MMM M

1

2

P; × P P PPP ww PPπP2 w w w PPP ww ω PPP w w ' 2 η X HH n6 P p n p n HH pp nnn HH ppp nnn HH p n p n γ HH p β nn ψ # xppp nnn V NNN Y NNN vv vv NNN v v ε NNN vv α N& v{ v Q

where α and β are blow ups of the line L ⊂ Q and its proper transform, respectively, the morphism γ is a blow up of the proper transform of the conic C, the morphism ψ is a P1 -bundle, the morphism ω is a birational contraction of a surface F ⊂ X to a curve such that C ∪ L ⊂ α ◦ γ F ⊂ Q, and α ◦ γ(F ) consists of all lines in Q ⊂ P4 that intersect L and C, the morphism φ is a fibration into del Pezzo surfaces of degree 7, the morphisms π1 and π2 are natural projections. Let E1 and E2 be exceptional surfaces of β and γ, respectively, let H1 , H2 ⊂ Q be general hyperplane sections that pass through the curves L and C, respectively. We have ¯ 1 + 2H ¯ 2 + E2 ∼ H ¯ 2 + 2H ¯ 1 + E1 , −KX ∼ H ¯1 ⊂ X ⊃ H ¯ 2 are proper transforms of H1 ⊂ Q ⊃ H2 , respectively. In particular, where H lct(X) 6 1/2. 74

We suppose that lct(X) < 1/2. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some positive rational number λ < 1/2. Let S be an irreducible surface on the threefold X. Put D = µS + Ω, where Ω is an effective Q-divisor such that S 6⊂ Supp(Ω). Then 1 ¯2 ¯ LCS H2 , ⊂ E1 ∩ H µS + Ω ¯2 2 H by Lemma 5.9. Thus, if µ 6 2, then either S = E1 , or S is a fiber of φ. Let Γ ∼ = P1 be a general fiber of the conic bundle ψ ◦ γ. Then 2 = D · Γ = µS · Γ + Ω · Γ > µS · Γ, which implies that µ 6 2 in the case when either S = E1 , or S is a fiber of φ. Therefore, we see that LCS(X, λD) does not contain surfaces. Applying Theorem 2.28 to the log pair (Y, λγ(D)) and ψ, we see that ¯ ∅ 6= LCS X, λD ( E2 ∪ L, ¯ ⊂ X is a curve such that γ(L) ¯ is a fiber of the conic bundle ψ. where P1 ∼ =L ¯ 6⊂ E1 and L ¯ ⊂ LCS(X, λD). Then Suppose that L ¯ ⊆ LCS Q, λα ◦ γ D ⊆ α ◦ γ L ¯ ∪ C ∪ L, α◦γ L which is impossible by Lemma 2.10. Hence by Theorem 2.7 we see that • either LCS(X, λD) ( E2 , ¯ and L ¯ ⊂ E1 . • or LCS(X, λD) ⊆ L ¯ ⊂ E1 . Note that E1 ∼ ¯·L ¯ = −1 on the surface E1 . We may assume that L = F1 . One has L 1 2 Applying Lemma 2.29 to the log pair (P × P , λω(D)), we see that LCS(X, λD) ⊂ F , because ω D ∼Q −KP1 ×P2 and λ < 1/2. Applying Lemma 2.26 to the log pair (V, λβ(D)) and the fibration η, we see that ∅ 6= LCS X, λD ( E1 ∪ Sφ , where Sφ is a singular fiber of φ, because lct(P1 × P1 ) = 1/2 (see Example 1.18). ¯ = ∅ and |F ∩ S¯φ ∩ E2 | < +∞. Thus, there is point P ∈ E2 such that We have F ∩ L LCS X, λD = P ∈ E2 by Theorem 2.7. But β(E1 ) ∩ β(P ) = ∅. Thus, it follows from Theorem 2.7 that LCS V, λβ D = β P . ˜1 ⊂ V ⊃ H ˜ 2 be the proper transforms of H1 ⊂ Q ⊃ H2 , respectively. Then Let H ˜ 2 + 2H ˜ 1 ∼Q β D , −KV ∼ H but it follows from the generality of H1 and H2 that 1 ˜ ˜ ˜ 1, H2 + 2H1 =β P ∪H LCS V, λβ D + 2 which is impossible by Theorem 2.7, because λ < 1/2. Lemma 9.16. Suppose that ‫(ג‬X) = 3.16. Then lct(X) = 1/2. 75

Proof. Let P1 ∼ = C ⊂ P3 be a twisted cubic curve, let O ∈ C be a point. There is a commutative diagram γ δ / P3 o P O ⊕ O 1 P E 2 2 V U fNNN P P pp8 7 NNN NNN N α NNN NNN

ω

P2 o

p ppp p p η p ppp β p p pp XA PF 2 AA AAυ AA A π2 π1

W

where E is a stable rank two vector bundle on P2 (see the proof of Lemma 8.13), and we have the following notation: • the morphism δ is a blow up of the point O; • the morphism γ contracts a surface G ⊂ U to the curve C ⊂ P3 ; • the morphism α contracts a surface E ∼ = F1 to the fiber of γ over the point O ∈ P3 ; • the morphism β is a blow up of the proper transform of the curve C; • the variety W is a smooth divisor on P2 × P2 of bi-degree (1, 1); • the morphisms π1 and π2 are natural projections; • the morphisms ω and η are natural P1 -bundles; • the morphism υ contracts a surface F ⊂ X to a curve P1 ∼ =Z⊂W such that ω ◦ α(E) = π1 (Z) and η ◦ β(G) = π2 (Z). Take general surfaces H1 ∈ |(ω ◦ α)∗ (OP2 (1))| and H2 ∈ |(η ◦ β)∗ (OP2 (1))|. Then −KX ∼ H1 + 2H2 , which implies that lct(X) 6 1/2. We suppose that lct(X) < 1/2. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some positive rational number λ < 1/2. Note that ∅ 6= LCS X, λD ⊆ E ∩ F, because lct(U ) = 1/2 by Lemma 8.11 and lct(W ) = 1/2 by Theorem 7.1. Applying Lemma 2.12 to the log pair (V7 , λβ(D)), we see that LCS X, λD = E ∩ F ∩ G, where |E ∩ F ∩ G| = 1. Then LCS X, λD + H2 = LCS X, λD ∪ H2 , where H2 ∩ LCS(X, λD) = ∅. But the divisor 1 1 K X + H1 − KX + λD + H2 ∼Q λ − 2 2 is ample, which is impossible by Theorem 2.7.

Lemma 9.17. Suppose that ‫(ג‬X) = 3.17. Then lct(X) = 1/2. Proof. The threefold X is a divisor in P1 × P1 × P2 of tri-degree (1, 1, 1). Take general surfaces H1 ∈ π1∗ OP1 1 , H2 ∈ π2∗ OP1 1 , H3 ∈ π3∗ OP2 1 , where πi is a natural projection of the threefold X onto the i-th factor of P1 × P1 × P2 . Then −KX ∼ H1 + H2 + 2H3 , 76

which implies that lct(X) 6 1/2. There is a commutative diagram υ2 υ1 5/ 1 P1 ×O P1 PO 1 iRo RRR lll PO RRR π l l l π2 ll RRR1 RRR ll ζ RRR lll l l RR l l l η1 η2 X RRRR lll

P1 × P2

l α1 llll l l l l u ll l ω1

π3

/ P2 o

RRR α RRR2 RRR RR) ω2

P1 × P2

where ωi , ηi and υi are natural projections, ζ is a P1 -bundle, and αi is a birational morphism that contracts a surface Ei ⊂ X to a smooth curve Ci ⊂ P1 × P2 such that ω1 (C1 ) = ω2 (C2 ) is a (irreducible) conic. Note that E2 ∼ H1 + H3 − H2 and E1 ∼ H2 + H3 − H1 . We suppose that lct(X) < 1/2. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some positive rational number λ < 1/2. Suppose that the set LCS(X, λD) contains a (irreducible) surface S ⊂ X. Put D = µS + Ω, where µ > 1/λ and Ω is an effective Q-divisor such that S 6⊂ Supp(Ω). Then 2 = D · Γ = µS · Γ + Ω · Γ > µS · Γ, where Γ ∼ = P1 is a general fiber of ζ. Hence S · Γ = 0, which implies that E2 6= S 6= E1 . One also has 2 = D · ∆ = µS · ∆ + Ω · ∆ > µS · ∆, 1 ∼ where ∆ = P is a general fiber of the conic bundle π2 . Hence S · ∆ = 0, which implies that S ∈ π3∗ OP2 m for some m ∈ Z>0 , because E2 6= S 6= E1 and S is an irreducible surface. Then 0 = S · Γ = m 6= 0, which is a contradiction. Thus, we see that the set LCS(X, λD) contains no surfaces. Applying Theorem 2.28 to ζ and using Theorem 2.7, we see that LCS X, λD = F ∼ = P1 , where F is a fiber of the P1 -bundle ζ. Applying Theorem 2.28 to the conic bundle π3 , we see that every fiber of the conic bundle π3 that intersects F must be reducible. This means that π3 F ⊂ ω1 C1 = ω2 C2 ⊂ P2 , which is impossible, because π3 (F ) is a line, and ω1 (C1 ) = ω2 (C2 ) is an irreducible conic.

Lemma 9.18. Suppose that ‫(ג‬X) = 3.18. Then lct(X) = 1/3. Proof. Let Q ⊂ P4 be a smooth quadric hypersurface, C ⊂ Q an irreducible conic, and O ∈ C a point. Then there is a commutative diagram Y fNNN NNN σ NNN NNN N

η

X

γ

τ

U υ

P3

/) 8 PG 1 q qq q α q φ qq / qQ ζ q z qq u q q ψ xq q j m ξ ] _ b d g β

/V

77

ω

where ζ is a blow up of the point O, the morphisms α and γ are blow ups of the conic C and its proper transform, respectively, β is a blow up of the fiber of the morphism α over the point O, the map ψ is a projection from O, the map φ is induced by the projection from the two-dimensional linear subspace that contains the conic C, the morphism τ is a blow up of the line ψ(C), the morphism υ is a blow up of an irreducible conic Z ⊂ P3 such that ψ C ∩ Z 6= ∅, and Z and ψ(C) are not contained in one plane, the morphism σ is a blow up of the proper transform of the conic Z, the map ξ is a projection from ψ(C), the morphism η is a P1 -bundle, and ω is a fibration into quadric surfaces. ¯ be a general fiber of ω ◦ β. Then H ¯ is a del Pezzo surface such that K 2¯ = 7, and Let H H ¯ + 2E + G, −KX ∼ 3H where G and E are the exceptional divisors of β and γ, respectively. In particular, lct(X) 6 1/3. We suppose that lct(X) < 1/3. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some positive rational number λ < 1/3. Note that ∅ 6= LCS X, λD ⊆ G, since lct(V ) = 1/3 by Lemma 8.15 and β(D) ∼Q −KV . Applying Lemma 2.26 to the del Pezzo fibration ω ◦ β and using Theorem 2.7, we see that there is a unique singular fiber S of the fibration ω ◦ β such that ∅ 6= LCS X, λD ⊆ G ∩ S, ¯ = 1/3 holds (see Example 1.18). because the equality lct(H) Let P ∈ G ∩ S be an arbitrary point of the locus LCS(X, λD). Put D = µS + Ω, where Ω is an effective Q-divisor such that S 6⊂ Supp(Ω). Then P ∈ LCS S, λΩ S

by Theorem 2.20. We can identify the surface β(S) with an irreducible quadric cone in P3 . Note that G ∩ S is an exceptional curve on S, so that there is a unique ruling of the cone β(S) intersecting the curve β(G). Let L ⊂ S be a proper transform of this ruling. Then L∩G 6= ∅ (moreover, |L∩G∩S| = 1), while L ∩ E = ∅. Hence P = L ∩ G by Lemma 5.10. We see that LCS(X, λD) = P . One has 2 ¯ ∪ P ∪ E, ¯ ¯ H ∪ P ⊆ LCS X, λD + H + E ⊆ H 3 ¯ is a general fiber of the fibration ω ◦ β. Therefore, the locus because H 2 ¯ + E ⊂X ∅ 6= LCS X, λD + H 3 ¯ and P 6∈ E. But must be disconnected, because P 6∈ H 2 ¯ + 2 E + G + λ − 1/3)KX ¯ − KX + λD + H + E ∼Q H 3 3 is an ample divisor, which is impossible by Theorem 2.7. The proof of Lemma 9.18 implies the following corollary. Corollary 9.19. Suppose that ‫(ג‬X) = 4.4 or ‫(ג‬X) = 5.1. Then lct(X) = 1/3. Lemma 9.20. Suppose that ‫(ג‬X) = 3.19. Then lct(X) = 1/3. 78

Proof. Let Q ⊂ P4 be a smooth quadric, and let L ⊂ P4 be a line such that L ∩ Q = P1 ∪ P2 , where P1 and P2 are different points. Let η : Q 99K P2 be the projection from L. The diagram s X @K@KK s~s~~ @@KKK s @@ KK ss~~ s @@ KKK ss ~~ s @@ KKK ss ~~~ s K β1 @@ ss ~~ β2 s @@ KKKK γ ~ γ1 sss KK2 @ ~ s @ KK s ~~~ α1 KK ss α2 s / o KK s Q U U s 1 2 K KK s s K KK ss s s K KK s s s K KK s s δ1 δ2 K KK ss s s ξ ξ K % ys 1 2 s K% ys π2 π1 / P3 o 3 η P OP2 ⊕ OP2 1 P O ⊕ O 1 2 2 P P K sP K WWWWW s gg g K ζ2 g WWWWW g ζ g s 1 K WW s ggggg K s ω2 WWWWWWW ggggg ω1 g g WWWWW K K g s g WWW%+ yssggggggg

P2

commutes, where αi is a blow up of the point Pi , the morphism βi contracts a surface P2 ∼ = Ei ⊂ X to the point that dominates Pi ∈ Q, the map ξi is a projection from Pi , the map ζi is a projection from the image of Pi , the morphism δi is a contraction of a surface F2 ∼ = Gi ⊂ Ui to a conic Ci ⊂ P3 , the morphism πi is a blow up of the image of Pi , the morphism γi contracts the proper transform of Gi to the proper transform of Ci , and ωi is a natural projection. The map γ1 ◦ γ2−1 is an elementary transformation of a conic bundle (see [166]), and δ1 ◦ β2 E1 ⊂ P3 ⊃ δ2 ◦ β1 E2 are planes that contain the conics C1 and C2 , respectively. Let H be a general hyperplane section of Q such that P1 ∈ H 3 P2 . Then ¯ + E1 + E2 , −KX ∼ 3H ¯ is the proper transform of H on the threefold X. We see that lct(X) 6 1/3. where H We suppose that lct(X) < 1/3. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some λ < 1/3. Note that ∅ 6= LCS X, λD ⊆ E1 ∪ E2 , because lct(Q) = 1/3. By Theorem 2.7, we may assume that ∅ 6= LCS X, λD ⊆ E1 . ¯ 2 ⊂ X be a proper transform of G2 . Then G ¯ 2 ∩E1 = ∅, because α2 (G2 ) ⊂ Q is a quadric Let G cone whose vertex is the point P2 , and the line L is not contained in Q. Hence ∅ 6= LCS P OP2 ⊕ OP2 1 , λγ2 D ⊆ γ2 E1 , where γ2 (E1 ) is a section of ω1 . Applying Theorem 2.28 to ω1 , we obtain a contradiction. Lemma 9.21. Suppose that ‫(ג‬X) = 3.20. Then lct(X) = 1/3. Proof. Let Q ⊂ P4 be a smooth quadric threefold, and let W ⊂ P2 × P2 79

be a smooth divisor of bi-degree (1, 1). Let L1 ⊂ Q ⊃ L2 be lines such that L1 ∩ L2 = ∅; then there exists a commutative diagram WO E EE yy EE y y ω EE y y EE y y EE y y EE y X y A EE y AA β } β2 }} EEυ2 1 A υ1 yyy A } AA EE yy }} A EE y } ~} EE yy y EE y V2 SSS y k V1 A k E y S k S k AA y π kkk } S S y } SSSπ2 EEEE 1 kk AA y } y } S k S k SSS EE yy kk }} α α1 AAA SSSE" |yukykkkkk ~}} 2 S _ o _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _/) P2 2 Q P ψ1

ψ2

where the morphisms αi and βi are blow ups of the line Li and its proper transform, respectively, the morphism ω is a blow up of a smooth curve C ⊂ W of bi-degree (1, 1), the morphisms υi and πi are natural P1 -bundles, and the map ψi is a linear projection from the line Li . ¯ be the exceptional divisor of ω, and let Ei be the exceptional divisor of βi . Then Let H ¯ + 2E1 + 2E2 , −KX ∼ 3H ¯ ⊂ Q is a hyperplane section that contains L1 and L2 . Hence lct(X) 6 1/3. because α2 ◦ β1 (H) We suppose that lct(X) < 1/3. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some positive rational number λ < 1/3. Note that ¯ = ∅, ∅ 6= LCS X, λD ⊆ E1 ∩ E2 ∩ H because lct(V1 ) = lct(V2 ) = 1/3 by Lemma 8.17 and lct(W ) = 1/2 by Theorem 7.1, which gives a contradiction. Lemma 9.22. Suppose that ‫(ג‬X) = 3.21. Then lct(X) = 1/3. Proof. Let π1 : P1 × P2 → P1 and π2 : P1 × P2 → P2 be natural projections. There is a morphism α : X −→ P1 × P2 that contracts a surface E to a curve C such that π1∗ (OP1 (1)) · C = 2 and π2∗ (OP2 (1)) · C = 1. The curve π2 (C) ⊂ P2 is a line. Therefore, there is a unique surface H2 ∈ π2∗ OP2 1 such that C ⊂ H2 . Let H1 be a fiber of the P2 -bundle π1 . Then ¯ 2 + 2E, −KX ∼ 2H¯1 + 3H ¯ i ⊂ X is a proper transform of the surface Hi . In particular, lct(X) 6 1/3. where H We suppose that lct(X) < 1/3. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some rational λ < 1/3. Note that LCS X, λD ⊆ E, because lct(P1 × P2 ) = 1/3 by Lemma 2.22. There is a commutative diagram n6 V hPPPP PPP δ2 nnn O n n PPP nn γ n PPP n nn PPP n n β1 β2 no n / U2 U1 X δ1

ω1

P1 o

α π1

P1 × P2

π2

ω2

/ P2

where V is a Fano threefold of index 2 with one ordinary double point O ∈ V such that ¯2 ∼ −KV3 = 40, the birational morphism βi is a contraction of the surface H = P1 × P1 to a smooth 80

¯ 2 ) to the point O ∈ V such that the rational curve, the morphism δi contracts the curve βi (H rational map δ2 ◦ δ1−1 : U1 99K U2 ¯ 2) ∼ is a standard flop in β1 (H = P1 , the morphism ω1 is a fibration whose general fiber is P1 × P1 , 1 ¯ 2) = O ∈ V . the morphism ω2 is a P -bundle, and γ is a birational morphism such that γ(H 9 The variety V is a section of Gr(2, 5) ⊂ P by a linear subspace of codimension 3. One has −KV ∼ 2 γ H¯1 + γ E , and the divisor γ(H¯1 ) + γ(E) is very ample. There is a commutative diagram γ

X α

P1 × P2

/V

η

ζ

/ P6 ξ

/ P5

such that the embedding ζ is given by the linear system |γ(H¯1 ) + γ(E)|, the map ξ is a linear projection from the point O, the embedding η is given by the linear system |H1 + H2 |. It follows from [85, Theorem 3.6] (see [86, Theorem 3.13]) that U2 ∼ = P E , where where E is a stable rank two vector bundle on P2 such that the sequence 0 −→ OP2 −→ E ⊗ OP2 1 −→ I ⊗ OP2 1 −→ 0 is exact, where I is an ideal sheaf of two general points on P2 . One has c1 (E) = −1 and c1 (E) = 2, and E is a Hulsbergen bundle (see [80]). It follows from [85, Theorem 3.5] that U1 ⊂ P OP1 ⊕ OP1 1 ⊕ OP1 1 ⊕ OP1 1 , and U1 ∈ |2T − F |, where T is a tautological bundle on P(OP1 ⊕ OP1 (1) ⊕ OP1 (1) ⊕ OP1 (1)), and F is a fiber of the projection P(OP1 ⊕ OP1 (1) ⊕ OP1 (1) ⊕ OP1 (1)) → P1 . ¯ 1 is a smooth del Pezzo surface such that K 2¯ = 7, or Either H H1 H1 ∩ C = 1, ¯ 1 , we see because H1 · C = 2. Applying Lemma 2.26 to the morphism ω1 ◦ β1 and the surface H that • either |H1 ∩ C| = 1, • or H1 ∩ LCS(X, λD) = ∅, ¯ 1 ) = 1/3 if H ¯ 1 is smooth. So, there is a fiber L of the projection E → C such that because lct(H LCS X, λD ⊆ L ¯ 2 ∩ E and P = L ∩ C. ¯ Applying Theorem 2.28 to ω2 and by Theorem 2.7. Put C¯ = H U2 , λβ2 D , we see that either LCS(X, λD) = P or LCS(X, λD) = L by Theorem 2.7. Suppose that LCS(X, λD) = L. Then LCS V, λγ D = γ L ⊂ V ⊂ P6 , where γ(L) ⊂ V ⊂ P6 is a line, because −KV · γ(L) = 2 and −KV ∼Q γ(D). We have Sing(V ) = O ∈ γ(L). Let S ⊂ V be a general hyperplane section of V ⊂ P6 such that γ(L) ⊂ S. Then • the surface S is a del Pezzo surface such that KS2 = 5, • the point O is an ordinary double point of the surface S, • the surface S is smooth outside of the point O ∈ γ(L), • the equivalence KS ∼ OP6 (1)|S holds, 81

which implies that S contains finitely many lines that intersect the line γ(L). Let H ⊂ V be a general hyperplane section of V ⊂ P6 . Put Q = γ(L) ∩ H. Then LCS H, λγ D = Q, H

by Remark 2.3, which contradicts Lemma 5.2, because λ < 1/3. ¯ Let F1 be a general fiber of π1 . Then Thus, we see that LCS(X, λD) = P ∈ C. F1 ∩ C = P1 ∪ P2 63 α P , where P1 6= P2 are two points of the curve C. One has P1 ∪ P2 ⊂ H2 ∩ F1 , because C ⊂ H2 . Let Z be a general line in F1 ∼ = P2 such that P1 ∈ Z. Then there is a surface F2 ∈ π2∗ OP2 1 such that Z ⊂ F2 . Let F¯1 ⊂ X ⊃ F¯2 be the proper transforms of F1 and F2 , respectively. Then P 6∈ F¯1 ∪ F¯2 . Let Z¯ ⊂ X be the proper transform of the curve Z. Then −KX · Z¯ = 2 and Z¯ ⊂ F¯1 ∩ F¯2 , ¯ 2 = ∅. Thus, the curve γ(Z) ¯ is a line on V ⊂ P6 such that Sing(V ) = O 6∈ γ(Z). ¯ but Z¯ ∩ H 6 ¯ ⊂ T . Then Let T be a general hyperplane section of the threefold V ⊂ P such that γ(Z) ¯2 + H ¯ 1 + E ∼ 2H ¯ 2 + F¯1 + E ∼ 2F¯2 + F¯1 − E, T¯ ∼ 2H where T¯ is the proper transform of the surface T on the threefold X. Hence ¯ 2 + 2H ¯ 1 + 2E ∼ −KX , F¯1 + F¯2 + T¯ ∼ 3F¯2 + 2F¯1 − E ∼ 2H and applying Theorem 2.7, we see that the locus 2 ¯ ¯ ¯ ¯ F1 + F2 + T P ∪ Z = LCS X, λD + 3 ¯ which is a contradiction. must be connected. But P 6∈ Z,

Lemma 9.23. Suppose that ‫(ג‬X) = 3.22. Then lct(X) = 1/3. Proof. Let π1 : P1 × P2 → P1 and π2 : P1 × P2 → P2 be natural projections. There is a morphism α : X −→ P1 × P2 that contracts a surface E to a curve C contained in a fiber H1 of π1 such that π2 (C) is a conic. We have E ∼ = F2 . Let H2 be a general surface in |π2∗ (OP2 (1))|. The equivalence −KX ∼ 2H¯1 + 3H¯2 + E ¯ i ⊂ X is a proper transform of the surface Hi . Hence lct(X) 6 1/3. holds, where H We suppose that lct(X) < 1/3. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some rational λ < 1/3. Note that LCS X, λD ⊆ E, since lct(P1 × P2 ) = 1/3 by Lemma 2.22. ¯ ⊂ X be the proper Let Q be the unique surface in |π2∗ (OP2 (2))| such that C ⊂ Q, and let Q ¯ ∩ H¯1 = ∅, and there is a commutative diagram transform of the surface Q. Then Q β / P O 2 ⊕O 2 2 X P P α

P1

SSSS SSSγS SSSS SSS) / P2 o_ _ _ _ _ _ _ _ _ _ P 1, 1, 1, 2 ψ

φ

× P2

π2

82

¯ to a curve, γ is a contraction of the surface β(H ¯ 1 ) to a point, such that β is a contraction of Q 1 the morphism φ is a natural P -bundle, and the map ψ is a natural projection. One has 5γ ◦ β E γ ◦ β D ∼Q ∼Q −KP(1,1,1,2) ∼Q OP(1,1,1,2) 5 , 2 which implies that E 6⊆ LCS(X, λD), because λ < 1/3. Applying Theorem 2.28 to φ, we see that there is a fiber F of the projection E → C such that ¯ ∪ F, ∅ 6= LCS X, λD ⊆ E ∩ Q ¯ including the possibility that LCS(X, λD) ⊂ E ∩ Q. ¯ Let M ⊂ P1 × P2 be a general surface in |H1 + H2 |, and Suppose that LCS(X, λD) ⊂ E ∩ Q. ¯ ⊂ X be the proper transform of the surface M . Then let M ¯ ∩ H¯1 = L, M where L is a line on H¯1 ∼ = P2 . Let R be the unique surface in |π2∗ (OP2 (1))| such that α(L) ⊂ R, ¯ be a proper transform of the surface R on the threefold X. Then and let R 2 ¯ ¯ 1, ¯ ¯ ¯ LCS X, λD ∪ L ⊆ LCS X, λD + ⊆ LCS X, λD ∪ L ∪ H M + H1 + R + H2 3 ¯=Q ¯∩H ¯ 1 = ∅ and −KX ∼ M ¯ + H¯1 + R ¯ + H¯2 , which contradicts Theorem 2.7. but L ∩ E ∩ Q ˘ ˘ = γ ◦ β(D). Then Therefore, we see that F ⊆ LCS(X, λD). Put F = γ ◦ β(F ) and D ˘ ⊆ C˘ ∪ F˘ , F˘ ⊆ LCS P 1, 1, 1, 2 , λD ¯ ⊂ P(1, 1, 1, 2) is a curve such that ψ(C) ˘ = π2 (C). where C˘ = γ ◦ β(Q) Let S be a general surface in |OP(1,1,1,2) (2)|. Then S ∼ = P2 and ˘ ⊆ C˘ ∪ F˘ ∩ S; F˘ ∩ S ⊆ LCS S, λD S

but 3D|S ∼Q −5KS , which is impossible by Lemma 2.8.

Lemma 9.24. Suppose that ‫(ג‬X) = 3.23. Then lct(X) = 1/4. Proof. Let O ∈ P3 be a point, let C ⊂ P3 be a conic such that O ∈ C, let Π ⊂ P3 be a unique plane such that C ⊂ Π, and let Q ⊂ P4 be a smooth quadric threefold. Then the diagram u X III uu II u II φ uu II uu u II η u δ uu II u U II u @ u @ } II u @@ υ ω }}} II uu @ u @ u II } @@ } uu II } u ~ } zu $ γ π / _ o _ _ _ _ _ _ o 2 Q V7 KK Y P Y2 s ζ KK

ss s KK 2 ss

KK 2 ss KK

ss KK 2 s K

ξ ss α KKK ψ 2 ss β

s KK s s KK 2 KK 2 ssss K% ys

P3

commutes, where we have the following notation: • the morphism α is a blow up of the point O with an exceptional divisor E; • the morphism π is a natural P1 -bundle; • the morphisms β and δ are blow ups of C and its proper transform, respectively; • the morphism γ contracts the proper transform of the plane Π to a point; • the morphism φ contracts the proper transform of the plane Π to a curve; • the morphism η contracts the proper transform of E to a curve L ⊂ Y such that γ Π ∈ γ L ⊂ Q ⊂ P4 and γ(L) is a line in P4 ; 83

• the morphism ω is a natural P1 -bundle; • the morphism υ is a blow up of the line γ(L); • the maps ψ, ξ and ζ are projections from O, γ(Π) and γ(L), respectively. Note that E is a section of π. ¯ ⊂ X be a proper transform of Π ⊂ P3 . Then lct(X) 6 1/4, because Let Π ¯ + 2E ¯ + 3G, −KX ∼ 4Π ¯ and G are exceptional surfaces of η and δ, respectively. where E We suppose that lct(X) < 1/4. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some positive rational number λ < 1/4. Note that ¯∩Π ¯ ∩ G, ∅ 6= LCS X, λD ⊆ E because lct(V7 ) = 1/4 by Lemma 9.26, lct(Y ) = 1/4 by Lemma 8.16 and lct(U ) = 1/3 by Lemma 8.17. Let R ⊂ P3 be a general cone over C whose vertex is P ∈ P3 , let H1 ⊂ P3 be a general plane such that O ∈ H1 3 P , and let H2 ⊂ P3 be a general plane such that P ∈ H2 . Then ¯ ∼ α◦δ ∗ R −E ¯ − G, H¯1 ∼ α ◦ δ ∗ H1 − E, ¯ H¯2 ∼ α ◦ δ ∗ H2 , R ¯ H ¯ 1, H ¯ 2 are proper transforms of R, H1 , H2 on the threefold X, respectively. One has where R, ¯+H ¯1 + H ¯ 2, −KX ∼ Q but it follows from the generality of R, H1 , H2 that the locus 3 ¯ ¯ ¯ Q + H1 + H2 = LCS X, λD ∪ P, LCS X, λD + 4 is disconnected, which is impossible by Theorem 2.7.

Lemma 9.25. Suppose that ‫(ג‬X) = 3.24. Then lct(X) = 1/3. Proof. Let W is a divisor of bi-degree (1, 1) on P2 × P2 . There is a commutative diagram X pp ζ pppp p π ppp p p xp F1 P1 o ξ

α

γ

/W

ω1

/ P2

where ω1 is a natural P1 -bundle, the morphism α contracts a smooth surface E∼ = P1 × P1 to a fiber L of ω1 , γ is a blow up of the point ω1 (L), the morphism ξ is a P1 -bundle, and ζ is a F1 -bundle. Let ω2 : X → P2 be a natural P1 -bundle that is different from ω1 . Then there is a surface G ∈ ω2∗ OP2 1 ¯ ⊂ X be a proper transform of G. Then such that L ⊂ G, because ω2 (L) is a line. Let G ¯ + 3E, −KX ∼ 2F + 2G where E is the exceptional divisor of α, and F is a fiber of ζ. We see that lct(X) 6 1/3. We suppose that lct(X) < 1/3. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some positive rational number λ < 1/3. Note that ∅ 6= LCS X, λD ⊆ E, since lct(W ) = 1/2 by Theorem 7.1. We may assume that F ∩ LCS(X, λD) 6= ∅. Then F1 ∼ = F ⊆ LCS X, λD ⊆ E ∼ = P1 × P1 by Lemma 2.26, because lct(F ) = 1/3 (see Example 1.18), which is a contradiction. 84

Lemma 9.26. Suppose that ‫(ג‬X) = 3.25. Then lct(X) = 1/3. Proof. Let L1 ⊂ P3 ⊃ L2 be lines such that L1 ∩ L2 = ∅. Then there is a commutative diagram 6 P3 hPPP PPP nnn PPαP2 PPP PPP β1 / V2 X

α1 nnnn

nn nnn β2 no nn V1 ω1

ω

P1 o

γ1

P1 × P1

γ2

ω2

/ P1

where the morphisms αi and βi are blow ups of the line Li and its proper transform, respectively, the morphism ωi is a natural P2 -bundle, the morphisms ω and γi are P1 -bundles. Note that V1 ∼ = P OP1 ⊕ OP1 ⊕ OP1 1 . = V2 ∼ Let H1 and H2 be proper transforms on X of planes in P3 such that Li ⊂ α(Hi ). Then −KX ∼ 2H1 + 2H2 + E1 + E2 ∼ 3H1 + H2 + 2E1 ∼ H1 + 3H2 + 2E2 , where Ei is an exceptional divisors of βi . Hence lct(X) 6 1/3. We suppose that lct(X) < 1/3. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some positive rational number λ < 1/3. Applying Lemma 2.26 to the F1 -fibrations ω2 ◦ β1 and ω1 ◦ β2 , we obtain a contradiction, because the equality lct(F1 ) = 1/3 holds (see Example 1.18). Remark 9.27. Actually, the result of Lemma 9.26 is contained in Corollary 6.4, but we still prefer to give a detailed proof that may have further applications. Lemma 9.28. Suppose that ‫(ג‬X) = 3.30. Then lct(X) = 1/4. Proof. Let O ∈ P3 be a point, and let γ : V7 → P3 be a blow up of the point O. Then there is a P1 -bundle π : V7 → P1 . Take a line O ∈ L ⊂ P3 ; then the diagram o F1 PPPP PPP τ ooo O o o PPP oo ζ o PPP o oo PP' o o w o 2 XA P!P 1 _@ > PN A ~ @@ } A ~ β ~ AAξ # @@ }} ~ AA }}π ~~ η @@@ A } ~ } ~ ) V V 7 @@ 2 } @@ }} @@ @ } }}γ α @@ } M φ V \ ^ 3 `~} b h q ψ υ

P

commutes, where α and ξ are blow ups of the line L and its proper transforms, respectively, the morphism η is is a natural P2 -bundle, the morphism β is a blow up of the curve ∼ P1 ∼ C ⊂ V P O ⊕ O ⊕ O 1 1 1 1 = = P P P such that β(C) = O, the morphisms ζ and υ are P1 -bundles, the maps φ and ψ are linear projections from L and O, respectively, and τ is a blow up of the point ψ(L) ∈ P2 . Let T be the proper transform on X of a general plane in P3 that passes through L ⊂ P3 , and let G be the exceptional divisor of the blow up β. Then −KX ∼ 4T + 3E + 2G, where E is the proper transform on X of the exceptional divisor of α. In particular, lct(X) < 1/4. We suppose that lct(X) < 1/4. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some rational number λ < 1/4. Note that ∅ 6= LCS X, λD ⊆ G, 85

because lct(V ) = 1/4 by Lemma 7.2. However, every fiber of the morphism η ◦ β is isomorphic to F1 , which is impossible by Lemma 2.26, because lct(F1 ) = 1/3 by Example 1.18. The proof of Lemma 9.28 implies the following corollary (cf. [176, Example 3.3]). Corollary 9.29. Suppose that ‫(ג‬X) = 4.12. Then lct(X) = 1/4. Remark 9.30. Actually, the results of Lemma 9.28 and Corollary 9.29 are contained in Corollary 6.4, but we still prefer to give a detailed proof that may have further applications. 10. Fano threefolds with ρ > 4 We use the assumptions and notation introduced in section 1. Lemma 10.1. Suppose that ‫(ג‬X) = 4.1. Then lct(X) = 1/2. Proof. The threefold X is a divisor on P1 × P1 × P1 × P1 of multidegree (1, 1, 1, 1). Let h i x1 : y1 , x2 : y2 , x3 : y3 , x4 : y4 be coordinates on P1 × P1 × P1 × P1 . Then X is given by the equation F x1 , y1 , x2 , y2 , x3 , y3 , x4 , y4 = 0, where F is a of multidegree (1, 1, 1, 1). Let π1 : X → P1 × P1 × P1 be a projection given by h h i i x1 : y1 , x2 : y2 , x3 : y3 , x4 : y4 7→ x2 : y2 , x3 : y3 , x4 : y4 ∈ P1 × P1 × P1 , and let π2 , π3 and π4 : X → P1 × P1 × P1 be projections defined in a similar way. Put F = x1 G x2 , y2 , x3 , y3 , x4 , y4 + y1 H x2 , y2 , x3 , y3 , x4 , y4 , where G(x2 , y2 , x3 , y3 , x4 , y4 ) and H(x2 , y2 , x3 , y3 , x4 , y4 ) are multi-linear forms that do not depend on x1 and y1 . Then π1 is a blow up of a curve C1 ⊂ P1 × P1 × P1 given by the equations G x2 , y2 , x3 , y3 , x4 , y4 = H x2 , y2 , x3 , y3 , x4 , y4 = 0, which define a surface E1 ⊂ P1 × P1 × P1 × P1 that is contracted by π1 . The equations x1 = H x2 , y2 , x3 , y3 , x4 , y4 = 0 define a divisor H1 ⊂ X such that −KX ∼ 2H1 + E1 , which implies that lct(X) 6 1/2. We suppose that lct(X) < 1/2. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some positive rational number λ < 1/2. Let E2 , E3 , E4 be surfaces in X defined in a way similar to E1 . Then ∅ 6= LCS X, λD ⊆ E1 ∩ E2 ∩ E3 ∩ E4 , because lct(P1 × P1 × P1 ) = 1/2 by Lemma 2.22. But Ei ⊂ P1 × P1 × P1 × P1 is given by ∂F x1 , y1 , x2 , y2 , x3 , y3 , x4 , y4 ∂F x1 , y1 , x2 , y2 , x3 , y3 , x4 , y4 = = 0, ∂xi ∂yi which implies that the intersection E1 ∩ E2 ∩ E3 ∩ E4 is given by the equations ∂F ∂F ∂F ∂F ∂F ∂F ∂F ∂F = = = = = = = = 0. ∂x1 ∂y1 ∂x2 ∂y2 ∂x3 ∂y3 ∂x4 ∂y4 Hence E1 ∩ E2 ∩ E3 ∩ E4 = Sing(X) = ∅, and LCS(X, λD) = ∅. Lemma 10.2. Suppose that ‫(ג‬X) = 4.2. Then lct(X) = 1/2. 86

Proof. Let Q1 ⊂ P4 ⊃ Q2 be quadric cones, whose vertices are O1 ∈ P4 3 O2 , respectively. Let O1 6∈ S1 ⊂ Q1 ⊂ P4 be a hyperplane section of Q1 , and let C1 ⊂ |−KS1 | be a smooth elliptic curve. Then the diagram X QQQ mmm QQQ β QQQ2 QQQ QQQ (

β1 mmmm

ζ1

U1

mm mmm m m m v

ζ2

U2

λ > 1/2 7 2 2 by Lemma 2.24, and hence ε > 4/3. ¯ or T¯1 6⊆ Supp(D) ¯ by Remark 2.23. We may assume that either E 6⊆ Supp(D) ¯ Suppose that E 6⊆ Supp(D). Let Z be a general fiber of the projection E → C. Then ¯ · Z = ε + Ω · Z > ε, 1 = −KU · Z = D ¯ which is a contradiction, because ε > 4/3. Thus, we see that T¯1 6⊆ Supp(D). 2 ¯ ⊂ T¯1 be a proper transform of a general line in T1 ∼ Let ∆ = P that passes through Q1 . Then ¯ > 1/λ > 7/3, ¯ =D ¯ ·∆ ¯ > multΓ D 2 = −KU · ∆ 1

¯ 6⊂ Supp(D) ¯ and ∆ ¯ ∩ Γ1 6= ∅. The obtained contradiction completes the proof. because ∆

Lemma 10.5. Suppose that ‫(ג‬X) = 4.6. Then lct(X) = 1/2. Proof. There is a birational morphism α : X → P3 that blows up three disjoint lines L1 , L2 , L3 . Let Hi be the proper transform on X of a general plane in P3 such that Li ⊂ α(Hi ). Then −KX ∼ 2H1 + E1 + H2 + H3 ∼ 2H2 + E2 + H1 + H3 ∼ 2H3 + E3 + H1 + H2 , where Ei is the exceptional divisor of α such that α(Ei ) = Li . In particular, we see that lct(X) 6 1/2. We suppose that lct(X) < 1/2. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some positive rational number λ < 1/2. 2 = 7, the linear system |H | has The surface Hi is a smooth del Pezzo surface such that KH i i 1 no base points and induces a smooth morphism φi : X → P , whose fibers are isomorphic to Hi . Suppose that |LCS(X, λD)| < +∞. We may assume that LCS(X, λD) 6⊆ E1 . Then the set 1 LCS X, λD + H1 + E1 2 is disconnected, which is impossible by Theorem 2.7, because H2 + H3 + (λ − 1/2)KX is ample. We may assume that H1 ∩ LCS(X, λD) 6= ∅. Then ∅ 6= H1 ∩ LCS X, λD ⊆ LCS H1 , λD H1

by Remark 2.3. Put C2 = E2 |H1 and C3 = E3 |H1 . Then and there is a unique curve that

P1

C2 · C2 = C3 · C3 = −1, ∼ = C ⊂ H1 such that C · C2 = C · C3 = 1 and C · C = −1. Note LCS H1 , λD =C H1

by Lemma 5.9. There is a unique smooth quadric Q ⊂ P3 that contains L1 , L2 , L3 . Note that ¯ ∩ H1 = C, Q ¯ ⊂ X is a proper transform of the surface Q. where Q ¯ to a curve of tri-degree (1, 1, 1). There is a morphism σ : X → P1 × P1 × P1 contracting Q ¯ Since Q ∩ H1 = C, one obtains (see Remark 2.3) that ¯ LCS X, λD ⊃ Q, ¯ because lct(P1 × P1 × P1 ) = 1/2. Put and hence LCS(X, λD) = Q, ¯ + Ω, D = µQ 91

¯⊂ where µ > 1/λ > 2, and Ω is an effective Q-divisor such that Q 6 Supp(Ω). Then α D = µQ + α Ω , which is impossible, because α(D) ∼Q 2Q ∼ −KP3 and µ > 2.

Lemma 10.6. Suppose that ‫(ג‬X) = 4.7. Then lct(X) = 1/2. Proof. There is a birational morphism α : X → W such that • the variety W is a smooth divisor of bi-degree (1, 1) on P2 × P2 ; • the morphism α contracts two (irreducible) surfaces E1 6= E2 to two disjoint curves L1 and L2 ; • the curves Li are fibers of one natural P1 -bundle W → P2 . There is a surface H ⊂ W such that −KX ∼ 2H and L1 ⊂ H ⊃ L2 . Then ¯ + E1 + E2 , −KX ∼ 2H ¯ is a proper transform of H on the threefold X. In particular, lct(X) 6 1/2. where H We suppose that lct(X) < 1/2. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some λ < 1/2. Then ∅ 6= LCS X, λD ⊆ E1 ∪ E2 , since lct(W ) = 1/2 by Theorem 7.1 and α(D) ∼Q −KW . We may assume that LCS(X, λD) ∩ E1 6= ∅. Let β : X → Y be a contraction of E2 . Then LCS Y, λβ D 6= ∅ and β(D) ∼Q −KY , which contradicts Lemma 9.25.

Lemma 10.7. Suppose that ‫(ג‬X) = 4.8. Then lct(X) = 1/3. Proof. There is blow up α : X → P1 × P1 × P1 of a curve C ⊂ P1 × P1 × P1 such that C ⊂ F1 and C · F2 = C · F3 = 1, where Fi is a fiber of the projection of P1 × P1 × P1 to its i-th factor. There is a surface P1 × P1 ∼ = G ∈ F2 + F3 such that C ⊂ G. Let E be the exceptional divisor of α. Then ¯ + 3E, −KX ∼ 2F¯1 + 2G ¯ are proper transforms of F1 and G, respectively. In particular, lct(X) 6 1/3. where F¯1 and G We suppose that lct(X) < 1/3. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some positive rational number λ < 1/3. Note that ∅ 6= LCS X, λD ⊆ E, because lct(P1 × P1 × P1 ) = 1/2 and α(D) ∼Q −KP1 ×P1 ×P1 . Let Q be a quadric cone in P4 . Then there is a commutative diagram ggg X TTTTT TTT γ ggggg TTT g TTT β g g g g TTT g g g g sg TTT ) 1 1 1 V U P × P × PS M M w M SSSS MMM w SSSS π www δ M MMM SSSS ww MMM ξ φ SS) {ww & _ o _ _ _ _ _ _ _ _ _ 1 1 Q P ×P αggggggg

ψ

where we have the following notations: • V is a variety with ‫(ג‬V ) = 3.31; ¯ to a curve; • the morphism β is a contraction of the surface G 1 1 ∼ ¯ • the morphism γ is a contraction of F1 = P × P to an ordinary double point; 92

the morphism δ is a blow up of the vertex of the quadric cone Q ⊂ P4 ; the morphism ξ is a blow up of a smooth conic in Q; the map ψ is a projection from the vertex of the cone Q; the morphism φ is a projection that is given by |F2 +F3 |, i. e. the projection of P1 ×P1 ×P1 onto the product of the last two factors; • the morphism π is a natural P1 -bundle. It follows from Corollary 6.4 that lct(V ) = 1/3. On the other hand, lct(U ) = 1/3 by Lemma 2.27. Hence ¯ ∩ F¯1 = ∅, ∅ 6= LCS X, λD ⊆ E ∩ G • • • •

which is a contradiction.

Lemma 10.8. Suppose that ‫(ג‬X) = 4.9. Then lct(X) = 1/3. Proof. There exists a point O ∈ P3 , and there exist lines L1 ⊂ P3 ⊃ L2 such that L1 ∩ L2 = ∅, the line L1 passes through the point O, and there is a commutative diagram φ

/ P2 o `@@ @@ π @@ @@

F1 cG GG

GG GG ξ GGG

Z obE EE

ι1

= V7 `@U@UUUUUU UUUU ε1 zzz @@ EE UUUU @@ EE zz UUUU z EE ζ1 @ z ε UU @ 2 z EE @@ EE z @ z @@ EE zz @@ EE zz @ E z z υ2 υ1 o / U2 U1 X

χ

P9 2 × P1

ρ

ζ2

τ

v vv vv v v vv v {v P1 cGG GG GG γ1 GGG

P1

×

Y n nn β2 nnnn n n n n vnnnn ω1 V1 A AA AA α1 AA AA i AAA i i ii AA i i i AA ii i i i AA iiiη i i A i i t ii δ1

ν

P1

P3

γ2

δ2 NNN NNN β1 NNN NNN N&

σ ι2

V2

α2 kkk kkk k k k kkk kkk ukkkkkk

ω2

/ P1

that uses the following notation: • the morphism σ is a blow up of the point O; • the morphism π is a natural P1 -bundle; • the morphism αi is a blow up of the line Li ; • the morphisms βi , υi and εi are blow ups of the proper transforms of the line Li ; • the morphisms τ and δ1 are blow ups of curves that are the preimages of the point O; • the morphism δ2 is a blow up of the point that dominates the point O; • the morphisms ω1 and ω2 are natural P2 -bundles, where V1 ∼ = V2 ∼ = P OP1 ⊕ OP1 ⊕ OP1 1 ; • the morphisms ι1 , ι2 , γ1 and γ2 are natural projections; ¯ 1 ), where L ¯ 1 ⊂ V7 is a proper transform of • the morphism φ is a blow up of the point π(L L1 ; 93

• the morphisms ρ and ζi contract the proper transforms of the plane Π ⊂ P3 such that L2 ⊂ Π 3 O; • the morphisms η, ν, ξ, χ, γ1 , γ2 and ι2 are natural P1 -bundles. Let Hi ⊂ X be the proper transform of a general plane in P3 that contains Li . Then −KX ∼ 3H1 + H2 + 2E1 + G, where Ei and G be the exceptional divisor of υi and τ , respectively. Thus, we have lct(X) 6 1/3. We suppose that lct(X) < 1/3. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some positive rational number λ < 1/3. Note that ∅ 6= LCS X, λD ⊆ G, since lct(Y ) = 1/3 by Lemma 9.26. But the surface G is not a fiber of the smooth morphism ω1 ◦ β2 ◦ τ : X −→ P1 , so we obtain a contradiction applying Lemma 2.26 to the morphism ω1 ◦ β2 ◦ τ .

The proof of Lemma 10.8 implies the following. Corollary 10.9. Suppose that ‫(ג‬X) = 5.2. Then lct(X) = 1/3. Remark 10.10. Actually, the results of Lemma 10.8 and Corollary 10.9 are contained in Corollary 6.4, but we still prefer to give a detailed proof that may have further applications. The following result is implied by Corollaries 9.19 and 10.9, Lemma 2.30 and Example 1.18. Corollary 10.11. Suppose that ρ > 5. Then ( 1/3 whenever ‫(ג‬X) ∈ {5.1, 5.2}, lct X = 1/2 in the remaining cases. Lemma 10.12. Suppose that ‫(ג‬X) = 4.13 and X is general. Then lct(X) = 1/2. Proof. Let F1 ∼ = F2 ∼ = F3 ∼ = P1 × P1 be fibers of three different projections P1 × P1 × P1 −→ P1 , respectively. There is a contraction α : X → P1 × P1 × P1 of a surface E ⊂ X to a curve C ⊂ P1 × P1 × P1 such that C · F1 = C · F2 = 1 and C · F3 = 3. Then there is a smooth surface P1 × P1 ∼ = G ∈ F1 + F2 such that C ⊂ G. In particular, we see that ¯ + E + 2F¯3 , −KX ∼ 2G ¯ are proper transforms of F3 and G, respectively. Hence lct(X) 6 1/2. where F¯3 and G We suppose that lct(X) < 1/2. Then there exists an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some positive rational number λ < 1/2. Note that ∅ 6= LCS X, λD ⊆ E ∼ = F4 , because lct(P1 × P1 × P1 ) = 1/2 and α(D) ∼ Q −KP1 ×P1 ×P1 . There are smooth surfaces H1 ∈ |3F1 + F3 and H2 ∈ |3F2 + F3 such that C = G · H1 = G · H2 , ¯ i be a proper transform of Hi on the threefold X. Then and H1 ∼ = H2 ∼ = P1 × P1 . Let H ¯1 ∩ G ¯=H ¯2 ∩ G ¯ = ∅. H 94

There is a commutative diagram gg X VVVVVV gggggvvv VVVV γ g g g g VVVV2 v ggg g g v g VVVV g v v gggg VVVV g v g g g v VVV* g g v g sgg v α U2 U1/ vv v vv β // v vv // v vv // {vv // 1 × P1 × P1 V P // BB BB // rrr r r φ2 φ1 / BB rr BB ξ2 // rrr ξ1 r B // BB ζ v BB // vv π v B v BB // vv B! {vv ' γ1

P1 × P1

P1 × P1

P1 × P1

¯ and H ¯ i to a smooth curves, the morphisms such that β and γi are contractions of the surfaces G π and φi are P1 -bundles, the morphisms ζ and ξi are projections that are given by the linear systems |F1 + F2 | and |Fi + F3 |, respectively. ¯1 ∩ G ¯ = ∅ that It follows from H • either the log pair (V, λβ(D)) is not log canonical, • of the log pair (U1 , λγ1 (D)) is not log canonical. Applying Theorem 2.28 to (V, λβ(D)) or (U1 , λγ1 (D)) (and the fibration π or φ1 ) and using Theorem 2.7, we see that LCS X, λD = Γ, where Γ is a fiber of the natural projection E → C. We may assume that α(Γ) ∈ F3 . Let F¯3 ⊂ X be the proper transform of the surface F3 . Put D = µF¯3 + Ω, where Ω is an effective Q-divisor on X such that F¯3 6⊂ Supp(Ω). Then µF3 + α Ω ∼Q 2 F1 + F2 + F3 , which gives µ 6 2. The log pair (F¯3 , λΩ|F¯3 ) is not log canonical along Γ ⊂ F¯3 by Theorem 2.20. One has Ω ∼Q −KF¯3 , ¯ F3 KF2¯ 3

and F¯3 is a del Pezzo surface such that = 5. Note that F¯3 may be singular. Namely, we have Sing F¯3 = ∅ ⇐⇒ C ∩ F3 = F3 · C = 3, and Sing(F¯3 ) ⊂ Γ. The following cases are possible: • the surface F¯3 is smooth and |C ∩ F3 | = 3; • the surface F¯3 has one ordinary double point and |C ∩ F3 | = 2; • the surface F¯3 has a singular point of type A2 and |C ∩ F3 | = 1. We have lct(F¯3 ) 6 λ < 1/2. Thus, it follows from Examples 1.18 and 5.3 that |C ∩ F3 | = 1, which is impossible if the threefold X is sufficiently general. 11. Upper bounds We use the assumptions and notation introduced in section 1. The purpose of this section is to find upper bounds for the global log canonical thresholds of the varieties X with o n ‫ ג‬X ∈ 1.1, 1.2, . . . , 1.17, 2.1, . . . , 2.36, 3.1, . . . , 3.31, 4.1, . . . , 4.13, 5.1, . . . , 5.7, 5.8 . Lemma 11.1. Suppose that ‫(ג‬X) = 1.8. Then lct(X) 6 6/7. 95

Proof. The linear system | − KX | does not have base points and induces an embedding X ⊂ P10 , and the threefold X contains a line L ⊂ X (see [168], [178]). It follows from [98, Theorem 4.3.3] (see [45], [178]) that there is a commutative diagram ρ U _ _ _ _ _ _/ W α

β

X _ _ _ψ_ _ _/ P3 where α is a blow up of the line L, the map ρ is a composition of flops, the morphism β is a blow up of a smooth curve of degree 7 and genus 3, and ψ is a double projection from L. Let S ⊂ X be the proper transform of the exceptional surface of β. Then multL S = 7 and S ∼ −3KX , which implies that lct(X) 6 6/7.

Lemma 11.2. Suppose that ‫(ג‬X) = 1.9. Then lct(X) 6 4/5. Proof. The linear system | − KX | does not have base points and induces an embedding X ⊂ P11 , and the threefold X contains a line L ⊂ X (see [168], [178]). It follows from [98, Theorem 4.3.3] (see [45], [178]) that there is a commutative diagram ρ U _ _ _ _ _ _/ W α

β

X _ _ _ψ_ _ _/ Q where Q ⊂ P4 is a smooth quadric threefold, α is a blow up of the line L, the map ρ is a composition of flops, the morphism β is a blow up along a smooth curve of degree 7 and genus 2, and ψ is a double projection from the line L. Let S ⊂ X be the proper transform of the exceptional surface of β. Then multL S = 5 and S ∼ −2KX , which implies that lct(X) 6 4/5.

Lemma 11.3. Suppose that ‫(ג‬X) = 1.10. Then lct(X) 6 2/3. Proof. The linear system | − KX | does not have base points and induces an embedding X ⊂ P13 , and the threefold X contains a line L ⊂ X (see [168], [178]). It follows from [98, Theorem 4.3.3] (see [45], [178]) that the diagram ρ U _ _ _ _ _ _/ W α

β

X _ _ _ψ_ _ _/ V5 commutes, where V5 is a smooth section of Gr(2, 5) ⊂ P9 by a linear subspace of dimension 6, the morphism α is a blow up of the line L, the map ρ is a composition of flops, the morphism β is a blow up of a smooth rational curve of degree 5, and ψ is a double projection from L. Let S ⊂ X be the proper transform of the exceptional surface of β. Then multL S = 3 and S ∼ −KX , which implies that lct(X) 6 2/3. Lemma 11.4. Suppose that ‫(ג‬X) = 2.2. Then lct(X) 6 13/14. 96

Proof. There is a smooth divisor B ⊂ P1 × P2 of bi-degree (2, 4) such that the diagram X PPP PPP φ nnn PPP2 π PPP PPP (/ 2 P1 × P2 P

φ1 nnnn

n nnn nnn v n P1 o π1

π2

commutes, where π is a double cover branched along B, the morphisms π1 and π2 are natural projections, φ1 is a fibration into del Pezzo surfaces of degree 2, and φ2 is a conic bundle. ¯ 1 = π(H1 ). Then the intersection Let H1 be a general fiber of φ1 . Put H ¯1 ∩ B ⊂ H ¯1 ∼ C=H = P2 is a smooth quartic curve. There is a point P ∈ C such that multP C · L > 3, ¯1 ∼ where L ⊂ H = P2 is a line that is tangent to C at the point P . The curve π2 (L) is a line. Thus, there is a unique surface H2 ∈ φ∗2 OP2 1 such that φ2 (H2 ) = π2 (L). Hence −KX ∼ H1 + H2 . ¯ 2 = π(H2 ). Then Let us show that lct(X, H1 + H2 ) 6 13/14. Put H 13 13 ¯ 1 2 1 ¯ 6= ∅ ⇐⇒ LCS P × P , B + 6= ∅ LCS X, H1 + H2 H1 + H2 14 2 14 by [105, Proposition 3.16]. Let α : V → P1 × P2 be a blow up of the curve C. Then 3 1 ˜ 13 ˜ 1 13 ∗ ˜ 2 + E ∼Q α KP1 ×P2 + B + ¯1 + H ¯2 KV + B + H1 + H H , 2 14 7 2 14 ˜ 2 ⊂ V are proper transforms of B, H ¯ 1, H ¯ 2 , respectively. But the log pair ˜ H ˜ 1, H where B, 13 ˜ 3 V, H2 + E 14 7 is not log terminal along the fiber Γ of the morphism α such that α(Γ) = P , because ¯ 2 > multP C · L > 3 ˜ 2 · E = multP C · H multΓ H due to the choice of the fiber H1 . We see that 3 3 1 ˜ 13 ˜ 13 ˜ ˜ Γ ⊆ LCS V, H2 + E ⊆ LCS V, B + H1 + H2 + E , 14 7 2 14 7 which implies that lct(X, H1 + H2 ) 6 13/14. Hence the inequality lct(X) 6 13/14 holds.

Remark 11.5. It follows from Lemmas 2.26 and 5.1 that lct(X) > 2/3 if ‫(ג‬X) = 2.2 and the threefold X satisfies the following generality condition: any fiber of φ1 satisfies the hypotheses of Lemma 5.1. Lemma 11.6. Suppose that ‫(ג‬X) = 2.7. Then lct(X) 6 2/3. Proof. There is a commutative diagram XA AA β ~~ AA ~ ~ AA ~ A ~ ~ _ _ _ _ _ _ _ / P1 Q α

ψ

where Q ⊂ P4 is a smooth quadric threefold, α is a blow up of a smooth curve that is a complete intersection of two divisors S1 , S2 ∈ OP4 2 , Q

97

the morphism β is a fibration into del Pezzo surfaces of degree 4, and ψ is a rational map that is induced by the pencil generated by the surfaces S1 and S2 . Then lct(X) 6 2/3, because 3¯ 1 S1 + E, 2 2

−KX ∼Q

where S¯1 ⊂ X is a proper transform of the surface S1 , and E is the exceptional divisor of α.

Lemma 11.7. Suppose that ‫(ג‬X) = 2.9. Then lct(X) 6 3/4. Proof. There is a commutative diagram X } AAA β } } AA }} AA } ~} P3 _ _ _ _ _ _ _/ P2 α

ψ

where α is a blow up of a smooth curve C ⊂ P3 of degree 7 and genus 5 that is a scheme-theoretic intersection of cubic surfaces in P3 , the morphism β is a conic bundle, and ψ is a rational map that is given by the linear system of cubic surfaces that contain C. One has −KX ∼Q

1 4 S + E, 3 3

where S ∈ |β ∗ (OP2 (1))|, and E is the exceptional divisor of α. We see that lct(X) 6 3/4.

Lemma 11.8. Suppose that ‫(ג‬X) = 2.12. Then lct(X) 6 3/4. Proof. There is a commutative diagram X } AAA β } } AA }} AA } ~} P3 _ _ _ ψ_ _ _ _/ P3 α

where α and β are blow ups of smooth curves C ⊂ P3 and Z ⊂ P3 of degree 6 and genus 3 that are scheme-theoretic intersections of cubic surfaces in P3 , and ψ is a birational map that is given by the linear system of cubic surfaces that contain C. Then −KX ∼Q

1 4 S + E, 3 3

where S ∈ |β ∗ (OP3 (1))|, and E is the exceptional divisor of α. We see that lct(X) 6 3/4.

Lemma 11.9. Suppose that ‫(ג‬X) = 2.13. Then lct(X) 6 2/3. Proof. There is a commutative diagram X ~ AAA β ~ AA α ~ AA ~~ A ~ ~ _ _ _ _ _ _ _ / P2 Q ψ where Q ⊂ P4 is a smooth quadric threefold, α is a blow up of a smooth curve C ⊂ Q of degree 6 and genus 2, the morphism β is a conic bundle, and ψ is a rational map that is given by the linear system of surfaces in |OP4 (2)|Q | that contain the curve C. One has −KX ∼Q

3 1 S + E, 2 2

where S ∈ |β ∗ (OP2 (1))|, and E is the exceptional divisor of α. We see that lct(X) 6 2/3. Lemma 11.10. Suppose that ‫(ג‬X) = 2.16. Then lct(X) 6 1/2. 98

Proof. There is a commutative diagram X } AAA β } } AA }} AA }~ } V4 _ _ _ _ _ _ _/ P2 α

ψ

where V4 ⊂ P5 is a smooth complete intersection of two quadric hypersurfaces, α is a blow up of an irreducible conic C ⊂ V4 , the morphism β is a conic bundle, and ψ is a rational map that is given by the linear system of surfaces in |OP5 (1)|V4 | that contain C. One has −KX ∼ 2S + E, where S ∈

|β ∗ (O

P2 (1))|,

and E is the exceptional divisor of α. We see that lct(X) 6 1/2.

Lemma 11.11. Suppose that ‫(ג‬X) = 2.17. Then lct(X) 6 2/3. Proof. There is a commutative diagram X ~ AAA β ~ AA ~ AA ~~ A ~ ~ _ _ _ _ _ _ _ / P3 Q α

ψ

where Q ⊂ P4 is a smooth quadric threefold, the morphisms α and β are blow ups of smooth elliptic curves C ⊂ Q and Z ⊂ P3 of degree 5, respectively, and the map ψ is given by the linear system of surfaces in |OP4 (2)|Q | that contain the curve C. One has 1 3 −KX ∼Q S + E, 2 2 ∗ where S ∈ |β (OP3 (1))|, and E is the exceptional divisor of α. We see that lct(X) 6 2/3. Lemma 11.12. Suppose that ‫(ג‬X) = 2.20. Then lct(X) 6 1/2. Proof. There is a commutative diagram XA AA β }} } AA } } AA ~}} _ _ _ _ _ _ _/ P2 V5 α

ψ

where V5 ⊂ P6 is a smooth intersection of Gr(2, 5) ⊂ P9 with a linear subspace of dimension 6, the morphism α is a blow up of a cubic curve P1 ∼ = C ⊂ V5 , the morphism β is a conic bundle, and ψ is given by the linear system of surfaces in |OP6 (1)|V5 | that contain the curve C. One has −KX ∼ 2S + E, where S ∈ |β ∗ (OP2 (1))|, and E is the exceptional divisor of α. We see that lct(X) 6 1/2.

Lemma 11.13. Suppose that ‫(ג‬X) = 2.21. Then lct(X) 6 2/3. Proof. There is a commutative diagram X@ @@ β ~~ ~ @@ ~ ~ @@ ~~ _ _ _ _ _ _ _/ Q Q α

ψ

P4

where Q ⊂ is a smooth quadric threefold, the morphisms α and β are blow ups of smooth rational curves C ⊂ Q and Z ⊂ Q of degree 4, and ψ is a birational map that is given by the linear system of surfaces in |OP4 (2)|Q | that contain the curve C. One has 1 3 −KX ∼Q S + E, 2 2 where S ∈ |β ∗ (OP4 (1))|Q |, and E is the exceptional divisor of α. We see that lct(X) 6 2/3. 99

Lemma 11.14. Suppose that ‫(ג‬X) = 2.22. Then lct(X) 6 1/2. Proof. There is a commutative diagram X } AAA β } α } AA }} AA ~}} _ _ _ _ _ _ _/ P3 V5 ψ

where V5 ⊂ P6 is a smooth intersection of Gr(2, 5) ⊂ P9 with a linear subspace of dimension 6, the morphisms α and β are blow ups of a conic C ⊂ V5 and a rational (not linearly normal) quartic Z ⊂ P3 , respectively, and ψ is given by the linear system of surfaces in |OP6 (1)|V5 | that contain the curve C. One has −KX ∼ 2S + E, ∗ where S ∈ |β (OP3 (1))|, and E is the exceptional divisor of α. We see that lct(X) 6 1/2. Lemma 11.15. Suppose that ‫(ג‬X) = 3.13. Then lct(X) 6 1/2. Proof. There is a commutative diagram h4 2 Vj VVVV VVVV hhhh PO h h h VVVVβ3 α2 hhhh h VVVV h hhh VVVV h h h h VVVV h h hh φ1 W2 jVVVV h4 W3 VVVV hhhh h h VVVV h h VVV hhhh hhhh π3 π2 VVVVVV h h h VVVV h hhhh α3 β2 h X VVVVV h h h VVVV hhh h h V h V VVVV hhh VVVV hhhhφ3 VVVV φ2 hhhhhhh VV* h t 2 π 1 h4 2 P jVVVVVV hhhh P VVVV h h h h VVVV hhhh V hhhhβ1 α1 VVVVVV h h h h VVVV hhhh

W1

such that Wi ⊂ P2 × P2 is a divisor of bi-degree (1, 1), the morphisms αi and βi are P1 -bundles, πi is a blow up of a smooth curve Ci ⊂ Wi of bi-degree (2, 2) such that αi Ci ⊂ P2 ⊃ βi Ci are irreducible conics, and φi is a conic bundle. Let Ei be the exceptional divisor of πi . Then −KX ∼ 2H1 + E1 ∼ 2H2 + E2 ∼ 2H3 + E3 ∼ E1 + E2 + E3 , where Hi ∈ |φ∗i (OP2 (1))|. We see that lct(X) 6 1/2.

Remark 11.16. Let us use the notation of the proof of Lemma 11.15 and assume that lct(X) < 1/2. Then there is an effective Q-divisor D ∼Q −KX such that the log pair (X, λD) is not log canonical for some λ < 1/2. Since lct(Wi ) = 1/2 by Theorem 7.1, one has ∅ 6= LCS(X, λD) ⊂ E1 ∩ E2 ∩ E3 . In particular, by Theorem 2.7 the locus LCS(X, λD) consists of a single point P ; note that P is an intersection P = F1 ∩ F2 ∩ F3 of three curves Fi such that F2 ∪ F3 (resp., F1 ∪ F3 , F1 ∪ F2 ) is a reducible fiber of the conic bundle φ1 (resp., φ2 , φ3 ). Appendix A. By Jean-Pierre Demailly. On Tian’s invariant and log canonical thresholds The goal of this appendix is to relate log canonical thresholds with the α-invariant introduced by G. Tian [179] for the study of the existence of Kähler–Einstein metrics. The approximation technique of closed positive (1, 1)-currents introduced in [48] is used to show that the α-invariant actually coincides with the log canonical threshold. 100

Algebraic geometers have been aware of this fact after [49] appeared, and several papers have used it consistently in the latter years (see e.g. [81], [12]). However, it turns out that the required result is stated only in a local analytic form in [49], in a language which may not be easily recognizable by algebraically minded people. Therefore, we will repair here the lack of a proper reference by stating and proving the statements required for the applications to projective varieties, e.g. existence of K¨ ahler–Einstein metrics on Fano varieties and Fano orbifolds. Usually, in these applications, only the case of the anticanonical line bundle L = −KX is considered. Here we will consider more generally the case of an arbitrary line bundle L (or Qline bundle L) on a complex manifold X, with some additional restrictions which will be stated later. Assume that L is equipped with a singular hermitian metric h (see e.g. [47]). Locally, L admits trivializations θ : L|U ' U × C, and on U the metric h is given by a weight function ϕ such that kξk2h = |ξ|2 e−2ϕ(z) for all z ∈ U, ξ ∈ Lz , when ξ ∈ Lz is identified with a complex number. We are interested in the case where ϕ is (at the very least) a locally integrable function for the Lebesgue measure, since it is then possible to compute the curvature form i ΘL,h = ∂∂ϕ π in the sense of distributions. We have ΘL,h > 0 as a (1, 1)-current if and only if the weights ϕ are plurisubharmonic functions. In the sequel we will be interested only in that case. Let us first introduce the concept of complex singularity exponent for singular hermitian metrics, following e.g. [184], [185], [4] and [49]. Definition A.1. If K is a compact subset of X, we define the complex singularity exponent cK (h) of the metric h, written locally as h = e−2ϕ , to be the supremum of all positive numbers c such that hc = e−2cϕ is integrable in a neighborhood of every point z0 ∈ K, with respect to the Lebesgue measure in holomorphic coordinates centered at z0 . Now, we introduce a generalized version of Tian’s invariant α, as defined in [179] (see also [173]). Definition A.2. Assume that X is a compact manifold and that L is a pseudo-effective line bundle, i. e. L admits a singular hermitian metric h0 with ΘL,h0 > 0. If K is a compact subset of X, we put αK (L) = inf cK (h) {h, ΘL,h >0}

where h runs over all singular hermitian metrics on L such that ΘL,h > 0. In algebraic geometry, it is more usual to look instead at linear systems defined by a family of linearly independent sections σ0 , σ1 , . . . , σN ∈ H 0 (X, L⊗m ). We denote by Σ the vector subspace generated by these sections and by |Σ| := P (Σ) ⊂ |mL| := P (H 0 (X, L⊗m )) the corresponding linear system. Such an (N + 1)-tuple of sections σ = (σj )06j6N defines a singular hermitian metric h on L by putting in any trivialization kξk2h =

|ξ|2 P

j

|σj (z)|2

1/m

=

|ξ|2 for ξ ∈ Lz , |σ(z)|2/m

hence h(z) = |σ(z)|−2/m with ϕ(z) =

X 1 1 log |σ(z)| = log |σj (z)|2 m 2m j

as the associated weight function. Therefore, we are interested in the number cK (|σ|−2/m ). In the case of a single section σ0 (corresponding to a linear system containing a single divisor), this 1 is the same as the log canonical threshold lctK (X, m D) of the where D is a divisor corresponding 101

1 to σ0 . We will also use the formal notation lctK (X, m |Σ|) in the case of a higher dimensional linear system |Σ| ⊂ |mL|. Now, recall that the line bundle L is said to be big if the Kodaira–Iitaka dimension κ(L) equals n = dimC (X). The main result of this appendix is

Theorem A.3. Let L be a big line bundle on a compact complex manifold X. Then for every compact set K in X we have 1 inf lctK X, D . αK (L) = inf cK (h) = inf m∈Z>0 D∈|mL| m {h, ΘL,h >0} Observe that the inequality 1 lctK X, D ≥ inf cK (h) m∈Z>0 D∈|mL| m {h, ΘL,h >0} inf

inf

is trivial, since any divisor D ∈ |mL| gives rise to a singular hermitian metric h. The converse inequality will follow from the approximation technique of [48] and some elementary analysis. The proof is parallel to the proof of [49, Theorem 4.2], although the language used there was somewhat different. In any case, we use again the crucial concept of multiplier ideal sheaves: if h is a singular hermitian metric with local plurisubharmonic weights ϕ, the multiplier ideal sheaf I(h) ⊂ OX (also denoted by I(ϕ)) is the ideal sheaf defined by I(h)z =

f ∈ OX,z | there exists a neighborhood V 3 z, Z 2 −2ϕ(x) such that |f (x)| e dλ(x) < +∞ , V

where λ is the Lebesgue measure. By Nadel (see [132]), this is a coherent analytic sheaf on X. Theorem A.3 has a more precise version which can be stated as follows. Theorem A.4. Let L be a line bundle on a compact complex manifold X possessing a singular hermitian metric h with ΘL,h > εω for some ε > 0 and some smooth positive definite hermitian (1, 1)-form ω on X. For every real number m > 0, consider the space Hm = H 0 (X, L⊗m ⊗I(hm )) of holomorphic sections σ of L⊗m on X such that Z Z |σ|2 e−2mϕ dVω < +∞, |σ|2hm dVω = X

X 1 m m! ω

is the hermitian volume form. Then for m 1, Hm is a non zero finite where dVω = dimensional Hilbert space and we consider the closed positive (1, 1)-current 1 1 X X i i Tm = ∂∂ log |gm,k |2 = ∂∂ log |gm,k |2h + ΘL,h 2π 2m 2π 2m k

k

where (gm,k )16k6N (m) is an orthonormal basis of Hm . The following statements hold. (i) For every trivialization L|U ' U × C on a cordinate open set U of X and every compact set K ⊂ U , there are constants C1 , C2 > 0 independent of m and ϕ such that X C1 1 1 C2 ϕ(z) − 6 ψm (z) := log |gm,k (z)|2 6 sup ϕ(x) + log n m 2m m r |x−z| m 2m which is the left hand part of statement (i). (ii) The first inequality in (i) implies ν(ψm , z) 6 ν(ϕ, z). In the opposite direction, we find sup ψm (x) 6 |x−z| cK (h). For the converse inequality, since we have c ∪Kj (h) = minj cKj (h), we can assume without loss of generality that K is contained in a trivializing open patch U of L. Let us take c < cK (ψm ). Then, 103

by definition, if V ⊂ X is a sufficiently small open neighborhood of K, the Hölder inequality for c the conjugate exponents p = 1 + mc−1 and q = 1 + m−1 c implies, thanks to equality p1 = mq , Z X Z −c/mq 1/p X dVω 6 |gm,k |2 |gm,k |2 e−2mϕ e−2(m/p)ϕ dVω = V

V

 Z  6

X

16k6N (m)

16k6N (m)

1/p  Z 2 −2mϕ   |gm,k | e dVω

X 16k6N (m)

V

 Z 1/p  = N (m) V

1/q X

|gm,k |2

−c/m

dVω 

=

16k6N (m)

1/q X

|gm,k |2

−c/m

dVω 

< +∞.

16k6N (m)

From this we infer cK (h) > m/p, i.e., cK (h)−1 6 p/m = 1/m + c−1 . As c < cK (ψm ) was arbitrary, we get cK (h)−1 6 1/m + cK (hm )−1 and the inequalities of (iii) are proved. ˜ →X Proof of Theorem A.3. Given a big line bundle L on X, there exists a modification µ : X ∗ ˜ of X such that X is projective and µ L = O(A + E) where A is an ample divisor and E an effective divisor with rational coefficients. By pushing forward by µ a smooth metric hA with positive curvature on A, we get a singular hermitian metric h1 on L such that ΘL,h1 > µ∗ ΘA,hA > εω on X. Then for any δ > 0 and any singular hermitian metric h on L with ΘL,h > 0, the interpolated metric hδ = hδ1 h1−δ satisfies ΘL,hδ > δεω. Since h1 is bounded away from 0, it follows that cK (h) > (1 − δ)cK (hδ ) by monotonicity. By Theorem A.4 (iii) applied to hδ , we infer cK (hδ ) = lim cK (hδ,m ), m→+∞

and we also have cK (hδ,m ) > lctK

1

Dδ,m

m for any divisor Dδ,m associated with a section σ ∈ H 0 (X, L⊗m ⊗ I(hm δ )), since the metric hδ,m P is given by hδ,m = ( k |gm,k |2 )−1/m for an orthornormal basis of such sections. This clearly implies 1 1 inf lctK Dδ,m > inf D . cK (h) > lim inf lim inf lctK m∈Z>0 D∈|mL| δ→0 m→+∞ m m In the applications, it is frequent to have a finite or compact group G of automorphisms of X and to look at G-invariant objects, namely G-equivariant metrics on G-equivariant line bundles L; in the case of a reductive algebraic group G we simply consider a compact real form GR instead of G itself. One then gets an α invariant αG,K (L) by looking only at G-equivariant metrics in Definition A.2. All contructions made are then G-equivariant, especially Hm ⊂ |mL| is a G-invariant linear system. For every G-invariant compact set K in X, we thus infer 1 (A.5) αG,K (L) = inf cK (h) = inf inf lctK |Σ| . m∈Z>0 |Σ|⊂|mL|, ΣG =Σ m {h is G−equivariant, ΘL,h >0} When G is a finite group, one can pick for m large enough a G-invariant divisor Dδ,m associated with a G-invariant section σ, possibly after multiplying m by the order of G. One then gets the slightly simpler equality 1 (A.6) αG,K (L) = inf inf lctK D . m∈Z>0 D∈|mL|G m In a similar manner, one can work on an orbifold X rather than on a non singular variety. The L2 techniques work in this setting with almost no change (L2 estimates are essentially insensitive to singularities, since one can just use an orbifold metric on the open set of regular points). 104

Appendix B. The Big Table This appendix contains the list of nonsingular Fano threefolds. We follow the notation and the numeration of these in [98], [126], [127]. We also assume the following conventions • the symbol Vi denotes a smooth Fano threefold such that −KX ∼ 2H and Pic Vi = Z H , • • • •

where H is a Cartier divisor on Vi , and H 3 = 8i ∈ {8, 16, . . . , 40}, the symbol W denotes a divisor on P2 × P2 of bidegree (1, 1) (or, that is the same, the variety P(TP2 )), the symbol V7 denotes a blow up of P3 at a point (or, that is the same, the variety P(OP2 ⊕ OP2 (1))), the symbol Q denotes a smooth quadric hypersurface in P4 , the symbol Si denotes a smooth del Pezzo surface such that KS2i = i ∈ 1, . . . , 8 , where S8 ∼ 6 P1 × P1 . =

The fourth column of Table 1 contains the values of global log canonical thresholds of the corresponding Fano varieties. The symbol ? near a number means that lct(X) is calculated for a general X with a given deformation type. If we know only the upper bound lct(X) 6 α, we write 6 α instead of the exact value of lct(X), and the symbol ? means that we don’t know any reasonable upper bound (apart from a trivial lct(X) 6 1). Table 1: Smooth Fano threefolds 3 ‫(ג‬X) −KX

Brief description

lct(X)

1.1

2

a hypersurface in P(1, 1, 1, 1, 3) of degree 6

1?

1.2

4

a hypersurface in P4 of degree 4 or a double cover of smooth quadric in P4 branched over a surface of degree 8

?

1.3

6

a complete intersection of a quadric and a cubic in P5

?

1.4

8

a complete intersection of three quadrics P6

?

1.5

10

a section of Gr(2, 5) ⊂ P9 by quadric and linear subspace of dimension 7

?

1.6

12

a section of the Hermitian symmetric space M = G/P ⊂ P15 of type DIII by a linear subspace of dimension 8

?

1.7

14

a section of Gr(2, 6) ⊂ P14 by a linear subspace of codimension 5

?

19

1.8

16

a section of the Hermitian symmetric space M = G/P ⊂ P of type CI by a linear subspace of dimension 10

6 6/7

1.9

18

6 4/5

1.10

22

a section of the 5-dimensional rational homogeneous contact manifold G2 /P ⊂ P13 by a linear subspace of dimension 11 V2 a zero locus of three sections of the rank 3 vector bundle Q, where Q is the universal quotient bundle on Gr(7, 3)

1.11

8

V1 that is a hypersurface in P(1, 1, 1, 2, 3) of degree 6

1/2

1.12

16

V2 that is a hypersurface in P(1, 1, 1, 1, 2) of degree 4

1/2

1.13

24

V3 that is a hypersurface in P4 of degree 3

1/2

1.14

32

V4 that is a complete intersection of two quadrics in P5

1/2

1.15

40

V5 that is a section of Gr(2, 5) ⊂ P9 by linear subspace of codimension 3

1/2

1.16

54

Q that is a hypersurface in P4 of degree 2

1/3

105

6 2/3

1.17

64

P3

1/4

2.1

4

a blow up of the Fano threefold V1 along an elliptic curve that is an intersection of two divisors from | − 21 KV1 |

1/2

2.2

6

a double cover of P1 × P2 whose branch locus is a divisor of bidegree (2, 4)

2.3

8

the blow up of the Fano threefold V2 along an elliptic curve that is an intersection of two divisors from | − 21 KV2 |

1/2

2.4

10

the blow up of P3 along an intersection of two cubics

3/4?

2.5

12

the blow up of the threefold V3 ⊂ P4 along a plane cubic

1/2?

2.6

12

a divisor on P2 × P2 of bidegree (2, 2) or a double cover of W whose branch locus is a surface in | − KW |

2.7

14

the blow up of Q along the intersection of two divisors from |OQ (2)|

6 2/3

2.8

14

a double cover of V7 whose branch locus is a surface in | − KV7 |

1/2?

2.9

16

the blow up of P3 along a curve of degree 7 and genus 5 which is an intersection of cubics

6 3/4

2.10

16

the blow up of V4 ⊂ P5 along an elliptic curve which is an intersection of two hyperplane sections

1/2?

2.11

18

the blow up of V3 along a line

1/2?

2.12

20

the blow up of P3 along a curve of degree 6 and genus 3 which is an intersection of cubics

6 3/4

2.13

20

the blow up of Q ⊂ P4 along a curve of degree 6 and genus 2

6 2/3

2.14

20

the blow up of V5 ⊂ P6 along an elliptic curve which is an intersection of two hyperplane sections

1/2?

2.15

22

the blow up of P3 along the intersection of a quadric and a cubic surfaces

1/2?

2.16

22

the blow up of V4 ⊂ P5 along a conic

6 1/2

2.17

24

the blow up of Q ⊂ P4 along an elliptic curve of degree 5

6 2/3

2.18

24

a double cover of P1 × P2 whose branch locus is a divisor of bidegree (2, 2)

1/2

2.19

26

the blow up of V4 ⊂ P5 along a line

1/2?

2.20

26

the blow up of V5 ⊂ P6 along a twisted cubic

6 1/2

2.21

28

the blow up of Q ⊂ P4 along a twisted quartic

6 2/3

2.22

30

the blow up of V5 ⊂ P6 along a conic

6 1/2

2.23

30

the blow up of Q ⊂ P4 along a curve of degree 4 that is an intersection of a surface in |OP4 (1)|Q | and a surface in |OP4 (2)|Q |

1/3?

2.24

30

a divisor on P2 × P2 of bidegree (1, 2)

1/2?

2.25

32

the blow up of P3 along an elliptic curve which is an intersection of two quadrics

1/2

2.26

34

the blow up of the threefold V5 ⊂ P6 along a line

1/2?

2.27

38

the blow up of P3 along a twisted cubic

1/2

2.28

40

the blow up of P3 along a plane cubic

1/4

4

6 13/14

?

2.29

40

the blow up of Q ⊂ P along a conic

1/3

2.30

46

the blow up of P3 along a conic

1/4 106

2.31

46

the blow up of Q ⊂ P4 along a line

1/3

2.32

48

W that is a divisor on P2 × P2 of bidegree (1, 1)

1/2

2.33

54

the blow up of P3 along a line

1/4

2.34

54

P1 × P2

1/3

2.35

56

V7 ∼ = P(OP2 ⊕ OP2 (1))

1/4

2.36

62

P(OP2 ⊕ OP2 (2))

1/5

3.1

12

a double cover of P1 × P1 × P1 branched in a divisor of tridegree (2, 2, 2)

3/4?

3.2

14

a divisor on a P2 -bundle P(OP1 ×P1 ⊕ OP1 ×P1 (−1, −1) ⊕ OP1 ×P1 (−1, −1)) such that X ∈ |L⊗2 ⊗ OP1 ×P1 (2, 3)|, where L is the tautological line bundle

1/2?

3.3

18

a divisor on P1 × P1 × P2 of tridegree (1, 1, 2)

2/3?

3.4

18

the blow up of the Fano threefold Y with ‫(ג‬Y ) = 2.18 along a smooth fiber of the composition Y → P1 × P2 → P2 of the double cover with the projection

1/2

3.5

20

the blow up of P1 × P2 along a curve C of bidegree (5, 2) such that the composition C ,→ P1 × P2 → P2 is an embedding

1/2?

3.6

22

the blow up of P3 along a disjoint union of a line and an elliptic curve of degree 4

1/2?

3.7

24

the blow up of the threefold W along an elliptic curve that is an intersection of two divisors from | − 12 KW |

1/2?

3.8

24

a divisor in |(α ◦ π1 )∗ (OP2 (1)) ⊗ π2∗ (OP2 (2))|, where π1 : F1 × P2 → F1 and π2 : F1 × P2 → P2 are projections, and α : F1 → P2 is a blow up of a point

1/2?

3.9

26

the blow up of a cone W4 ⊂ P6 over the Veronese surface R4 ⊂ P5 with center in a disjoint union of the vertex and a quartic on R4 ∼ = P2

1/3

3.10

26

the blow up of Q ⊂ P4 along a disjoint union of two conics

1/2

3.11

28

the blow up of the threefold V7 along an elliptic curve that is an intersection of two divisors from | − 12 KV7 |

1/2

3.12

28

the blow up of P3 along a disjoint union of a line and a twisted cubic

1/2

3.13

30

the blow up of W ⊂ P2 × P2 along a curve C of bidegree (2, 2) such that π1 (C) ⊂ P2 and π2 (C) ⊂ P2 are irreducible conics, where π1 : W → P2 and π2 : W → P2 are natural projections

3.14

32

the blow up of P3 along a disjoint union of a plane cubic curve that is contained in a plane Π ⊂ P3 and a point that is not contained in Π

1/2

3.15

32

the blow up of Q ⊂ P4 along a disjoint union of a line and a conic

1/2

3.16

34

the blow up of V7 along a proper transform via the blow up α : V7 → P3 of a twisted cubic passing through the center of the blow up α

1/2

3.17

36

a divisor on P1 × P1 × P2 of tridegree (1, 1, 1)

1/2

3.18

36

the blow up of P3 along a disjoint union of a line and a conic

1/3

3.19

38

the blow up of Q ⊂ P4 at two non-collinear points

1/3

3.20

38

the blow up of Q ⊂ P4 along a disjoint union of two lines

1/3

3.21

38

the blow up of P1 × P2 along a curve of bidegree (2, 1)

1/3

3.22

40

the blow up of P1 × P2 along a conic in a fiber of the projection P1 × P2 → P1

1/3

107

6 1/2

3.23

42

the blow up of V7 along a proper transform via the blow up α : V7 → P3 of an irreducible conic passing through the center of the blow up α

1/4

3.24

42

W ×P2 F1 , where W → P2 is a P1 -bundle and F1 → P2 is the blow up

1/3

3.25

44

the blow up of P3 along a disjoint union of two lines

1/3

3.26

46

the blow up of P3 with center in a disjoint union of a point and a line

1/4

3.27

48

P1 × P1 × P1

1/2

3.28

48

P1 × F1

1/3

3.29

50

the blow up of the Fano threefold V7 along a line in E ∼ = P2 , where E is the exceptional divisor of the blow up V7 → P3

1/5

3.30

50

the blow up of V7 along a proper transform via the blow up α : V7 → P3 of a line that passes through the center of the blow up α

1/4

3.31

52

the blow up of a cone over a smooth quadric in P3 at the vertex

1/3

4.1

24

divisor on P1 × P1 × P1 × P1 of multidegree (1, 1, 1, 1)

1/2

4.2

28

the blow up of the cone over a smooth quadric S ⊂ P3 along a disjoint union of the vertex and an elliptic curve on S

1/2

4.3

30

the blow up of P1 × P1 × P1 along a curve of tridegree (1, 1, 2)

1/2

4.4

32

the blow up of the smooth Fano threefold Y with ‫(ג‬Y ) = 3.19 along the proper transform of a conic on the quadric Q ⊂ P4 that passes through the both centers of the blow up Y → Q

1/3

4.5

32

the blow up of P1 × P2 along a disjoint union of two irreducible curves of bidegree (2, 1) and (1, 0)

3/7

4.6

34

the blow up of P3 along a disjoint union of three lines

1/2

4.7

36

the blow up of W ⊂ P2 × P2 along a disjoint union of two curves of bidegree (0, 1) and (1, 0)

1/2

4.8

38


1/3

4.9

40

the blow up of the smooth Fano threefold Y with ‫(ג‬Y ) = 3.25 along a curve that is contracted by the blow up Y → P3

1/3

4.10

42

P1 × S7

1/3

4.11

44

the blow up of P1 × F1 along a curve C ∼ = P1 such that C is contained 1 ∼ in a fiber F = F1 of the projection P × F1 → P1 and C · C = −1 on F

1/3

4.12

46

the blow up of the smooth Fano threefold Y with ‫(ג‬Y ) = 2.33 along two curves that are contracted by the blow up Y → P3

1/4

4.13

26


1/2?

5.1

28

the blow up of the smooth Fano threefold Y with ‫(ג‬Y ) = 2.29 along three curves that are contracted by the blow up Y → Q

1/3

5.2

36

the blow up of the smooth Fano threefold Y with ‫(ג‬Y ) = 3.25 along two curves C1 6= C2 that are contracted by the blow up φ : Y → P3 and that are contained in the same exceptional divisor of the blow up φ

1/3

5.3

36

P1 × S6

1/2

5.4

30

P1 × S5

1/2

5.5

24

P1 × S4

1/2 108

5.6

18

P1 × S3

1/2

5.7

12

P1 × S2

1/2

5.8

6

P1 × S1

1/2 References

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