Log canonical thresholds of certain Fano hypersurfaces
aa r X i v : . [ m a t h . AG ] J a n Log canonical thresholds of certain Fano hypersurfaces
Ivan Cheltsov and Jihun Park Abstract.
We study log canonical thresholds on quartic threefolds, quintic fourfolds, and double spaces.As an important application, we show that they have K¨ahler-Einstein metrics if they are general.
Keywords : Anticanonical linear system; Fano variety; K¨ahler-Einstein metric; Log canonical threshold.
Mathematics Subject Classification (2000) : 14J45; 14C20; 53C25.
All varieties are defined over C . The multiplicity of a nonzero polynomial f ∈ C [ z , · · · , z n ] at a point P ∈ C n is the nonnegative integer m such that f ∈ m mP \ m m +1 P , where m P is the maximal ideal of polynomials vanishing at the point P in C [ z , · · · , z n ]. It can be also defined by derivatives. The multiplicity of f at the point P is the numbermult P ( f ) = min (cid:26) m (cid:12)(cid:12)(cid:12) ∂ m f∂ m z ∂ m z · · · ∂ m n z n ( P ) = 0 (cid:27) . On the other hand, we have a similar invariant that is defined by integrations. This invariant, whichis called the log canonical threshold of f at the point P , is given by c P ( f ) = sup n c (cid:12)(cid:12)(cid:12) | f | − c is locally L near the point P ∈ C n o . This number appears in many places. For instance, the log canonical threshold of the polynomial f atthe origin is the same as the absolute value of the largest root of the Bernstein-Sato polynomial of f .Even though log canonical threshold was implicitly known and extensively studied under different namesby J. H. M Steenbrink, A. Varchenko and so forth, it was formally introduced to birational geometry byV. Shokurov in [31] as follows. Let X be a Q -factorial variety with at worst log canonical singularities, Z ⊂ X a closed subvariety, and D an effective Q -divisor on X . The log canonical threshold of D along Z is the number c Z ( X, D ) = sup n c (cid:12)(cid:12)(cid:12) the log pair ( X, cD ) is log canonical along Z o . For the case Z = X we use the notation c ( X, D ) instead of c X ( X, D ). Because log canonicity is a localproperty, we see that c Z ( X, D ) = inf P ∈ Z { c P ( X, D ) } . If X = C n and D = ( f = 0), then we also use the notation c ( f ) for the log canonical threshold of D atthe origin.Even though several methods have been invented in order to compute log canonical thresholds, it isnot easy to compute them in general. However, many problems in birational geometry are related to School of Mathematics, The University of Edinburgh, Edinburgh, EH9 3JZ, UK;
[email protected] IBS Center for Geometry and Physics; Department of Mathematics, POSTECH, Pohang, Kyungbuk 790-784,Korea; [email protected]
I. Cheltsov, J. Park, J. Won log canonical thresholds. The log canonical thresholds play a significant role in the study on birationalgeometry. They show many interesting properties (see [10], [11], [12], [17], [19], [20], [21], [23], [24], [25],[26]).We occasionally find it useful to consider the smallest value of log canonical thresholds of effectivedivisors linearly equivalent to a given divisor, in particular, an anticanonical divisor (for instance, see[23]).
Definition 1.1.1.
Let X be a Q -factorial Fano variety with at worst log terminal singularities. For anatural number m >
0, we define the m -th global log canonical threshold of X by the numberlct m ( X ) = inf (cid:26) c (cid:18) X, m H (cid:19) (cid:12)(cid:12)(cid:12) H ∈ (cid:12)(cid:12) − mK X (cid:12)(cid:12)(cid:27) . Note that the number lct m ( X ) is defined to be ∞ if the linear system | − mK X | is empty. Also, we definethe global log canonical threshold of X by the numberlct( X ) = inf n ∈ N n lct m ( X ) o . We can immediately seelct( X ) = sup ( c (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) the log pair ( X, cD ) is log canonical forevery effective Q -divisor D with D ≡ − K X ) . To see the simplest case, let S be a smooth del Pezzo surface. It follows from [5, Theorem 1.7] and [23,Section 3] thatlct ( S ) = lct ( S ) = / S ∼ = F or K S ∈ { , } , / S ∼ = P × P or K S ∈ { , } , / K S = 4 , / S is a cubic in P with an Eckardt point , / S is a cubic in P without Eckardt points , / K S = 2 and | − K S | has a tacnodal curve , / K S = 2 and | − K S | has no tacnodal curves , / K S = 1 and | − K S | has a cuspidal curve , K S = 1 and | − K S | has no cuspidal curves . (1.1.2)For a quasismooth hypersurface X in P ( a , . . . , a ) of degree P i =0 a i −
1, where a ≤ · · · ≤ a , one canfind lct( X ) > for 1936 values of ( a , a , a , a , a ) (see [16, Corollary 3.4]). Moreover, for a quasismoothhypersurface X in P (1 , a , . . . , a ) of degree P i =1 a i having terminal singularities, there are exactly 95possible quadruples ( a , a , a , a ) found in [14] and [16]. It follows from [4, Theorem 1.3] that lct( X ) = 1if ( a , a , a , a ) n (1 , , , , (1 , , , , (1 , , , , (1 , , , o and the hypersurface X is sufficiently general.It is proved that the global log canonical threshold of a rational homogeneous space of Picard rank 1and Fano index r is r (see [13, Theorem 2]). Example 1.1.3.
Let X be a smooth hypersurface of degree n ≥ P n . Thenlct m ( X ) ≥ n − n due to [3, Theorem 1.3] and [6, Theorem 3.3]. Furthermore, lct m ( X ) = n − n if and only if X containsa cone of dimension n − X ) ≥ n − n . However, it is shown that lct( X ) = 1 if X is general and n ≥ og canonical thresholds of certain Fano hypersurfaces From the results of [30], it is natural to expect the following:
Conjecture 1.1.4.
The global log canonical thresholds of a general quartic threefold and a general quinticfourfold are . This conjecture has been proposed for canonical thresholds in [30, Conjecture 2].For an evidence of the conjecture, we can consider the first global log canonical threshold of a generalhypersurface. It is not hard to show that the first global log canonical threshold of a general hypersurfaceof degree n ≥ P n is one (see Proposition 2.1.1). In the case of smooth quartic threefolds, we can findall the first global log canonical thresholds (see Proposition 2.1.2).For the global log canonical thresholds, we prove the following: Theorem 1.1.5.
Let X be a general hypersurface of degree n = 4 or in P n . Then lct ( X ) ≥ for n = 4 ; for n = 5 . The global log canonical threshold of a Fano variety is an algebraic counterpart of the α -invariantintroduced in [32]. One of the most interesting applications of the global log canonical thresholds of Fanovarieties is the following result proved in [9, p. 549] (see also [22] and [32]). Theorem 1.1.6.
Let X be an d -dimensional Fano variety with at most quotient singularities. The variety X has an orbifold K¨ahler–Einstein metric if the inequality lct ( X ) > dd + 1 holds. The inequality in Example 1.1.3 is not strong enough to apply Theorem 1.1.6 to a smooth hypersurfaceof degree n in P n . However, we see that (1.1.2) enables Theorem 1.1.6 to imply the existence of a K¨ahler–Einstein metric on a general cubic surface and that [30, Theorem 2] enables Theorem 1.1.6 to imply theexistence of a K¨ahler–Einstein metric on a general hypersurface of degree n ≥ P n . Even thoughTheorem 1.1.5 is much weaker than Conjecture 1.1.4, they are strong enough to imply the existence of aK¨ahler–Einstein metric. Consequently, we can obtain the following: Corollary 1.1.7.
A general hypersurface of degree n ≥ in P n has a K¨ahler–Einstein metric. In fact, a smooth conic in P has a K¨ahler-Einstein metric because it is isomorphic to P and theFubini-Study metric of a projective space is K¨ahler-Einstein. Furthermore, a smooth cubic surface alwaysadmits a K¨ahler–Einstein metric (see [33, Section 2]). Meanwhile, it is proved that a K¨ahler–Einsteinmetric exists on a smooth hypersurface in P n defined by a homogeneous polynomial equation of the form z n + z n + f n ( z , · · · , z n ) = 0, where n ≥ f n is a homogeneous polynomial of degree n in variables z , · · · , z n (see [1, Proposition 3.1]).Also, in this paper, we will study log canonical thresholds on double spaces, i.e. , double covers of P n ,and obtain similar results as what we have on Fano hypersurfaces in P n . For instance, we will provethat the first global log canonical threshold of a smooth double space is equal to its global log canonicalthreshold (see Proposition 3.2.1) and that every smooth double cover of P n ramified along a hypersurfaceof degree 2 n admits a K¨ahler-Einstein metric.Let us close this section by a conjecture inspired by [34, Question 1]. Conjecture 1.1.8.
For a smooth Fano variety X , lct( X ) = lct m ( X ) for some natural number m ≥ . I. Cheltsov, J. Park, J. Won n in P n . As we mentioned, one can consider the first global log canonical threshold of a general hypersurface ofdegree n ≥ P n in behalf of Conjecture 1.1.4. Proposition 2.1.1.
Let X be a general hypersurface of degree n ≥ in P n . Then lct ( X ) = 1 .Proof. Consider the space S n = P n × P (cid:0) H ( P n , O P n ( n )) (cid:1) with the natural projections p : S n → P n and q : S n → P (cid:0) H ( P n , O P n ( n )) (cid:1) . Put I n = n ( O, F ) ∈ S n (cid:12)(cid:12)(cid:12) F ( O ) = 0 and F = 0 is smooth . o .Let ( O, F ) be a pair in I n . Suppose that O = [1 : 0 : · · · : 0]. Then F can be given by a polynomial ofthe form z n − z n + z n − q ( z , · · · , z n ) + · · · + z q n − ( z , · · · , z n ) + q n ( z , · · · , z n ) ∈ C [ z , · · · , z n ] , where q i is a homogeneous polynomial of degree i .We say that the point O is bad on the hypersurface F = 0 if one of the following condition holds:(1) q ( z , · · · , z n − , ≡ q ( z , · · · , z n − ,
0) = { l ( z , · · · , z n − ) } for some linear form l ( z , · · · , z n − ) and if we as-sume l ( z , · · · , z n − ) = z n − , either q ( z , · · · , z n − , , ≡ q ( z , · · · , z n − , ,
0) = { m ( z , · · · , z n − ) } for some linear form m ( z , · · · , z n − ).Then consider a subset of I n , Y n = n ( O, F ) ∈ I n (cid:12)(cid:12)(cid:12) the point O is bad on the quartic F = 0 o . One can see that for a given point P on P n , the dimension of p − ( P ) ∩ Y n is strictly smaller than h ( P n , O P n ( n )) − ( n + 1), and hence the dimension of the space Y n is smaller than the dimension of P (cid:0) H ( P n , O P n ( n )) (cid:1) . Therefore, the image of the regular map q | Y n : Y n → P (cid:0) H ( P n , O P n ( n )) (cid:1) is aproper closed subset of P (cid:0) H ( P n , O P n ( n )) (cid:1) . So, a general hypersurface of degree n in P n has no badpoint.Let X be a general hypersurface of degree n in P n and H be a divisor in | − K X | . We claim that thepair ( X, H ) is log canonical at every point P on X . By a suitable coordinate change we may assume thatthe point P = [1 : 0 : · · · : 0]. We may also assume that the hypersurface X is defined by the equation z n − z n + z n − q ( z , · · · , z n ) + · · · + z q n − ( z , · · · , z n ) + q n ( z , · · · , z n ) = 0 , where q i is a homogeneous polynomial of degree i . Unless the hyperplane section H is given by thetangent hyperplane at the point P , the divisor H is smooth at the point P , and hence the pair ( X, H )is log canonical at the point P . Now we suppose that H is given by the tangent hyperplane T at thepoint P . The hyperplane T in P n is defined by z n = 0 in our case. Since both X and T are smooth and H = T ∩ X , we obtain c P ( X, H ) = c P ( T, H ) from [11, Theorem 3.1]. Furthermore, c P ( T, H ) = c ( f ),where f = q ( z , · · · , z n − ,
0) + · · · + q n − ( z , · · · , z n − ,
0) + q n ( z , · · · , z n − , P is not a badpoint on X , the polynomial q ( z , · · · , z n − ,
0) is not zero polynomial. If the rank of the quadraticpolynomial q ( z , · · · , z n − ,
0) is at least 2, then c ( f ) = 1 by [17, Lemma 8.10 (8.10.3)]. If the rankof the quadratic polynomial q ( z , · · · , z n − ,
0) is 1, we may assume that q ( z , · · · , z n − ,
0) = z n − .Consider the polynomial f with weights wt( z ) = · · · = wt( z n − ) = 2, wt( z n − ) = 3. The leadingterm of f with respect to the weights is f w = z n − + q ( z , · · · , z n − , , e q = q ( z , · · · , z n − , ,
0) is neither zero polynomial nor a cube of a linear polynomial, we obtain c ( e q ) ≥ ,and hence c ( f w ) = max { + c ( e q ) , } =1. By [19, Proposition 2.1], we have c ( f ) ≥ c ( f w ) = 1.Therefore, the pair ( X, H ) is log canonical at every point on X . Consequently, lct ( X ) = 1.For smooth quartic threefolds, one can compute all the possible first global log canonical thresholds bystudying normal quartic surfaces. Here we only list them and the brief idea to compute them as follows: Proposition 2.1.2.
Let X be a smooth quartic threefold in P . The first global log canonical threshold lct ( X ) is one of the following: n , , , , , , , , , , , , , , , , , , , , , , , , o . Furthermore, for each number µ in the set above, there is a smooth quartic threefold X with lct ( X ) = µ . og canonical thresholds of certain Fano hypersurfaces Its proof goes as follows. A divisor S ∈ | − K X | is given by the intersection of X and a hyperplane H in P . Because the log canonical threshold c ( X, S ) is equal to the log canonical threshold c ( H, S ) (see[11, Theorem 3.1]), the result above can be obtained by investigating log canonical thresholds of normalquartic surfaces H in P . Note that a hyperplane section of a smooth hypersurface in P n , n ≥ P n − can be attained by a hyperplane section of a smooth hypersurfacein P n (see [15]). Let S be a normal surface in P defined by a homogeneous quartic polynomial F . Wesuppose that S has a singular point at [0:0:0:1]. We then consider the log pair ( C , D ), where D is thefourth affine piece of S that is defined by the polynomial f ( x, y, z ) = F ( x, y, z, C , D ) instead of ( X, S ). For thedetail of the computation, see [35].Before we prove Theorem 1.1.5, let us explain our generality condition. Let X d be a hypersurface ofdegree d in P n , d ≥ n ≥
4. Let P be an arbitrary point on X d . By suitable coordinate changes, weassume that P = [1 : 0 : · · · : 0]. Then the hypersurface X d is defined by z n − q ( z , · · · , z n ) + z n − q ( z , · · · , z n ) + · · · + z q n − ( z , · · · , z n ) + q d ( z , · · · , z n ) = 0 , where q i are homogeneous polynomials of degrees i in variables z , · · · , z n . Definition 2.1.3.
The hypersurface X d is said to be k -regular at the point P , where 0 ≤ k ≤ d , if thehomogenous polynomials q , q , · · · , q k form a regular sequence in C [ z , · · · , z n ]. The hypersurface X d is said to be k -regular if it is k -regulareverywhere. Proposition 2.1.4.
A general hypersurface of degree n in P n is ( n − -regular.Proof. See [28, Proposition 1].To prove Theorem 1.1.5 we need a linear system on X d that has a big multiplicity at a given point buta small base locus. Put f i ( z , · · · , z n ) = i X j =1 z i − j q j ( z , · · · , z n )for each 1 ≤ i ≤ d . Definition 2.1.5.
The m -th hypertangent linear system M at the point P is the linear subsystem of | − mK X d | consisting of the divisors cut by hypersurfaces m X i =1 f i ( z , · · · , z n ) p m − i ( z , · · · , z n ) = 0 , where p j ( z , · · · , z n ) is a homogeneous polynomial of degree j .Note that mult P ( M ) ≥ m + 1 for each divisor M in the m -th hypertangent linear system on X d . Lemma 2.1.6.
Suppose that the hypersurface X d is ( n − -regular at a point P . Then the followinghold.1. There are finitely many lines (possibly none) on X d passing through the point P .2. The base locus of the ( n − -th hypertangent linear system M at the point P consists of linespassing through the point P on X d .Proof. There is a one-to-one correspondence between the set of lines passing through the point P and thezero locus of the polynomials q = · · · = q d = 0 in P n − . Since the homogeneous polynomials q , · · · , q n − form a regular sequence in C [ z , · · · , z n ], they defines a finite set in P n − . This proves the first assertion.The base locus of the linear system M is defined by the equations f = · · · = f n − = 0. Therefore, itis cut out by the equations q = q = · · · = q n − = 0. This shows the second assertion. I. Cheltsov, J. Park, J. Won
We close this section by the following useful lemma.
Lemma 2.1.7.
Let X be a smooth hypersurface of degree n in P n and D be an effective Q -divisornumerically equivalent to − K X . For a non-negative number λ ≤ , there is a point P ∈ X such that ( X, λD ) is log canonical on X \ P .Proof. The log pair (
X, λD ) is log canonical in the outside of finitely many points of the smooth hyper-surface X (see [27, Theorem 2] or [28, Section 3]). Suppose that there are two points at which the logpair ( X, λD ) is not log canonical. Then for sufficiently small ǫ > X, ( λ − ǫ ) D ) is not logcanonical at the two points either. Since the divisor − ( K X + ( λ − ǫ ) D ) is nef and big, it follows fromfrom the connectedness principle of Shokurov (see [18, Theorem 17.4]) that the locus of non-Kawamatalog terminal singularities of the log pair ( X, ( λ − ǫ ) D ) is connected. This is a contradiction. Let X be a smooth quartic hypersurface in P such that the following general conditions hold: • the threefold X is 3-regular; • every line on the hypersurface X has normal bundle O P ( − ⊕ O P ; • the intersection of X with a two-dimensional linear subspace of P cannot be a double conic curve. Remark . A line on the quartic X has normal bundle O P ( − ⊕O P if and only if no two-dimensionallinear subspace of P is tangent to the quartic X along the line (see [7, Theorem 1.9]). Remark . It follows from Proposition 2.1.2 and [6] that lct ( X ) ≥ . To avoid the long proof ofProposition 2.1.2, we can use instead Proposition 2.1.1 by adding extra generality conditions. Remark . Let B and B ′ be effective Q -Cartier Q -divisors on a variety V . Then( V, αB + (1 − α ) B ′ )is log canonical if both ( V, B ) and (
V, B ′ ) are log canonical, where 0 ≤ α ≤ n = 4. Put λ = . Let D be an effective Q -divisor on X suchthat D ≡ − K X . To prove Theorem 1.1.5, we have to show that ( X, λD ) is log canonical.Suppose that (
X, λD ) is not log canonical. Due to Remarks 2.2.2 and 2.2.3, we may assume that D = n R where R is an irreducible divisor with R ∼ − nK X for some natural number n >
1. ByLemma 2.1.7, the log pair (
X, λD ) is not log canonical only at a single point P .The threefold X can be given by v x + v q ( x, y, z, u ) + vq ( x, y, z, u ) + q ( x, y, z, u ) = 0 ⊂ P ∼ = Proj (cid:0) C (cid:2) x, y, z, u, v (cid:3)(cid:1) , where q i ( x, y, z, u ) is a homogeneous polynomial of degree i . Furthermore, we may assume that the point P is located at [0 : 0 : 0 : 0 : 1]. Let T be the surface on X cut out by x = 0. Lemma 2.2.4.
The multiplicity of D at the point P is at most .Proof. The statement immediately follows from the inequalities4 = H · T · D ≥ mult P ( T ∩ D ) ≥ mult P ( T ) mult P ( D ) ≥ P ( D ) , where H is a general hyperplane section of X passing through the point P .Let π : U → X be the blow up at the point P with the exceptional divisor E . Then¯ D ≡ π ∗ ( D ) − mult P ( D ) E, where ¯ D is the proper transform of the divisor D via the morphism π .It follows from [8, Corollary 3.5] or [30, Proposition 3] that there is a line L ⊂ E such thatmult P ( D ) + mult L (cid:0) ¯ D (cid:1) > λ . og canonical thresholds of certain Fano hypersurfaces Recall that E is isomorphic to P .Let L be the linear system of hyperplane sections of X such that S ∈ L ⇐⇒ either L ⊂ ¯ S or S = T, where ¯ S is the proper transform of S via the birational morphism π . There is a two-dimensional linearsubspace Π ⊂ P such that the base locus of L consists of the intersection Π ∩ X .Let S be a general surface in L . Then S is a smooth K3 surface. Put T S = T (cid:12)(cid:12) S = r X i =1 Z i , where each Z i is an irreducible curve. The generality conditions imply that the curve T S is reduced (seeRemark 2.2.1). Then P ri =1 deg( Z i ) = 4. It follows thatmult P ( S ∩ D ) ≥ mult P ( S ) mult P ( D ) + mult L (cid:0) ¯ S ∩ ¯ D (cid:1) ≥ mult P ( D ) + mult L (cid:0) ¯ D (cid:1) > λ . Put D S = D (cid:12)(cid:12) S = r X i =1 m i Z i + ∆ , where m i is a non-negative rational number and ∆ is an effective one-cycle on S whose support does notcontain the curves Z , . . . , Z r . Then r X i =1 m i mult P ( Z i ) + mult P (∆) = mult P ( D S ) > λ , and the support of the cycle ∆ does not contain any component of the cycle T S . We have4 = T S · D S = r X i =1 m i deg ( Z i ) + T S · ∆ ≥ r X i =1 m i deg ( Z i ) + mult P ( T S ) λ − r X i =1 m i mult P ( Z i ) ! . Remark . The equality m i = mult Z i ( D ) holds for every i because X | Π is reduced.It follows from the 3-regularity of X that mult P ( T S ) ≤ Lemma 2.2.6.
Suppose that mult P ( T S ) = 3 . Then > λ + deg ( Z k ) m k , for Z k that is not a line passing through the point P .Proof. Let ¯ T be the proper transform of the surface T via the birational morphism π . Then3 = mult P ( T S ) = mult P ( T ∩ S ) = mult P ( T ) mult P ( S ) + mult L (cid:0) ¯ T ∩ ¯ S (cid:1) . Hence, we see that L ⊂ ¯ T . Since mult P ( D ) > λ and mult P ( T ) = 2, it follows thatmult P ( T ∩ D ) ≥ mult P ( T ) mult P ( D ) + mult L (cid:0) ¯ T ∩ ¯ D (cid:1) ≥ P ( D ) + mult L (cid:0) ¯ D (cid:1) > λ . Let L , . . . , L m be all the lines on X that pass through the point P . Put T ∩ D = m X i =1 ǫ i L i + ¯ m k Z k + Υ , where ǫ i and ¯ m k are non-negative rational numbers, and Υ is an effective one-cycle on X whose supportdoes not contain the lines L , . . . , L m . Then ¯ m k ≥ m k by Remark 2.2.5. I. Cheltsov, J. Park, J. Won
Taking the intersection with a general hyperplane section of X , we see that4 ≥ r X i =1 ǫ i + ¯ m k deg ( Z k ) , but ¯ m k mult P ( Z k ) + mult P (Υ) > λ − P ri =1 ǫ i .Take a general member M in the third hypertangent linear system M at the point P . Note that thebase locus of M consists of the lines L , . . . , L m by Lemma 2.1.6. Hence, we have12 = M · T · D ≥ r X i =1 ǫ i + M · ( ¯ m k Z k + Υ) > r X i =1 ǫ i + 4 λ − r X i =1 ǫ i ! = 12 λ − r X i =1 ǫ i . This implies 16 > /λ + deg( Z k ) m k since 4 ≥ P ri =1 ǫ i + ¯ m k deg( Z k ) and ¯ m k ≥ m k .From now on, in order to describe the reduced curve T S , we will use the following notations: • C : an irreducible cubic not passing through the point P . • e C : an irreducible cubic that is smooth at the point P . • b C : an irreducible cubic that is singular at the point P .For i = 1 , • Q i : an irreducible quadric not passing through the point P . • f Q i : an irreducible quadric passing through the point P .For i = 1 , , , • L i : a line not passing through the point P . • f L i : a line passing through the point P .Then, the following are all the possible configuration of T S . In each case, we derive a contradictoryinequality from our assumptions so that the log pair ( X, λD ) should be log canonical. To obtain acontradictory inequality for each case, we start from the inequality4 = T S · D S = X m i Z i · T S + T S · ∆ ≥ X m i deg( Z i ) + mult P ( T S )mult P (∆) > X m i deg( Z i ) + mult P ( T S ) (cid:18) λ − X m i mult P ( Z i ) (cid:19) , and then we show that the number A := X m i deg( Z i ) + mult P ( T S ) (cid:18) λ − X m i mult P ( Z i ) (cid:19) is greater than 4. CASE A.
The curve T S is an irreducible quartic curve.1. mult P ( T S ) = 2. D S = mT S + ∆.A contradictory inequality: A = 4 m + 2 (cid:18) λ − m (cid:19) = 4 λ > . og canonical thresholds of certain Fano hypersurfaces
2. mult P ( T S ) = 3. D S = mT S + ∆.An auxiliary inequality: 16 > λ + 4 m by Lemma 2.2.6 . A contradictory inequality: A = 4 m + 3 (cid:18) λ − m (cid:19) = 6 λ − m > λ − (cid:18) − λ (cid:19) = 21 λ − > . CASE B.
The curve T S is reducible and contains no line passing through the point P .1. T S = b C + L . D S = m b C + m L + ∆.An auxiliary inequality : 1 = L · D S ≥ m − m . A contradictory inequality: A = 3 m + m + 2 (cid:18) λ − m (cid:19) = 4 λ + m − m ≥ λ + m − m > . T S = f Q + f Q . D S = m f Q + m f Q + ∆.A contradictory inequality: A = 2 m + 2 m + 2 (cid:18) λ − m − m (cid:19) = 4 λ > . CASE C.
The curve T S contains a unique line passing through the point P .1. T S = f Q + f L + L . D S = m f Q + m f L + m L + ∆.Auxiliary inequalities: 2 = Q · D S ≥ − m + 2 m + 2 m L · D S ≥ m + m − m ) ⇒ ≥ m A contradictory inequality: A = 2 m + m + m + 2 (cid:18) λ − m − m (cid:19) = 4 λ + m − m ≥ λ + m − > . T S = e C + f L . D S = m e C + m f L + ∆.An auxiliary inequality: 3 = e C · D S ≥ m . A contradictory inequality: A = 3 m + m + 2 (cid:18) λ − m − m (cid:19) = 4 λ + m − m ≥ λ + m − > . I. Cheltsov, J. Park, J. Won T S = b C + f L . D S = m b C + m f L + ∆.Auxiliary inequalities: 3 = b C · D S ≥ m > λ + 3 m by Lemma 2.2.6A contradictory inequality: A = 6 λ − m − m ≥ λ − − m > λ − − (cid:18) − λ (cid:19) = 18 λ − > . CASE D.
The curve T S contains two lines passing through the point P .1. T S = f Q + f L + f L . D S = m f Q + m f L + m f L + ∆,Auxiliary inequalities: 2 = f Q · D S ≥ − m + 2 m + 2 m > λ + 2 m by Lemma 2.2.6.A contradictory inequality: A = 6 λ − m − m − m ≥ λ − m − m ) > λ − − (cid:18) − λ (cid:19) = 24 λ − > . T S = f L + f L + L + L . D S = m f L + m f L + m L + m L + ∆, where we may assume that m ≥ m .An auxiliary inequality: 1 = L · D S ≥ m + m + m − m A contradictory inequality: A = 4 λ + m + m − ( m + m ) ≥ λ + m + m − m − (1 − m − m + 2 m ) ≥ λ + 2 m − m − > . T S = Q + f L + f L . D S = mQ + m f L + m f L + ∆.An auxiliary inequality:2 = Q · D S ≥ − m + 2 m + 2 m ⇒ m ≥ m + m . A contradictory inequality: A = 4 λ + 2 m − m − m ≥ λ + m − > . CASE E.
The curve T S contains three lines passing through the point P .1. T S = f L + f L + f L + L . D S = m f L + m f L + m f L + mL + ∆.Auxiliary inequalities: 1 = L · D S ≥ − m + m + m + m > λ + m by Lemma 2.2.6.A contradictory inequality: A = 6 λ + m − m + m + m ) ≥ λ + m − m ) > λ − − (cid:18) − λ (cid:19) = 42 λ −
50 = 4 . Therefore, Theorem 1.1.5 for n = 4 has been proved. og canonical thresholds of certain Fano hypersurfaces In this section, we prove Theorem 1.1.5 for n = 5.Let X be a quintic hypersurface in P such that the following generality conditions hold:G 1. The hypersurface X is 4-regular;G 2. For every 3-dimensional linear space Π in P , the intersection X ∩ Π is irreducible and reduced;G 3. For each point P ∈ X and each 3-dimensional linear space Π contained in the tangent hyperplaneat P and containing the point P , if the surface Z := X ∩ Π has multiplicity two at the point P ,then it satisfies the following:G 3.0. The surface Z G 3.0.1. cannot be singular along a line passing through the point P ;G 3.0.2. cannot contain four lines passing through the point P .G 3.1. If Z contains only one line L passing through the point P ,G 3.1.1. then the line L meets its residual curve by a general hyperplane section in Π either at atleast one smooth point or at at least two ordinary double points.G 3.2. If Z contains only two lines, L and L , passing through the point P , thenG 3.2.1. it has at most four singular points on L ∪ L ;G 3.2.2. if it has four singular points on L i , then all of them are ordinary double points;G 3.2.3. if it has three singular points on L i , then two of them are ordinary double points;G 3.2.4. if it has exactly three singular points on the line L i , then the line L i meets its residualcurve by a general hyperplane section in Π at one smooth point;G 3.2.5. if it has exactly two singular points on the line L i , then either P is a non-ordinary doublepoint and the line L i meets its residual curve by a general hyperplane section in Π attwo smooth points, or the point P is an ordinary double point and the line L i meets itsresidual curve by a general hyperplane section in Π at at least one smooth point.G 3.2.6. if it has no singular point other than P on the line L i , then the line L i meets its residualcurve by a general hyperplane section in Π at at least two smooth points.G 3.3. If Z contains three lines, L , L and L , passing through the point P ,G 3.3.1. if the three lines are coplanar, then it is smooth on ( L ∪ L ∪ L ) \ { P } and each line L i meets its residual curve by a general hyperplane section in Π at four points;G 3.3.2. if the three lines are not coplanar, then either P is a non-ordinary double point, thesurface Z is smooth at every point of ( L ∪ L ∪ L ) \ { P } and each line L i meets itsresidual curve by a general hyperplane section in Π at four points, or P is an ordinarydouble point and each line L i meets its residual curve by a general hyperplane sectionin Π at two smooth points. Lemma 2.3.1.
A general quintic hypersurface X in P satisfies the condition G 2.Proof. This follows directly from [2, Theorem 5.1].
Lemma 2.3.2.
A general quintic hypersurface X in P satisfies the condition G 3.Proof. See Appendix.Put λ = . Let D be an effective Q -divisor on X such that D ≡ − K X . We claim that the log pair( X, λD ) is log canonical.Suppose that the log pair (
X, λD ) is not log canonical. As in the case of quartic threefolds, we mayassume that D = n R where R is an irreducible divisor with R ∼ − nK X for some natural number n .Furthermore, the following lemma enables us to assume n >
1. We may use Proposition 2.1.1 with extragenerality conditions in order to assume n >
Lemma 2.3.3.
If a quintic hypersurface Y in P is -regular, then lct ( Y ) = 1 . I. Cheltsov, J. Park, J. Won
Proof.
The main idea of the proof is the same as that of Proposition 2.1.2. It is enough to prove c ( f ) = 1for a quintic polynomial f ( x, y, z, u ) ∈ C [ x, y, z, u ] obtained from the quintic polynomial defining thequintic Y . Using the 4-regular condition we can derive enough monomials from the polynomial f to have c ( f ) = 1. We omit the detailed computation. For the detail, see [35].It follows from Lemma 2.1.7 that there is a point P ∈ X such that the log pair ( X, λD ) is log canonicalon X \ P . Therefore, the log pair ( X, λD ) is not log canonical only at the point P .By suitable coordinate changes, we may assume that P = [0 : 0 : 0 : 0 : 0 : 1] and that the fourfold X is given by an equation w x + X i =2 w − i q i ( x, y, z, u, v ) = 0 ⊂ P ∼ = Proj (cid:0) C (cid:2) x, y, z, u, v, w (cid:3)(cid:1) , where q i ( x, y, z, u, v ) is a homogeneous polynomial of degree i . Let T be the threefold on X cut by x = 0.Let π : U → X be the blow up at the point P with the exceptional divisor E . Then¯ D ≡ π ∗ ( D ) − mult P ( D ) E, where ¯ D is the proper transform of the divisor D via the morphism π . Note that mult P ( D ) > λ . Itfollows from [30, Proposition 3] that either mult P ( D ) > λ or there is a plane Ω ⊂ E ∼ = P such thatmult P ( D ) + mult Ω (cid:0) ¯ D (cid:1) > λ . In the case when mult P ( D ) > λ , let L be a sufficiently general pencil of hyperplane sections of X thatpass through the point P . In the case when mult P ( D ) ≤ λ , let L be the pencil of hyperplane sections of X such that S ∈ L ⇐⇒ either Ω ⊂ ¯ S or S = T, where ¯ S is the proper transform of S via the birational morphism π . In both the cases, there is athree-dimensional linear subspace Π ⊂ P such that the base locus of L consists of the intersection Π ∩ X .Let S be a general threefold in L . Then S = T and mult P ( S ∩ D ) > λ .Put Z = X | Π . The surface Z is reduced and irreducible because X contains neither quadric surfacesnor planes by our initial assumption. The 4-regularity of X implies that mult P ( Z ) ≤ Lemma 2.3.4.
The multiplicity of Z at the point P is .Proof. Suppose that mult P ( Z ) ≤
2. Let M be the 4th hypertangent linear system at the point P andlet M be a general member in M . The base locus of M consists of finitely many lines on X that passthrough the point P .Put D ∩ S = mZ + Υ, where m is a non-negative rational number and Υ is a 2-cycle whose supportdoes not contain the surface Z . Then mult P (Υ) > λ − m but T does not contain components of Υ. Wetherefore have mult P ( T ∩ Υ) > λ − m. We then consider the one cycle T ∩ Υ. We may write T ∩ Υ = k X i =1 α i L i + ∆ , where L i is a line contained in Z and passing through the point P and the support of ∆ contain none ofthe lines L i ’s. We have M · ∆ = M · T · D · S − mT · Z − k X i =1 α i L i ! = 20 − m − k X α i and M · ∆ ≥ mult P ( M ) mult P (∆) > λ − m − k X i =1 α i , og canonical thresholds of certain Fano hypersurfaces and hence 4 = 20 λ − < k X i =1 α i . On the other hand, using our generality condition G 3, we obtain the opposite inequality P ki =1 α i ≤ P ( Z )=3.By our generality condition, we have k ≤
3. Note that we may regard T ∩ Υ as a divisor in |O Z (1 − m ) | on the quintic surface Z ⊂ Π ∼ = P since T ∩ S = Z . For each line L j we consider the hyperplane section A j of Z by a general hyperplane in Π passing through the line L j . The divisor A j on the surface Z consists of the line L j and the residual curve C j . On the surface Z , we have k X i =1 α i C j · L i ≤ C j · ( k X i =1 α i L i + ∆) = 4(1 − m ) ≤ . (2.3.5)On the surface Z , the local intersection number of C i and L j at an ordinary double point of Z iswell-defined and it is at least if these two curves intersect there. The local intersection number of C i and L j at a smooth point of Z is at least 1 if these two curves intersect there. CASE k = 1 . G 3.1.1 implies 1 ≤ C · L . Then the inequality (2.3.5) implies α ≤ CASE k = 2 . First we suppose that neither L nor L contains exactly two singular points of Z . Then it followsfrom the conditions G 3.2.1, 3.2.2, 3.2.3, 3.2.4, 3.2.6 that 2 ≤ C j · L j . This implies that α j ≤
2, and hence α + α ≤ L contains exactly two singular points of Z . One of them are the point P .Suppose that P is a non-ordinary double point. Then2 α ≤ C · ( α L + α L ) ≤ α ≤
2. On the other hand, it follows from G 3.2.3, G 3.2.4, G 3.2.5 and G 3.2.6that 2 α ≤ C · ( α L + α L ) ≤ , which implies that α ≤
2. Then α + α ≤ P is an ordinary double point. We obtain from G 3.2.5 that32 α + 12 α ≤ C · ( α L + α L ) ≤ . On the other hand, regardless of the number of the singular points on L , we see12 α + 32 α ≤ C · ( α L + α L ) ≤ P is an ordinary double point. These imply that α + α ≤ CASE k = 3 . Suppose that the three lines are coplanar. Then G 3.3.1 shows that for each j = 1, 2, 3 we have3 ≤ C j · L j , and hence α j ≤ . Therefore, we obtain α + α + α ≤ P is anordinary double point, and when the point P is not an ordinary double point.Suppose that P is not an ordinary double point. Then G 3.3.2 shows that for each j = 1, 2, 3 we have3 ≤ C j · L j , and hence α j ≤ . Therefore, we obtain α + α + α ≤ P is an ordinary double point. Then G 3.3.2 shows that for each i and j , C j · L i ≥
12 + 2 δ ij , I. Cheltsov, J. Park, J. Won where δ ij is the Kronecker-delta function, i.e., δ ij = 1 if i = j ; δ ij = 0 if i = j . Then the inequality (2.3.5)implies that for each 1 ≤ j ≤
3, we have12 ( α + α + α ) + 2 α j ≤ X i =1 α i C j · L i ≤ , and hence 3( α + α + α ) ≤
72 ( α + α + α ) ≤ X j =1 3 X i =1 α i C j · L i ≤ , which implies that α + α ≤
4. This completes the proof.Let ¯ T be the proper transform of T via the birational morphism π . Then3 = mult P ( Z ) = mult P ( T ∩ S ) = mult P ( T ) mult P ( S ) + mult Ω (cid:0) ¯ T ∩ ¯ S (cid:1) , which implies that Ω ⊂ ¯ T . Since mult P ( D ) > λ and mult P ( T ) = 2, it follows thatmult P ( T ∩ D ) ≥ mult P ( T ) mult P ( D ) + mult Ω (cid:0) ¯ T ∩ ¯ D (cid:1) ≥ P ( D ) + mult Ω (cid:0) ¯ D (cid:1) > λ . Now we restrict everything to a general hyperplane section of the fourfold X . Let H be a generalhyperplane in P passing through the point P . Put˜ X = H ∩ X, ˜ T = H ∩ T, ˜ S = H ∩ S, ˜ D = H ∩ D, ˜ Z = H ∩ Z, ˜Υ = H ∩ Υ . Let ˜ P = [0 : 0 : 0 : 0 : 1]. The threefold ˜ X is 4-regular at the point ˜ P . The divisor ˜ D is equivalent to O P (1) | ˜ X .We have mult ˜ P (cid:16) ˜ T (cid:17) = 2 , mult ˜ P ( ˜ Z ) = 3 , mult ˜ P (cid:16) ˜ T ∩ ˜ D (cid:17) > λ , mult ˜ P (cid:16) ˜ S ∩ ˜ D (cid:17) > λ . The intersection ˜ T ∩ ˜ S consists of the irreducible reduced curve ˜ Z . Put˜ T ∩ ˜ D = ¯ m ˜ Z + ∆ , where ¯ m is a non-negative rational number and ∆ is an effective one-cycle on ˜ X whose support does notcontain the curve ˜ Z . Then, mult ˜ P (∆) > λ − m .Let N be the third hypertangent linear system at the point ˜ P . Lemma 2.1.6 shows that the base locusof N does not contain any curves because the threefold ˜ X contains no lines passing through the point ˜ P .Hence, for a general member N in N we have15 = N · ˜ T · ˜ D ≥
15 ¯ m + N · ∆ >
15 ¯ m + 4 (cid:18) λ − m (cid:19) , which implies 15 > λ + 3 ¯ m . Since ˜ D ∩ ˜ S = m ˜ Z + ˜Υ and mult ˜ P ( ˜Υ) > λ − m , on the surface ˜ S we have5 − m = ˜ Z ∩ ˜Υ > mult ˜ P (cid:16) ˜ Z (cid:17) mult ˜ P (cid:16) ˜Υ (cid:17) > (cid:18) λ − m (cid:19) . Thus, we see that 4 m > λ − Z is reduced and ˜ S is a sufficiently general hyperplane section of ˜ X that contains the curve˜ Z . Thus, we have m = mult ˜ Z ( ˜ D ) ≤ mult ˜ Z ( ˜ T ∩ ˜ D ) = ¯ m, which implies 15 ≥ λ + 3 m > λ + 34 (cid:18) λ − (cid:19) . It contradicts λ = .The obtained contradiction completes the proof of Theorem 1.1.5. og canonical thresholds of certain Fano hypersurfaces The Picard group of a smooth Fano hypersurface of degree n ≥ P n is generated by an anticanonicaldivisor. Therefore, it is natural that we consider only plurianticanonical divisors when we define its globallog canonical threshold. However, in other varieties, it may not be enough. Therefore, we generalizes theglobal log canonical threshold as follows: Definition 3.1.1.
Let X be a Q -factorial variety with at worst log canonical singularities. For an integraldivisor D on the variety X and a natural number m >
0, we define the m -th global log canonical thresholdof the divisor D by the numberlct m ( X, D ) = inf (cid:26) c (cid:18) X, m H (cid:19) (cid:12)(cid:12)(cid:12) H ∈ (cid:12)(cid:12) mD (cid:12)(cid:12)(cid:27) , where the number lct m ( X, D ) is defined to be ∞ if the linear system | mD | is empty. Also, we define theglobal log canonical threshold of D by the numberlct( X, D ) = inf n ∈ N n lct n ( X, D ) o . Let π : V → P n be a smooth double cover ramified along a hypersurface S of degree 2 m in P n , n ≥ H be the pull-back of a hyperplane in P n by the covering map π . We can consider thedouble cover V as a smooth hypersurface of degree 2 m in P (1 n +1 , m ). Proposition 3.2.1.
The global log canonical threshold lct(
V, H ) is equal to the first global log canonicalthreshold lct ( V, H ) .Proof. Let us use the arguments in the proof of [30, Proposition 5].Suppose that there is a divisor D in the linear system | µH | for some integer µ ≥ c (cid:18) V, µ D (cid:19) < lct ( V, H ) ≤ . It follows from Remark 2.2.3 that we may assume that the support of the divisor D does not containdivisors of the linear system | H | .Choose a number λ such that c (cid:16) V, µ D (cid:17) < λ < lct ( V, H ). Then the log pair (cid:16) V, λµ D (cid:17) is not logcanonical. By [29, Proposition 4.3] we have the center of a non-log-canonical singularity of the log pair (cid:16) V, λµ D (cid:17) at a point P on V .Suppose that π ( P ) ∈ S . Let T be the unique divisor in the linear system | H | that is singular at thepoint P . Since we have mult P ( D ) > µ , we obtain an absurd inequality2 µ = D · T n − ≥ mult P ( D ∩ T ) > µ. Now, we suppose that π ( P ) S . Let ξ : W → V be the blow up at the point P and E ∼ = P n − be theexceptional divisor of the birational morphism ξ . Then, it follows from [30, Proposition 3] that there isa hyperplane Λ ⊂ E such that mult P ( D ) + mult Λ (cid:0) ¯ D (cid:1) > µ, where ¯ D is the proper transform of D on the variety W .Let G be a general divisor in | H | such that Λ ⊂ Supp( ¯ G ), where ¯ G is the proper transform of G onthe variety W . Then, we also obtain a contradictory inequality2 µ = D · G n − ≥ mult P ( D ∩ G ) > µ. I. Cheltsov, J. Park, J. Won
Now we are ready to prove the following result.
Proposition 3.2.2.
The following inequality holds: lct (
V, H ) ≥ min (cid:18) , m + n − m (cid:19) . Proof.
By Proposition 3.2.1, it is enough to consider the first global log canonical threshold lct ( V, H )instead of lct(
V, H ). Let D be a divisor in | H | .The double space V can be defined by a quasi-homogenous equation w = f ( x , . . . , x n ) in the weightedprojective space P (cid:0) n +1 , m (cid:1) ∼ = Proj ( C [ x , . . . , x n , w ]), where wt( x i ) = 1, wt( w ) = m , and f is a homo-geneous polynomial of degree 2 m . Note that the homogenous polynomial f defines the smooth hyper-surface S in P n since V is smooth. We may assume that the divisor D is cut out on V by the equation x = 0. The divisor D is a hypersurface in P (1 n , m ) ∼ = Proj ( C [ x , . . . , x n , w ]) defined by the equation w = f (0 , x , . . . , x n ). It has isolated singularities since the hypersurface D S := { f (0 , x , . . . , x n ) = 0 } ⊂ P n − ∼ = Proj ( C [ x , . . . , x n ]) , has isolated singularities (see [15]).It follows from [11, Theorem 3.1] that the log pair ( V, λD ) is log terminal if and only if ( P (1 n , m ) , λD )is log terminal because V is smooth and the divisor D is contained in the smooth locus of P (1 n , m ). Itthen follows from [17, Proposition 8.21] that c ( V, D ) = c ( P (1 n , m ) , D ) = 12 + c (cid:0) P n − , D S (cid:1) . We then see that [11, Theorem 3.1] and [6, Theorem 3.3] imply c (cid:0) P n − , D S (cid:1) = c ( S, D S ) ≥ n − m . This completes the proof.Let π : V → P n be a double cover ramified along a smooth hypersurface of degree 2 n ≥
4. It is a Fanovariety of Fano index 1 and the pull-back of a hyperplane in P n is an anticanonical divisor of V . It followsfrom Proposition 3.2.2 (for n ≥
3) and [5, Theorem 1.7] (for n = 2) thatlct ( V ) ≥ n − n , while [30, Theorem 2] shows that lct( V ) = 1 if V is general and n ≥
3. Therefore, we immediately obtainthe following result that has been proved by [1] in a different way.
Corollary 3.2.3.
A smooth double cover of P n ramified along a hypersurface of degree n ≥ admits aK¨ahler-Einstein metric.Remark . Combining the results of [3] and the proof of Proposition 3.2.2, we can easily obtain thefollowing. Let V be the smooth hypersurface in P (1 n +1 , m ) of degree 2 m ≥ n ≥ w = f ( x , . . . , x n ) ⊂ P (cid:0) n +1 , m (cid:1) ∼ = Proj ( C [ x , . . . , x n , w ]) , where wt( x i ) = 1, wt( w ) = m , and f is a homogeneous polynomial of degree 2 m . Suppose that c ( V, D ) = m + n − mµ , where D ∈ | µH | and µ ∈ N . Then D = µT , where T is a divisor that is cut out on the hypersurface V by an equation P ni =0 λ i x i = 0 such that the hypersurface f ( x , . . . , x n ) = n X i =0 λ i x i = 0 ⊂ P n − ∼ = Proj C [ x , . . . , x n ] . n X i =0 λ i x i !! is a cone over a smooth hypersurface in P n − of degree 2 m . og canonical thresholds of certain Fano hypersurfaces We can also give an easy proof of the following result that is a corollary of [30, Theorem 2].
Proposition 3.2.5.
Let V be the double cover of P n , n ≥ , ramified along a general hypersurface S ofdegree n in P n . Then lct( V, H ) = 1 .Proof.
We assume that for every hyperplane M ⊂ P n , the intersection S ∩ M has at most isolated doublepoints. This generality condition is obviously satisfied for a general hypersurface S because n ≥ D be a divisor in the linear system | H | . It follows from [17, Lemma 8.12] that the singularities ofthe log pair ( V, D ) are log canonical if and only if the singularities of the log pair (cid:18) P n , π ( D ) + 12 S (cid:19) are log canonical. Put M = π ( D ). It follows from [17, Theorem 7.5] that the singularities of the log pair( V, D ) are log canonical if and only if the log pair ( M, S | M ) is log canonical. But the log pair ( M, S | M )is log canonical because S | M has at most isolated double points.The generality assumption in Proposition 3.2.5 is weaker than that of [30, Theorem 2].Let V be the double cover of P ramified along a smooth sextic S ⊂ P . Note that the pull-back of ahyperplane in P is an anticanonical divisor. As we did for quartic threefolds, we are also able to find allthe possible first global log canonical thresholds of V . Proposition 3.2.6.
Let V be the smooth double cover of P ramified along a sextic. Then, the first globallog canonical threshold of the Fano variety V is one of the following: { , , , , , , , , , , , , , , , } . Furthermore, for each number µ in the set above, there is a smooth double cover V of P ramified alonga sextic with lct ( V ) = µ .Proof. For the proof, see [35]. Its brief idea is as follows. For a hyperplane H in P , we see that c ( V, π ∗ ( H )) = min (cid:26) ,
12 + c ( H, H ∩ S ) (cid:27) . The intersection H ∩ S is a reduced sextic plane curve on H ∼ = P . Therefore, for the first statementof Proposition 3.2.6, it is enough to consider all the possible values of c ( P , C ) for reduced sextic planecurves. Furthermore, we can consider only the values for c ( f ), where f is a reduced sextic polynomialvanishing at the origin.Because the first global log canonical thresholds coincide with the global log canonical thresholds ondouble spaces, Proposition 3.2.6 implies a stronger result as follows. Corollary 3.2.7.
Let V be a smooth double cover of P ramified along a sextic. Then, the global logcanonical threshold of the Fano variety V is one of the numbers in Proposition 3.2.6. Furthermore, foreach number µ in Proposition 3.2.6, there is a smooth double cover V of P ramified along a sextic with lct( V ) = µ . Let us finish the paper by an example of a smooth double cover of P ramified along a sextic surfacewith the global log canonical threshold 1. Example 3.2.8.
Let V be the smooth double cover of P ramified along the sextic surface S ⊂ P definedby the equation x + x + x + x + x x x x = 0 . Let C ⊂ P be the curve defined by the intersection of the surface S and the Hessian surface Hess( S ) of S . For the tangent hyperplane T P at a point P ∈ S , if the multiplicity of the curve T P ∩ S at the point P is at least 3, then the curve C is singular at the point P . Using the computer program, Singular , one cancheck that the curve C is smooth in the outside of the curves x i = x j = 0 with i = j . Furthermore, for I. Cheltsov, J. Park, J. Won a point P in S that belongs to the curves x i = x j = 0 with i = j , one can easily check that the log pair( S, H P ) is log canonical, where H P is the hyperplane section of S by the tangent hyperplane to S atthe point P . Consequently, lct( V ) = lct ( V ) = 1. The variety V is an explicit example of smooth Fanovariety with the following properties (We do not know any other explicit example of such a smooth Fanovariety). For each i = 1 , , · · · , r , let V i = V . Then, the paper [30] implies that the product V × · · · × V r is not rational and Bir( V × · · · × V r ) = Aut( V × · · · × V r ) . Moreover, for each dominant rational map ρ : V × · · · × V r Y whose general fiber is rationallyconnected, there is a subset { i , · · · , i k } ⊂ { , · · · , r } such that the diagram V × · · · × V rπ (cid:15) (cid:15) ρ ( ( ❘❘❘❘❘❘❘❘ V i × · · · × V i k ¯ ρ / / ❴❴❴❴❴❴ Y commutes, where π is the natural projection and ¯ ρ is a birational map. Acknowledgements.
The authors would like to express their sincere appreciation to the referee forthe invaluable comments. The referee’s comments enable us to improve their results as well as theirexposition. In particular, the referee pointed out a gap in the previous proof of Lemma 2.3.4. To fix thegap, the authors introduce new generality conditions to quintic fourfolds. This serious revision was donewhile the first two authors stay at Hausdorff Research Institute for Mathematics at Bonn, Germany forResearch in Groups Program from 1st of August to 4th of September 2012. Ivan Cheltsov and Jihun Parkwould like to thank the institute for their support. Jihun Park has been supported by the Research CenterProgram (Grant No. CA1205-02) of Institute for Basic Science and SRC-GAIA(Grant No. 2011-0030795)of the National Research Foundation in Korea.
Appendix
Let X F be a smooth quintic hypersurface in P that is given by zeroes of a section F ∈ H ( P , O P (5)).It follows from Proposition 2.1.4 that there exists a non-empty Zariski open subset U G ∈ H ( P , O P (5))such that X F is 4-regular whenever F ∈ U G . Similarly, it follows from Lemma 2.3.1 that there exists anon-empty Zariski open subset U G ∈ H ( P , O P (5)) such that for every 3-dimensional linear space Πin P , the intersection X F ∩ Π is irreducible and reduced if F ∈ U G .The purpose of this Appendix is to prove Lemma 2.3.2, i.e., to prove the existence of a non-emptyZariski open subset U G ∈ H ( P , O P (5)) such that for each F ∈ U G the hypersurface X F satisfies thecondition G 3 (see Section 2.3). Indeed, we prove the statement as follows: For each a (= 0 , , , and b (= 1 , , · · · , , there exists a non-empty Zariski open subset U in H ( P , O P (5)) such that if F ∈ U , then for each point P ∈ X and each -dimensional linear space Π contained in the tangent hyperplane at P and containing the point P , the surface Z := X ∩ Π satisfies the condition G 3. a . b . Since we use the same method in order to prove the statement for each a and b , we first explain how theproof goes and then show the required computations in each case G 3. a . b .The proof goes as follows.First we consider the space S = F × H (cid:0) P , O P (5) (cid:1) with the natural projections p : S → H ( P , O P (5)) and q : S → F . Here, F is a suitable flag varietyin P . Depending on the case, the flag F will be F lag (0 , , , , F lag (0 , , , F lag (0 , , ,
4) or
F lag (0 , , F lag ( n , · · · , n k ) is the flag variety that parametrizes k -tuples (Π n , · · · , Π n k ) of n i -dimensional linear spaces with Π n ⊂ · · · ⊂ Π n k ⊂ P . A 0-dimensional linear space will be denotedby P and a four dimensional linear space will be denoted by T . og canonical thresholds of certain Fano hypersurfaces We then put I = (cid:0) ( P, Π n , · · · , Π n k − , T ) , F (cid:1) ∈ S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F ( P ) = 0; T is the tangent hyperplane to X F at P ; X F satisfies the properties P G .a.b . , where the properties P G .a.b will be specified in the individual proofs. Then in each case, we will see thatit is easy to check that the morphism q | I : I → F is surjective.With this set up, we compute the codimension c of q | − I ( P, Π n , · · · , Π n k − , T ) in H ( P , O P (5)) for apoint ( P, Π n , · · · , Π n k − , T ) ∈ F . We may always assume that T is defined by x = 0, Π is defined by x = y = 0, Π by x = y = z = 0, Π by x = y = z = u = 0 and P = [0 : 0 : 0 : 0 : 0 : 1]. We write thequintic polynomial F as w q + w q ( x, y, z, u, v ) + w q ( x, y, z, u, v ) + w q ( x, y, z, u, v ) + wq ( x, y, z, u, v ) + q ( x, y, z, u, v ) , where q i is a homogeneous polynomial of degree i .The condition F ( P ) = 0 is equivalent to q = 0. The condition that T is the tangent hyperplane to X F at P is equivalent to q = λx for some λ ∈ C ∗ . These two conditions contribute to the codimension c by 5. For each a and b , we will show that that the properties P G .a.b makes another contribution to thecodimension c by more than dim F − q | − I ( P, Π n , · · · , Π n k − , T ) in H ( P , O P (5)) is morethan dim F . These implies that the morphism p | I cannot be surjective. Taking the properties P G .a.b into consideration, we can immediately notice that this non-surjectivity implies the statement.Therefore, to prove the statement for each case, it is enough to • specify the flag F with its dimension; • specify the property P G .a.b ; • show that the properties P G .a.b makes another contribution to the codimension c by more thandim F −
Lemma I.
The statement holds for G 3.0.1.Proof.
The flag F is F lag (0 , , , P G . . = { X F ∩ Π is singular along Π . } . The condition that X F ∩ Π contains the line Π is equivalent to the condition that for each i = 2 , , , q i contains no v i . For X F ∩ Π in order to be singular along L , for each i = 2 , , ,
5, thepolynomial q i must not contain the monomials zu r v i − r − , r = 0 , , · · · , i −
1. These altogether show thatthe properties P G . . is of codimension > Lemma II.
The statement holds for G 3.0.2.Proof.
The flag F is F lag (0 , , P G . . = { X F ∩ Π contains four lines. } . Since we may assume that q , q , q , q forms a regular sequence, X F ∩ Π containing four linesis equivalent to q ( x, y, z, u, v ) vanishing at four given points in q ( x, y, z, u, v ) = q ( x, y, z, u, v ) = q ( x, y, z, u, v ) = q ( x, y, z, u, v ) = 0 in P and q (0 , , z, u, v ) vanishing at four given points in q (0 , , z, u, v ) = q (0 , , z, u, v ) = 0 in P . These altogether show that the properties P G . . is ofcodimension 8. Lemma III.
The statement holds for G 3.1.1. I. Cheltsov, J. Park, J. Won
Proof.
The flag F is F lag (0 , , , P G . . = X F ∩ Π contains Π ;Π meets its residual curve by a general hyperplanesection of X F ∩ Π in Π only at singular points; X F ∩ Π has at most one ordinary double point on Π . . We write q i (0 , , z, u, v ) = X r + s + t = i A rst z r u s v t , where A rst ’s are constants.The condition that X F ∩ Π contains the line Π is equivalent to A t = 0 for t = 2, 3, 4 and 5 sincethe line Π is defined by x = y = z = u = 0.The surface X F ∩ Π has singular points on the line Π exactly where the polynomials A vw + A v w + A v w + A v and A vw + A v w + A v w + A v have common zeros in P . Thezero given by v = 0 corresponds to the singular point P . To see this, put ¯ F ( z, u, v, w ) = F (0 , , z, u, v, w ).Since A t = 0 for t = 2, 3, 4 and 5, we always have ∂ ¯ F∂v (0 , , v, w ) = ∂ ¯ F∂w (0 , , v, w ) = 0. The commonzeros of ∂ ¯ F∂z (0 , , v, w ) = v ( A w + A vw + A v w + A v ) ,∂ ¯ F∂u (0 , , v, w ) = v ( A w + A vw + A v w + A v )are the singular points of X F ∩ Π on the line Π . Note that Π and its residual curve by a generalhyperplane meet at every singular point of X F ∩ Π on the line Π . Therefore, the second condition isequivalent to the condition that the polynomials A vw + A v w + A v w + A v and A vw + A v w + A v w + A v have four common zeros in P with counting multiplicity, i.e., these twopolynomials are proportional. This imposes three additional independent conditions on the coefficientsof F .The condition that the polynomial A vw + A v w + A v w + A v has k zeros withoutcounting multiplicity imposes 4 − k additional independent conditions on the coefficients of F . Note that1 ≤ k ≤ k − F . Herewe verify the claim only for the case with k = 4. The other cases with k = 3 and 2 can be verified in thesame way.We write the homogenized Hessian matrix of the polynomial q (0 , , z, u, v ) + q (0 , , z, u, v ) + q (0 , , z, u, v ) + q (0 , , z, u, v ) along the line Π as follows: A w + A vw + A v w + A v ) A w + A vw + A v w + A v A w + 2 A vw + 3 A v w + 4 A v A w + A vw + A v w + A v A w + A vw + A v w + A v ) A w + 2 A vw + 3 A v w + 4 A v A w + 2 A vw + 3 A v w + 4 A v A w + 2 A vw + 3 A v w + 4 A v . Let H ( v, w ) be the determinant of the homogenized Hessian matrix. The condition that threeof the four singular points on Π is not ordinary double points is equivalent to the conditionthat H ( v, w ) vanishes at three points out of the four points defined by A w v + A v w + A v w + A v = 0 and A vw + A v w + A v w + A v = 0 in P . We claim thatit imposes three additional independent conditions on the coefficients of F . To verify the claim,we put A = 0 , A = 0 , A = 0 , A = 0 , A = 0 , A = 0 A = 0 , A = 0 A = 0 , A = 0 , A = 0 , A = 0 . Since A w v + A v = 0 and A vw + A v = 0 defines four points in P , we have[ λ : µ ] ∈ P with λ ( A , A ) = µ ( A , A ). We then see that in our restricted situation, thecondition is equivalent to the condition that A w v + A v = 0 has three common pointswith (cid:0) A w + 4 A v (cid:1) (cid:8) λ (cid:0) A w + A v (cid:1) + µ (cid:0) A w + A v (cid:1)(cid:9) = 0 og canonical thresholds of certain Fano hypersurfaces P . Since this is a condition of codimension 3 in the restricted situation, it verifies the claim.These altogether show that the properties P G . . is of codimension > Lemma IV.
The statement holds for G 3.2.1.Proof.
The flag F is F lag (0 , , , , P G . . = X F contains Π ; X F ∩ Π contains a line other than Π passing through P ; X F ∩ Π contains four singular points other than P on the two lines on X ∩ Π passing through the point P . . The condition that X F contains the line Π is equivalent to the condition that for each i = 2 , , ,
5, thepolynomial q i contains no v i . The condition that X F ∩ Π contains a line other than Π passing through P is equivalent to the condition that q (0 , , , u, v ), q (0 , , , u, v ) and q (0 , , , u, v ) vanish at the pointother than the point given by u = 0 in P where q (0 , , , u, v ) vanishes. For X F ∩ Π in order to havefour singular points other than P on the two lines on X F ∩ Π passing through the point P is a conditionof codimension 4. These altogether show that the properties P G . . is of codimension 11. Lemma V.
The statement holds for G 3.2.2.Proof.
The flag F is F lag (0 , , , , P G . . = X F contains Π ; X F ∩ Π contains two lines passing through P ; X F ∩ Π has three singular points other than P on Π ; X F ∩ Π has at least one singular point on Π that is not an ordinary double point. . The condition that Π ⊂ X F is equivalent to the fact that each q i ( x, y, z, u, v ) does not have v i monomial,which is condition of codimension 4. The condition that X F ∩ Π contains another line passing throughthe point P is equivalent to the condition that either q (0 , , , u, v ), q (0 , , , u, v ) and q (0 , , , u, v )vanish at the points in P where q (0 , , , u, v ) /u vanishes, or q (0 , , , u, v ) is a zero polynomial and q (0 , , , u, v ), q (0 , , , u, v ) and q (0 , , , u, v ) have common root in P . Thus, the condition that X F ∩ Π contains another line passing through the point P is a condition of codimension 3. For thesurface X F ∩ Π to have three singular points on Π other than P is a condition of codimension 3.Arguing as in the proof of Lemma III, we can see that the condition that one of the singular points of X F ∩ Π on Π is not an ordinary double point is a condition of codimension 1. These altogether showthat the properties P G . . is of codimension > Lemma VI.
The statement holds for G 3.2.3.Proof.
The flag F is F lag (0 , , , , P G . . = X F contains Π ; X F ∩ Π contains a line other than Π passing through P ; X F ∩ Π contains two non-ordinary singular points on Π . . The condition that X F contains the line Π is equivalent to the condition that for each i = 2 , , ,
5, thepolynomial q i contains no v i . The condition that X F ∩ Π contains a line other than Π passing through P is equivalent to the condition that q (0 , , , u, v ), q (0 , , , u, v ) and q (0 , , , u, v ) vanish at the pointother than the point given by u = 0 in P where q (0 , , , u, v ) vanishes. As in the proof of Lemma III,we can see that for X F ∩ Π to have two non-ordinary singular points on Π is a condition of codimension4. These altogether show that the properties P G . . is of codimension > Lemma VII.
The statement holds for G 3.2.4. I. Cheltsov, J. Park, J. Won
Proof.
The flag F is F lag (0 , , , , P G . . = X F contains Π ; X F ∩ Π contains a line other than Π passing through P ; X F ∩ Π contains two singular points other than P on the line Π ;Π meets its residual curve by a general hyperplane section of X F ∩ Π in Π only at three points. . We write q i (0 , , z, u, v ) = X r + s + t = i A rst z r u s v t , where A rst ’s are constants.The condition that X F ∩ Π contains the line Π is equivalent to A t = 0 for t = 2, 3, 4 and 5. Thecondition that X F ∩ Π contains a line other than Π passing through P is equivalent to the condition that q (0 , , , u, v ), q (0 , , , u, v ) and q (0 , , , u, v ) vanish at the point other than the point given by u = 0in P where q (0 , , , u, v ) vanishes. For X F ∩ Π in order to have two singular points other than P onthe line Π and for Π to meet its residual curve by a general hyperplane section of X F ∩ Π in Π only atthree point are equivalent to the condition that the polynomials A vw + A v w + A v w + A v and A vw + A v w + A v w + A v have four common zeros in P with counting multiplicity andthe polynomial A vw + A v w + A v w + A v has three zeros without counting multiplicity.This condition is of codimention 4. These altogether show that the properties P G . . is of codimension11. Lemma VIII.
The statement holds for G 3.2.5.Proof.
The flag F is F lag (0 , , , , P G . . = X F contains Π ; X F ∩ Π contains a line other than Π passing through P ; X F ∩ Π contains one singular points other than P on the line Π ;either Π meets its residual curve by a general hyperplane section in Π only at singular points orΠ meets its residual curve by a general hyperplane section in Π only at three points and X F ∩ Π has a non-ordinary singular point P . . The first two conditions imposes seven independent conditions on the coefficients of F as before. For thesurface X F ∩ Π to have a singular point on Π other than P and plus for Π to meet its residual curve bya general hyperplane section in Π only at singular points impose at least four independent conditions onthe coefficients of F . Meanwhile, for the surface X F ∩ Π to have a singular point on Π other than P andplus for Π to meet its residual curve by a general hyperplane section in Π only at three points imposeat least four independent conditions on the coefficients of F . However, the condition that X F ∩ Π hasa non-ordinary singular point P is of codimension 1. Therefore, the properties P G . . is of codimension11.These altogether show that the properties P G . . is of codimension 11. Lemma IX.
The statement holds for G 3.2.6.Proof.
The flag F is F lag (0 , , , , P G . . = X F contains Π ; X F ∩ Π contains a line other than Π passing through P ;Π meets its residual curve by a general hyperplane section of X F ∩ Π in Π at at most two points. . The first two conditions imposes seven independent conditions on the coefficients of F as before.For the last condition, we write q i (0 , , z, u, v ) = X r + s + t = i A rst z r u s v t , og canonical thresholds of certain Fano hypersurfaces where A rst ’s are constants.The last condition is equivalent to the condition that either the polynomials A vw + A v w + A v w + A v and A vw + A v w + A v w + A v have a common zero at v = 0 withmultiplicity at least 3 or they are proportional and have only two zeros (without counting multiplicities).The former and the latter are both a condition of codimension at least 4.Therefore, the properties P G . . is of codimension at least 11. Lemma X.
The statement holds for G 3.3.1.Proof.
The flag F is F lag (0 , , , P G . . = X F ∩ Π contains three lines passing through P ;either X F ∩ Π has a singular point on Π other than P or one of the lines L i meetsits residual curve by a general hyperplane section in Π at at most three points. . The condition that X F ∩ Π contains three lines passing through the point P is equivalent to thecondition that q (0 , , , u, v ) is identically zero; q (0 , , , u, v ) and q (0 , , , u, v ) vanish at the threepoints in P where q (0 , , , u, v ) vanishes. For the surface X F ∩ Π to have a singular point on Π otherthan P is a condition of codimension 1. For one of the lines L i to meet its residual curve by a generalhyperplane section in Π at at most three points is also a condition of codimension 1. These altogethershow that the properties P G . . is of codimension > Lemma XI.
The statement holds for G 3.3.2.Proof.
The flag F is F lag (0 , , P G . . = X F ∩ Π contains three lines passing through P ;either one of the lines L i meets its residual curve by a general hyperplane sectionin Π at at most one smooth point orone of the lines L i meets its residual curve by a general hyperplane section in Π at at most two smooth points and X F ∩ Π has a non-ordinary singular point at P . . The condition that X F ∩ Π contains three lines passing through the point P is equivalent to thecondition that q (0 , , z, u, v ) and q (0 , , z, u, v ) vanish at three points in P where both q (0 , , z, u, v )and q (0 , , z, u, v ) vanish.For one of the lines L i to meet its residual curve by a general hyperplane section in Π at at most onesmooth is also a condition of codimension at least 2. For one of the lines L i to meet its residual curve bya general hyperplane section in Π at at most two smooth is also a condition of codimension at least 1.The condition that X F ∩ Π has a non-ordinary singular point at P is equivalent to the condition that thequadratic polynomial q (0 , , z, u, v ) is singular in variables z , u , v . This is a condition of codimension 1.These altogether show that the properties P G . . is of codimension > References [1] C. Arezzo, A. Ghigi, P. Gian,
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