The Fano variety of lines and rationality problem for a cubic hypersurface
aa r X i v : . [ m a t h . AG ] J un THE FANO VARIETY OF LINES ANDRATIONALITY PROBLEM FOR A CUBIC HYPERSURFACE
SERGEY GALKIN, EVGENY SHINDER
Abstract.
We find a relation between a cubic hypersurface Y and itsFano variety of lines F ( Y ) in the Grothendieck ring of varieties. We usethis relation to study the Hodge structure of F ( Y ). Finally we proposea criterion for rationality of a smooth cubic hypersurface in terms of itsvariety of lines.In particular, we show that if the class of the affine line is not a zero-divisor in the Grothendieck ring, then the variety of lines on a smoothrational cubic fourfold is birational to a Hilbert scheme of two points on a K Contents
1. Introduction 12. The Grothendieck ring of varieties 52.1. Generalities 52.2. Realizations 62.3. The Grothendieck ring and rationality questions 82.4. The Cancellation conjecture 103. The Hilbert scheme of two points 104. The Fano variety of lines on a cubic 124.1. Definition and basic properties 124.2. Decomposability 124.3. Decomposability and the associated K Y - F ( Y ) relation 165.1. The relation in K ( V ar/k ) 165.2. Examples and immediate applications 175.3. The relation in K ( V ar/k )[ L − ] 196. Hodge structure of the Fano variety F ( Y ) 227. Rational cubic hypersurfaces 24References 261. Introduction
Let Y ⊂ P d +1 be a cubic hypersurface over a field k . The Fano variety F ( Y ) of lines on Y is defined as the closed subvariety of the Grassmannian Gr (1 , P d +1 ) = Gr (2 , d + 2) consisting of lines L ⊂ Y . See [BV78], [AK77] for details and Section 4.1 for a short summary on the Fano variety of lines on acubic.For the purpose of this Introduction assume that Y is smooth, in whichcase F ( Y ) is smooth projective of dimension 2 d −
4, and connected for d ≥ • d = 2: F ( Y ) consists of 27 isolated points. This has been discovered inthe correspondence between Cayley and Salmon and has been publishedin 1849 [Cay49, Sal49]. • d = 3: F ( Y ) is a surface of general type which has been introduced byFano [F04], and then studied by Bombieri and Swinnerton-Dyer [BS67]in their proof of the Weil conjectures for a smooth cubic threefolddefined over a finite field and by Clemens and Griffiths [CG72] in theirproof of irrationality of a smooth complex cubic threefold. • d = 4: F ( Y ) is a holomorphic symplectic fourfold [BD85]. For severaltypes of cubic fourfolds F ( Y ) is isomorphic to a Hilbert scheme of twopoints on a K F ( Y ) in terms of the geometry of Y and discuss applications to rationality of Y . In the most concise form therelation between the geometry of the Fano variety F ( Y ) and the cubic Y ,which we call the Y - F ( Y ) relation looks like:[ Y [2] ] = [ P d ][ Y ] + L [ F ( Y )] . Here Y [2] is the Hilbert scheme of length two subschemes on Y . The equalityholds in the Grothendieck ring of varieties, and encodes the basic fact that aline L intersecting the cubic Y in two points determines the third intersectionpoint unless L ⊂ Y . See Theorems 5.1, 5.6 for different ways of expressing the Y - F ( Y ) relation.Different kinds of invariants of the Fano variety may be computed using the Y - F ( Y ) relation. For example we can easily compute the number of lines onreal and complex smooth or singular cubic surfaces (see Examples 5.3, 5.4).On the other hand we can compute the Hodge structure of H ∗ ( F ( Y ) , Q ) fora smooth complex cubic Y of an arbitrary dimension d . It turns out thatthe Hodge structure of F ( Y ) is essentially the symmetric square of the Hodgestructure of Y (Theorem 6.1).A central question in studying cubics is that of rationality of a smooth cubic d -fold. This question is highly non-trivial in dimension d ≥ d ≥ d ≥ p = 2 [Mur72, Mur73, Mur74]. ANO VARIETY AND RATIONALITY FOR A CUBIC 3
In dimension 4 the situation is much more complicated. Let us give a briefoverview on rationality of smooth cubic fourfolds over k = C . First of all,there are examples of smooth rational cubic fourfolds: the simplest ones arecubic fourfolds containing two disjoint 2-planes and Pfaffian cubic fourfolds.These and other classes of rational cubic fourfolds have been studied by Morin[Mor40], Fano [F43], Tregub [Tr84], [Tr93], Beauville-Donagi [BD85] and Has-sett [Has99].Nevertheless a very general cubic fourfold is expected to be irrational. Moreprecisely, according to a conjecture made by Iskovskih in the early 1980s (see[Tr84]), the algebraic cycles inside H ( Y, Q ) of a rational cubic fourfold forma lattice of rank at least two. It is known that such cubic fourfolds form acountable union of divisors in the moduli space. Thus for a very general cubicfourfold algebraic classes in H ( Y, Q ) are multiples of h , where h ∈ H ( Y, Q )is the class of the hyperplane section [Zar90, Has00]In light of this discussion it is quite remarkable that no irrational cubic four-fold is known at the moment. Kulikov deduced irrationality of a general cubicfourfold from a certain conjectural indecomposability of Hodge structure ofsurfaces [Kul08]; the latter version of indecomposability however was recentlyshown to be false [ABB13].Hassett called cubic fourfolds with an extra algebraic class in H ( Y, Q ) spe-cial , and studied them in detail, giving complete classification of special cubicfourfolds into a countable union of divisors inside the moduli space of all cubicfourfolds [Has00]. Hassett also classified those special cubic fourfolds Y thathave an associated K S . This basically means that the primitivecohomology lattices of Y and S are isomorphic (see Definition 4.7 for details).It is expected that rational smooth cubic fourfolds are not only special, butalso have associated K K S associated to Y : the derived category of coherent sheaveson S must be embedded into the derived category of coherent sheaves on Y .Furthemore, Addington and Thomas showed that this criterion is genericallyequivalent to the Hodge-theoretic criterion of Hassett [AT12].A different but related conjectural necessary condition for rationality of cu-bic fourfolds is given by Shen: according to [Sh12, Conjecture 1.6] a smoothrational cubic fourfold Y , F ( Y ) must have a certain algebraic class in themiddle cohomology H ( F ( Y ) , Z ).In any even dimension d = 2 r there exist smooth rational cubics. To con-struct one we start with an r -dimensional subvariety W ⊂ P r +1 with oneapparent double point . This means that through a general point p ∈ P r +1 there is a unique secant line L p to W , i.e. a line which intersects W in twopoints. A simple geometric construction which has been used by Morin [Mor40]and Fano [F43] shows that any cubic Y containing such a W is rational [Rus00, SERGEY GALKIN, EVGENY SHINDER
Prop.9]. The case of 2 r -dimensional cubics containing two disjoint r -planes isa particular case of this construction.Smooth connected varieties with one apparent double point exist for anydimension r [Bab31, Ed32, Rus00], and such varieties are classified in smalldimensions. For example, for r = 1 it is the twisted cubic in P , and for r = 2there are two degree four rational normal scrolls and a del Pezzo surface ofdegree five [Sev01, Rus00].We finish this overview of known results on rationality of cubics by notingthat there is no examples of smooth rational cubics of odd dimension.In the direction of irrationality of cubic fourfolds we prove the following (seeTheorem 7.5): let k be a field of characteristic zero and assume the CancellationConjecture: L = [ A ] is not a zero divisor in the Grothendieck ring K ( V ar/k )of varieties. If Y is a smooth rational cubic fourfold, then F ( Y ) is birationallyequivalent to S [2] for some K S . In particular Y is special and S isassociated to Y in the sense of Hodge structure (Proposition 4.5). Accordingto the discussion above this in particular implies that a very general smoothcomplex cubic fourfold Y is irrational.Modulo the same assumption ( L being not a zero-divisor) we give a shorterproof for the result of Clemens and Griffiths on irrationality of smooth cubicthreefolds (Theorem 7.4).We also get a criterion for irrationality of higher-dimensional smooth cubics:the Fano variety of lines on a smooth rational cubic must be stably decompos-able (see Definition 4.2 and Theorem 7.1). However it is not clear at themoment whether this criterion gives an obstruction to rationality in dimension d ≥ char ( k ) = 0. Roughly speaking the theorem of Larsen andLunts says that in the quotient ring K ( V ar ) / ( L ) each class has a uniquedecomposition into classes of stable birational equivalence. The same sort ofuniqueness lies in the heart of the proof of irrationality of cubic threefolds byClemens and Griffiths [CG72] who consider decompositions of the principallypolarized intermediate Jacobian of a cubic threefold.Starting with a rational smooth cubic d -fold Y we write its class in the form[ Y ] = [ P d ] + L · M Y , for some M Y ∈ K ( V ar/k ) (Corollary 2.2) and plug this into the Y - F ( Y ) re-lation. Assuming the Cancellation Conjecture we may divide by L and thendeduce using the theorem of Larsen and Lunts that F ( Y ) is stably decompos-able. This is not possible in dimension d = 3 and in dimension d = 4 yields abirational equivalence between F ( Y ) and S [2] . Our approach is especially effi-cient in these two dimensions as the Fano of variety of lines has non-negative ANO VARIETY AND RATIONALITY FOR A CUBIC 5
Kodaira dimension for d ≤
4, and for such varieties the notion of stable bira-tional equivalence coincides with birational equivalence due to existence of theMRC fibration [KMM92, Kol96, GHS03] (Lemma 2.6).Let us now briefly explain the structure of the paper. Sections 2, 3, 4 containmaterial on the Grothendieck ring of varieties, the Hilbert scheme of lengthtwo subschemes and the Fano variety of lines on a cubic. Most of this is well-known except possibly the discussion of decomposability of the Fano varietyof lines in 4.2 and 4.3.In Section 5 we prove several versions of the Y - F ( Y ) relation for a pos-sibly singular cubic hypersurface over an arbitrary field and deduce simpleconsequences of this relation.In Section 6 we express the Hodge structure of F ( Y ) with rational coef-ficients in terms of the Hodge structure of Y for a smooth complex cubichypersurface. In particular this recovers known results for cubic threefolds[CG72] and fourfolds [BD85].Section 7 contains applications of the Y - F ( Y ) relation to rationality of cu-bics.We would like to thank our friends and colleagues Arend Bayer, PaoloCascini, Sergey Finashin, Sergey Gorchinskiy, Alexander Kuznetsov, Fran¸coisLoeser, Yuri Prokhorov, Francesco Russo, Nick Shepherd-Barron, ConstantinShramov, Evgeny Shustin, Nicolo Sibilla, Maxim Smirnov, Fedor LazarevichZak for discussions, references and their interest in our work. Special thanksgo to Constantin Shramov, Andrey Soldatenkov and Ziyu Zhang for their com-ments on a draft of this paper. The second named author is greatly indebtedto Daniel Huybrechts’ seminar on cubic hypersurfaces in Bonn in the Summersemester of 2013 which has been a great opportunity to learn about cubics.2. The Grothendieck ring of varieties
Detailed references on the Grothendieck ring of varieties are [L02, Bit04,DL02].2.1.
Generalities.
Throughout the paper we work in the Grothendieck ring K ( V ar/k ) of varieties over k , which as an abelian group is generated by classes[ X ] for quasi-projective varieties X over k with relations[ X ] = [ U ] + [ Z ]for any closed Z ⊂ X with open complement U . K ( V ar/k ) becomes a ringwith the product defined on generators as[ X ] · [ Y ] = [ X × Y ] . Note that 1 = [ pt ]. We write L = [ A ] ∈ K ( V ar/k ) for the Lefschetz class. If the field k is not perfect we take the class of X × Y with the reduced scheme structure. SERGEY GALKIN, EVGENY SHINDER
For each n ≥ X Sym n ( X ) = X ( n ) := X n / Σ n descend to K ( V ar/k ) and satisfy(2.1)
Sym n ( α + β ) = X i + j = n Sym i α · Sym j β, α, β ∈ K ( V ar/k ) . In particular if for m ≥ m ∈ K ( V ar/k ) for the class of a disjointunion of m points, one can prove that Sym n ( m ) = (cid:18) n + m − n (cid:19) (= dim k Sym n ( k m )) , m ≥ . We also have(2.2)
Sym n ( L m α ) = L mn Sym n α, α ∈ K ( V ar/k )([G01, Lemma 4.4]).The symmetric powers are put together in the definition of Kapranov’s “mo-tivic” zeta function: Z Kap ( X, t ) = X n ≥ [ Sym n ( X )] t n ∈ K ( V ar/k )[[ t ]] . We will need the following two useful formulas in K ( V ar/k ): • Let X → S be a Zarisky locally-trivial fibration with fiber F . Then(2.3) [ X ] = [ F ] · [ S ]This is proved by induction on the dimension of S . • Let X to be a smooth variety and Z ⊂ X be a smooth closed subvarietyof codimension c . Then(2.4) [ Bl Z ( X )] − [ P ( N Z/X )] = [ X ] − [ Z ]This follows from definitions. Note that by (2.3) [ P ( N Z/X )] = [ P c − ][ Z ].If char ( k ) = 0, then there is an alternative description of the Grothendieckring K ( V ar/k ) due to Bittner: the generators are classes of smooth projectiveconnected varieties and the relations are of the form (2.4) [Bit04].2.2.
Realizations.
We will call a ring morphism µ from K ( V ar/k ) to anyring R a realization homomorphism with values in R . We list some well-knownexamples of realization homomorphisms together with realizations of the zeta-function. • Counting points: k = F q is the finite field of q elements, µ ( X ) = X ( F q ) ∈ Z . Then the realization Z Kap ( X, t )) = Y x ∈ X − t [ k ( x ): k ] = exp (cid:16)X m ≥ X ( F q m ) m t m (cid:17) , ANO VARIETY AND RATIONALITY FOR A CUBIC 7 is the Hasse-Weil zeta-function ( X denotes the set of closed points of X ). In particular we have(2.5) X (2) ( F q ) = X ( F q ) + X ( F q )2 . • Etale Euler characteristic: k any field, µ = χ , with χ ( X ) = P p ≥ ( − p dim H pet,c ( X ¯ k , Q l ) ∈ Z being the geometric etale Euler char-acteristic with compact supports. Here l = char ( k ) is any prime. Thestandard comparison theorems imply that if k ⊂ C , then χ ( X ) = χ C ( X ) := χ c ( X ( C )). We have χ ( Z Kap ( X, t )) = (cid:16) − t (cid:17) χ ( X ) , and in particular(2.6) χ ( X (2) ) = χ ( X )( χ ( X ) + 1)2 . • Real Euler characteristic: k ⊂ R , µ = χ R with χ R ( X ) = χ c ( X ( R )) ∈ Z . Then χ R ( Z Kap ( X, t )) = (cid:16) − t (cid:17) χ C ( X ) − χ R ( X )2 (cid:16) − t (cid:17) χ R ( X ) , and in particular(2.7) χ R ( X (2) ) = χ R ( X ) + χ C ( X )2 . • Hodge polynomials: k ⊂ C , µ ([ X ]) = E ( X C , u, v ) = X p,q ≥ e p,q ( X C ) u p v q ∈ Z [ u, v ]is the virtual Hodge-Deligne polynomial of X C . We have e p,q ( X C ) =( − p + q h p,q ( X C ) when X is smooth and projective [DK87, Section 1].Note that E ( X C , ,
1) = χ C ( X ). We have E ( Z Kap ( X, t )) = Y p,q (cid:16) − u p v q t (cid:17) e p,q ( X ) ([C96, C98], see also [GLM07]). Sometimes it is convenient to considerthe appropriate truncation of the Hodge-Deligne polynomial to make itinvariant under birational equvalence (and even under stable birationalequivalence, cf [LL03], Definition 3.4). Thus we considerΨ X ( t ) := E ( X C , − t, ∈ Z [ t ] . If X is smooth and projective, then(2.8) Ψ X ( t ) = X p ≥ h p, ( X C ) t p . SERGEY GALKIN, EVGENY SHINDER • Hodge realization: k ⊂ C . We can encode more information aboutthe Hodge structure than in the Hodge polynomials by considering thefull Hodge realization µ Hdg : K ( V ar/k ) → K ( HS ) , where K ( HS ) denotes the Grothendieck ring of polarizable pure ratio-nal Hodge structures. For a smooth projective X , µ Hdg ( X ) = [ H ∗ ( X C , Q )].This gives rise to a well-defined realization using the main result of[Bit04].Note that the Hodge polynomial E is the composition of the Hodgerealization and the natural homomorphism K ( HS ) → Z [ u, v ]which maps a pure Hodge structure H to P p,q ( − p + q h p,q ( H ).For a smooth projective variety, H ∗ ( Sym k ( X ) , Q ) is a pure Hodgestructure isomorphic to Sym k ( H ∗ ( X, Q )). This implies that the homo-morphism µ Hdg is compatible with taking symmetric powers.2.3.
The Grothendieck ring and rationality questions.
In this section k is a field of characteristic zero. Lemma 2.1.
Let X , X ′ be smooth birationally equivalent varieties. Then wehave an equality in the Grothendieck ring K ( V ar/k ) : [ X ′ ] − [ X ] = L · M where M is a linear combination of classes of smooth projective varieties ofdimension d − .Proof. The Weak Factorization Theorem [W03, AKMW02] says that X ′ canbe obtained from X using a finite number of blow ups and blow downs withsmooth centers; thus to prove the theorem we may assume X ′ = Bl Z ( X ) where Z is a smooth subvariety of X of codimension c ≥ X ′ ] − [ X ] = [ P ( N Z/X )] − [ Z ] = ([ P c − ] − · [ Z ] = L · [ P c − × Z ] . (cid:3) Corollary 2.2. If X is a rational smooth d -dimensional variety, then [ X ] = [ P d ] + L · M X where M X is a linear combination of classes of smooth projective varieties ofdimension d − . We are led to the following definition:
Definition 2.3.
Let
X/k be an irreducible d -dimensional variety. We call theclass M X := [ X ] − [ P d ] L ∈ K ( V ar/k )[ L − ] the rational defect of X . ANO VARIETY AND RATIONALITY FOR A CUBIC 9
Example 2.4.
Let
X/k be a smooth hypersurface of dimension d . Then bythe Weak Lefschetz theorem there is an isomorphism of Hodge structures H ∗ ( X, Q ) = H ∗ ( P d , Q ) ⊕ H d ( X, Q ) prim , and by construction the Hodge realization of the rational defect M X is theHodge structrue of weight ( d − obtained by the Tate twist of H d ( X, Q ) prim : µ Hdg ( M X ) = [ H d ( X, Q ) prim (1)] ∈ K ( HS ) . In our study of rationality of cubics the rational defect M X is an analogof the intermediate Jacobian considered by Clemens-Griffiths [CG72] and theClemens-Griffiths component of the derived category introduced by Kuznetsov(see [KuzECM, KuzCM, BBS12, Kuz13]).By Corollary 2.2, if X is smooth and rational, then the rational defect M X can be lifted to an element of K ( V ar/k ). A weak version of the conversestatement follows from Theorem 2.5 below.We recall that contrary to our intuition it is an open question whether[ X ] = [ Y ] implies that X and Y are birationally equivalent [Grom99], [LL03,Question 1.2], [LiS10], [LaS11], [Lit12].There is however the following powerful result due to Larsen and Lunts. Re-call that two smooth projective varieties X and Y are called stably birationallyequivalent if for some m, n ≥ X × P m is birationally equivalent to Y × P n . Theorem 2.5. [LL03]
Let k be a field of characteristic zero. The quotient-ring K ( V ar/k ) (cid:14) ( L ) is naturally isomorphic to the free abelian group generated byclasses of stable birational equivalence of smooth projective connected varietiestogether with its natural ring structure.In particular, if X and Y , . . . Y m are smooth projective connected varietiesand [ X ] ≡ m X j =1 n j [ Y j ] ( mod L ) , for some n j ∈ Z , then X is stably birationally equivalent to one of the Y j .Proof. See [LL03, Proposition 2.7, Corollary 2.6]. Larsen and Lunts have k = C . However, their proof only relies on the Weak Factorization Theorem whichholds true for any field of characteristic zero [W03, AKMW02]. See also [Bit04]. (cid:3) In general stable birational equivalence is weaker than birational equivalence[BCSS85]. However, for varieties of non-negative Kodaira dimension these twonotions coincide:
Lemma 2.6. If X and Y are stably birationally equivalent varieties of thesame dimension. If X is not uniruled then X and Y are birational.Proof. This fact follows from the existence of the MRC fibration [KMM92,Kol96, GHS03]. See [LiS10, Corollary 1] for a slightly different proof. (cid:3)
The Cancellation conjecture.
In Section 7 we rely on the followingConjecture:
Conjecture 2.7. L is not a zero divisor in K ( V ar/k ) . At the moment this is not known for any field k . Validity of this conjecturehas been discussed in [DL02, 3.3], [LiS10], [LaS11] and [Lit12, Conjecture 14].Note that it is known that K ( V ar/k ) is not an integral domain [P02],[Kol05]. 3.
The Hilbert scheme of two points
In this section
X/k is a reduced quasi-projective variety with all irreduciblecomponents of dimension d . One defines X [2] = Hilb ( X ) as the Hilbertscheme of subschemes of X of length two. X [2] admits an open subvariety X [2] , parametrizing reduced length two subschemes, i.e. pairs of distinct k -rational or Galois conjugate points on X defined over a quadratic extension of k . Thus we have an isomorphism X [2] , ≃ X (2) − X. Points of the closed complement of X [2] , parametrize points on X togetherwith a tangent direction.It is well-known that if X is smooth, then X [2] is also smooth and has apresentation X [2] ≃ Bl ∆ ( X × X ) Z / ⊂ X × X is the diagonal, and the action of Z / p ≥
0, we introduce a locally closed subvariety
Sing ( X ) p ⊂ X and a closed subvariety Sing ( X ) ≥ p ⊂ X : Sing ( X ) p = { x ∈ X : dim T x,X = d + p } ,Sing ( X ) ≥ p = { x ∈ X : dim T x,X ≥ d + p } . We endow these subvarieties with the reduced subscheme structure in the case X is non-reduced. We have Sing ( X ) = X − Sing ( X ) Sing ( X ) ≥ = Sing ( X ) . On each stratum
Sing ( X ) p the tangent sheaf T X restricts to a sheaf T p ofconstant fiber dimension; thus T p is locally free of rank d + p [Har77, ExerciseII.5.8]. Lemma 3.1.
The complement X [2] − X [2] , admits a stratification by locallyclosed subvarieties Z p , p ≥ d with Z p ≃ P Sing ( X ) p ( T p ) . ANO VARIETY AND RATIONALITY FOR A CUBIC 11
Proof. X [2] − X [2]0 parametrizes non-reduced subschemes of length 2 on X ; welet Z p to denote the locus of those subschemes whose support is contained in Sing ( X ) p . The natural morphism Z p → Sing ( X ) p is the projectivization of T p . (cid:3) Corollary 3.2.
1) We have the following formula in K ( V ar/k ) : (3.1) [ X [2] ] = [ X (2) ] + ([ P d − ] − X ] + X q ≥ L d + q − · [ Sing ( X ) ≥ q ] .
2) In particular, if X is a hypersurface in a smooth variety V of dimension d + 1 , then (3.2) [ X [2] ] = [ X (2) ] + ([ P d − ] − X ] + L d · [ Sing ( X )] . Proof.
1) We make a straightforward computation based on Lemma 3.1:[ X [2] ] = ([ X (2) ] − [ X ]) + X p ≥ [ Z p ] == ([ X (2) ] − [ X ]) + X p ≥ ([ P d − ] + p X q =1 L d + q − ) · [ Sing ( X ) p ] == [ X (2) ] + ([ P d − ] − X ] + X p ≥ p X q =1 L d + q − · [ Sing ( X ) p ] == [ X (2) ] + ([ P d − ] − X ] + X q ≥ X p ≥ q L d + q − · [ Sing ( X ) p ] == [ X (2) ] + ([ P d − ] − X ] + X q ≥ L d + q − · [ Sing ( X ) ≥ q ] .
2) As T x,X ⊂ T x,V and V is smooth, we have dim T x,X ≤ d + 1, thus Sing ( X ) p = ∅ for p > (cid:3) Finally we need the following Lemma:
Lemma 3.3.
Let X be a complex smooth projective variety. The number ofholomorphic one and two-forms of X [2] are given by h , ( X [2] ) = h , ( X ) h , ( X [2] ) = h , ( X ) + h , ( X )( h , ( X ) − . Proof.
We have H p, ( X [2] ) ≃ (cid:16) Sym ( H ∗ ( X )) (cid:17) p, . In particular, H , ( X [2] ) ≃ (cid:16) Sym ( H ∗ ( X )) (cid:17) , = H , ( X ) H , ( X [2] ) ≃ (cid:16) Sym ( H ∗ ( X )) (cid:17) , = H , ( X ) ⊕ Λ H , ( X ) . (cid:3) The Fano variety of lines on a cubic
Definition and basic properties.
In this section k is an arbitrary field, Y a cubic d -fold in P d +1 = P ( V ), dim k ( V ) = d + 2. Let the equation of Y be G ∈ Γ( P d +1 , O (3)) = Sym ( V ∗ ) . We allow Y to have arbitrary singularities.We consider the Grassmannian Gr (2 , V ) of lines on P d +1 and its universalrank two bundle U ⊂ O Gr (2 ,d +2) ⊗ V . The section G gives rise to a section e G ∈ Γ( Gr (2 , d + 2) , Sym ( U ∗ )). One defines the Fano scheme of lines on Y asthe zero locus of this section(4.1) Z ( e G ) ⊂ Gr (2 , V ) . The Fano scheme could have non-reduced components (see [AK77, Remark1.20 (i)]). In this paper we ignore the nonreduced structure of Z ( e G ) and let F ( Y ) := Z ( e G ) red ⊂ Gr (2 , V )to be the Fano variety. F ( Y ) is connected if d ≥
3. If we don’t assume Y to be smooth, F ( Y ) maybe singular or reducible. If Y is smooth, then F ( Y ) is smooth of dimension2 d −
4, and F ( Y ) is irreducible if d ≥
3. This is a particular case of thefollowing:
Proposition 4.1. If Y is non-singular along a line L ⊂ Y , then L representsa smooth point of F ( Y ) on an irreducible component of codimension d − .Proof. See [AK77, BV78]. (cid:3)
Finally we recall that in the smooth case the canonical class of F ( Y ) is givenby(4.2) ω F ( Y ) = O (4 − d )where O (1) is induced from the Pl¨ucker embedding F ( Y ) ⊂ Gr (2 , d + 2) ⊂ P ( d +22 ) − . Thus we see that K F ( Y ) ≥ d ≤ Decomposability.
In this section k is a field of characteristic zero and Y /k is a smooth cubic d -fold.For our study of rationality of cubics in Section 7 the following property ofthe Fano variety of lines will be relevant: Definition 4.2.
Let W be an irreducible k -dimensional variety. We call W decomposable (resp. stably decomposable) if W is birationally equivalent (resp.stably birationally equivalent) to either V × V ′ or V [2] for some k -dimensionalvarieties V , V ′ . ANO VARIETY AND RATIONALITY FOR A CUBIC 13 As char ( k ) = 0, we may resolve singularities and so we will assume that V and V ′ are smooth and projective.If W is not uniruled, then by Lemma 2.6 stable decomposability is the sameas decomposability. This applies in particular to the Fano variety F ( Y ) ofsmooth cubic threefolds and fourfolds, as by (4.2) F ( Y ) have non-negativecanonical class for d = 3 ,
4, and hence are not uniruled.Furthermore in dimensions d = 3 and d = 4 we can effectively solve thequestion of decomposability. Below we show that cubic threefolds have inde-composable Fano variety and for cubic fourfolds F ( Y ) can be only decomposedas S [2] where S is a K d ≥ F ( Y ) are potentially different as F ( Y ) has negative canonical class,so in particular is uniruled and even rationally connected. The Ψ-polynomialof F ( Y ) is zero for d ≥
5, so that Hodge numbers do not give any control ondecomposability.
Proposition 4.3.
For a smooth cubic threefold
Y /k the Fano variety F ( Y ) isnot (stably) decomposable.Proof. It is sufficient to prove the claim over the algebraic closure of k . By theLefschetz principle we may assume k = C .We need to show that the surface F ( Y ) is not birationally equivalent to C × C ′ or C (2) where C and C ′ are smooth projective curves.The Ψ-polynomial (2.8) of the Fano surface F ( Y ) isΨ F ( Y ) ( t ) = 1 + 5 t + 10 t (see Example 6.3).On the other hand the Ψ-polynomial of C × C ′ is equal to(1 + g ( C ) t )(1 + g ( C ′ ) t ) . As Ψ F ( Y ) ( t ) does not admit a non-trivial integer (or even real) factorization, F ( Y ) is not birational to C × C ′ .By Lemma 3.3, the Ψ-polynomial of C (2) is1 + g ( C ) t + g ( C )( g ( C ) − t , thus if F ( Y ) is birational to C (2) , then g ( C ) = 5.By Lemma 4.4, C (2) is a minimal surface. Minimal models for surfaces ofgeneral type are unique, so that if F ( Y ) and C (2) are birationally equivalent,then there exists a morphism F ( Y ) → C (2) which is a composition of contractions of ( − h , ( F ( Y )) > h , ( C (2) ) . This is a contradiction since h , ( F ( Y )) = 25 (Example 6.3), h , ( C (2) ) = g ( C ) + 1 = 26 (this can be shown as in the proof of Lemma 3.3). (cid:3) Lemma 4.4.
Let C be a complex smooth projective curve of genus g > . If Γ ⊂ C (2) is a smooth rational curve, then deg Γ = 1 − g. In particular C (2) does not contain ( − -curves unless g = 2 .Proof. In what follows we identify points of C (2) with effective degree twodivisors on C , in particular Γ is a 1-parameter family of such divisors.We first show that Γ parametrizes fibers of a degree 2 covering C → P sothat C is necessarily a hyperelliptic curve.Fix a point c ∈ C . The Abel-Jacobi map C (2) → J ac ( C ) c + c
7→ O ( c + c − c )contracts the rational curve Γ. Hence all degree two divisors parametrized byΓ are rationally equivalent. Let L denote the corresponding complete linearsystem.We have Γ ⊂ L and it is easy to see that in fact Γ = L : otherwise the surface C (2) would contain a linear projective subspace |L| of dimension >
1. Finally, L has no fixed components: if c + c and c ′ + c ′ are rationally equivalentdivisors and c = c ′ , then two points c and c ′ are rationally equivalent; since g ( C ) > c = c ′ .Thus we have shown that Γ corresponds to a complete linear system ofdegree 2 and dimension 1, which gives rise to a 2 : 1 covering C → P and theassociated hyperelliptic involution τ : C → C . In these terms:Γ = (cid:8) { x, τ ( x ) } , x ∈ C (cid:9) . Let π : C → C (2) denote the natural degree two covering. Consider thepreimage e Γ = π ∗ (Γ) of Γ in C . Using the projection formula and the fact that π ∗ is multiplicative we get2 · Γ = π ∗ π ∗ (Γ ) = π ∗ e Γ , and after taking degrees we obtain(4.3) deg Γ = 12 deg e Γ . As e Γ = { ( x, τ ( x )) : x ∈ C } is the image of the diagonal ∆ ⊂ C under theautomorphism id × τ , we have(4.4) deg e Γ = deg ∆ = deg c ( C ) = 2 − g. The claim follows from (4.3) and (4.4). (cid:3)
ANO VARIETY AND RATIONALITY FOR A CUBIC 15
Decomposability and the associated K surface for cubic four-folds.Proposition 4.5. If Y /k is a smooth cubic fourfold, and F ( Y ) is (stably)decomposable, then F ( Y ) is birationally equivalent to a Hilbert scheme of twopoints on a K surface.Proof. The proof is similar to that of Proposition 4.3. The Ψ-polynomial ofthe Fano variety is given by:Ψ F ( Y ) ( t ) = 1 + t + t (see Example 6.4).If F ( Y ) is birationally equivalent to a product of two surfaces S , S ′ , then1 + t + t = Ψ S ( t ) · Ψ S ′ ( t ) , deg Ψ S ( t ) , deg Ψ S ′ ( t ) ≤ t + t does not admit a non-trivial factorization into a prod-uct of two polynomials with positive integer coefficients, such decompositionis not possible.Assume now that F ( Y ) is birationally equivalent to S [2] for a smooth pro-jective surface S . Let q = h , ( S ), p g = h , ( S ). By Lemma 3.3 for Ψ S [2] tomatch Ψ F ( Y ) we must have q = 0, p g = 1.It can be proved directly or applying [AA02, Corollary 1], that S has Kodairadimension κ S = 0. We may replace S by its minimal model; so we assume S is a minimal surface. Thus by the Enriques-Kodaira classification of surfaces S can be a K
3, an abelian surface, an Enriques surface or a hyperellipticsurface. Among these four types of surfaces q = 0, p g = 1 only holds for a K (cid:3) The following two definitions are given by Hassett [Has00]:
Definition 4.6.
A smooth complex cubic fourfold Y is called special if the rankof the sublattice of algebraic cycles in H ( Y, Z ) is at least two. Note that the sublattice of algebraic classes in H ( Y, Z ) coincides with theHodge lattice H , ( Y, C ) ∩ H ( Y, Z ) since the integral Hodge conjecture isknown for cubic fourfolds [Zuc77, Mur77, V07]. Definition 4.7.
A polarized K surface ( S, H ) is associated to Y if for somealgebraic cycle T ∈ H ( Y, Z ) we have a Hodge isometry between primitiveHodge lattices (cid:10) h , T (cid:11) ⊥ ⊂ H ( Y, Z )(1) and h H i ⊥ ⊂ H ( S, Z ) with its Beauville-Bogomolov form [B83] . Proposition 4.8.
Let
Y / C be a smooth cubic fourfold. If the Fano variety F ( Y ) is decomposable, then Y is special and the K surface from Proposition4.5 is associated to Y in terms of Hodge structure. Proof.
By Proposition 4.5, F ( Y ) is birationally equivalent to S [2] for a K S . Now the result follows e.g. from [Ad14, Theorem 2]. (cid:3) Remark 4.9.
The condition of decomposability of F ( Y ) is strictly strongerthan the condition of Y having an associated K surface [Ad14] . Seventy-fouris the smallest discriminant for which a special cubic fourfold has an associated K surface but F ( Y ) is generically not birational to a Hilbert scheme [Ad14] . Corollary 4.10.
For a very general smooth cubic fourfold
Y / C the Fano va-riety F ( Y ) is not decomposable.Proof. Special cubic fourfolds form a countable union of divisors in the modulispace of all cubics [Has00]. (cid:3) The Y - F ( Y ) relation The relation in K ( V ar/k ) .Theorem 5.1. Let Y be a reduced cubic hypersurface of dimension d . Wehave the following relations in K ( V ar/k ) : (5.1) [ Y [2] ] = [ P d ][ Y ] + L [ F ( Y )](5.2) [ Y (2) ] = (1 + L d )[ Y ] + L [ F ( Y )] − L d [ Sing ( Y )] where Sing ( Y ) is the singular locus of Y .Proof. We first note that (5.2) follows from (5.1) using the formula (3.2) ofCorollary 3.2.Let us now prove (5.1). Consider the incidence variety W := { ( x ∈ L ) : L ⊂ P d +1 , x ∈ Y } . In other words W is the projectivization of the vector bundle T P d +1 (cid:12)(cid:12) Y on Y .Let(5.3) φ : Y [2] W := { ( x ∈ L ) : L ⊂ P d +1 , x ∈ Y } be a rational morphism which is defined as follows. A point τ ∈ Y [2] corre-sponds to a length 2 subscheme of Y : τ can be a pair of k -rational points, a k -rational point with a tangent direction, or a pair of Galois conjugate points.In any case there is a unique k -rational line L = L τ passing through τ . Forgeneral τ , the intersection ξ = L ∩ Y is a length 3 scheme and there is a third k -rational intersection point x ∈ L ∩ Y . We define φ ( τ ) := ( x ∈ L τ ).In fact φ is a birational isomorphism and φ − is defined by mapping ( x ∈ L )to the subscheme of length 2 obtained as the residual intersection of L with Y . ANO VARIETY AND RATIONALITY FOR A CUBIC 17
The morphism φ fits into the following diagram U (cid:127) _ (cid:15) (cid:15) ≃ / / U ′ (cid:127) _ (cid:15) (cid:15) Y [2] φ / / ❴❴❴❴❴❴❴❴❴ WZ ?(cid:31) O O q ●●●●●●●●● Z ′ ?(cid:31) O O q ′ | | ②②②②②②②②② F ( Y )Here Z ⊂ Y [2] is the closed subvariety consisting of those τ ∈ Y [2] that thecorresponding line L τ is contained in Y and Z ′ ⊂ W is the closed subvarietyconsisting of ( x ∈ L ) with L is contained in Y . U and U ′ are the opencomplements to Z and Z ′ respectively.Note that W is a P d -bundle over Y . Furthermore q ′ : Z ′ → F ( Y ) is a P -bundle over F ( Y ) and similarly, q : Z → F ( Y ) is a Sym ( P ) = P -bundleover F ( Y ). Thus the Fiber Bundle Formula (2.3) implies that[ W ] = [ P d ][ Y ][ Z ] = [ P ][ F ( Y )][ Z ′ ] = [ P ][ F ( Y )] . Putting everything together we obtain:[ Y [2] ] − [ P ][ F ( Y )] = [ P d ][ Y ] − [ P ][ F ( Y )]or equivalently [ Y [2] ] = [ P d ][ Y ] + L [ F ( Y )] . (cid:3) Examples and immediate applications.Corollary 5.2.
1) Let Y be a cubic hypersurface over an arbitrary field. Thenfor the etale Euler characteristic we have χ ( F ( Y )) = χ ( Y )( χ ( Y ) − χ ( Sing ( Y )) .
2) Let Y be a real cubic hypersurface. Then χ R ( F ( Y )) = ( χ R ( Y ) + χ C ( Y )) − χ R ( Sing ( Y )) , d odd (cid:0) χ R ( Y )( χ R ( Y ) −
4) + χ C ( Y ) (cid:1) + χ R ( Sing ( Y )) , d even
3) Let k = F q , the finite field and let N = Y ( F q ) , N = Y ( F q ) , N s = Sing ( Y )( F q ) . Then F ( Y )( F q ) = N − q d ) N + N q d + q d − N s . Proof.
The formulas follow by applying respective realization homomorphismsfrom Section 2 to the formula for Y (2) of Theorem 5.1 and formulas (2.5), (2.6),(2.7) using that L k ) = q k , χ C ( L k ) = 1, χ R ( L k ) = ( − k . (cid:3) Example 5.3.
Let k be an algebraically closed field. Let Y /k be a cubic surfacewith r isolated du Val singularities with the sum of Milnor numbers of thesingular points equal to n .Let e Y be the minimal desingularization of Y . We know that χ ( e Y ) = 9 (forexample because e Y is a smooth rational surface with K e Y = K Y = 3 ). On theother hand we have χ ( e Y ) = χ ( Y ) + n so that χ ( Y ) = 9 − n and Corollary 5.2 (1) implies that (5.4) F ( Y )( k ) = χ ( F ( Y )) = (9 − n )(6 − n )2 + r In particular, a smooth cubic surface has 27 lines and a cubic surface withan ordinary double point has 21 lines.The same formula (5.4) has been obtained in [BW79] , page 255, over thecomplex numbers using case by case analysis (see also [Dol12, Table 9.1] );another proof is given in [KM87, Satz 1.1] . Example 5.4.
The structure of real cubic surfaces and the number of lines onthem is a classical subject initiated by Schl¨afli [Sch63] in 1863. In particularSchl¨afli proves that there can be , , or lines on a smooth real cubicsurface. See [Seg42, KM87, Sil89, Kol97, PT08] for a discussion on real cubicsurfaces.Recall that a smooth real cubic surface Y is either isomorphic to a blow of RP in k pairs of complex conjugate points and − k real points, where k is , , or in which case Y is a rational surface or is an irrational surface with Y ( R ) homeomorphic to a disjoint union of RP and a two-sphere S .If Y is a smooth rational real cubic surface, then χ R ( Y ) = χ R ( RP ) + (6 − k ) χ R ( L ) = 1 − (6 − k ) = 2 k − and Corollary 5.2 (2) implies that there are F ( Y )( R ) = χ R ( F ( Y )) = (2 k − k −
9) + 92 = 2 k − k +27 = , k = 015 , k = 17 , k = 23 , k = 3 real lines on Y .In the case of irrational Y we have χ R ( Y ) = χ R ( RP ) + χ R ( S ) = 1 + 2 = 3 ANO VARIETY AND RATIONALITY FOR A CUBIC 19 and thus there are F ( Y )( R ) = χ R ( F ( Y )) = − real lines on Y .Similarly one can deduce a more general formula for the number of lines ona real cubic surface with du Val singularities which has been also computed in [KM87, Satz 2.8] . Example 5.5.
Let Y be a cone over a ( d − -dimensional cubic ¯ Y . Then wehave [ Y ] = 1 + L · [ ¯ Y ] Sym [ Y ] = 1 + L · [ ¯ Y ] + L · Sym [ ¯ Y ][ Sing ( Y )] = 1 + L · [ Sing ( ¯ Y )][ F ( Y )] = [ ¯ Y ] + L [ F ( ¯ Y )] (for the last equality note that the set of lines on Y that pass through thevertex of the cone are parametrized by ¯ Y , whereas the rest of the lines projectisomorphically onto a line on ¯ Y and the fiber of Y over each such line is a -plane contained in Y ).In this case one can see that the formula of Theorem 5.1 (2) for ¯ Y impliesthe same kind of formula for Y . The relation in K ( V ar/k )[ L − ].Theorem 5.6. Let Y be a cubic d -fold and let M Y be its rational defect (seeDefinition 2.3). There is the following relation in K ( V ar/k )[ L − ] : [ F ( Y )] = Sym ( M Y + [ P d − ]) − L d − (1 − [ Sing ( Y )]) . Proof.
We compute the symmetric square of[ Y ] = [ P d ] + L · M Y ∈ K ( V ar/k )[ L − ]using identities (2.1), (2.2):[ Y (2) ] = Sym [ P d ] + L · [ P d ] · M Y + L · Sym ( M Y ) . Substituting this into Theorem 5.1(2) gives:(5.5) L · [ F ( Y )] = [ Y (2) ] − (1 + L d )[ Y ] + L d · [ Sing ( Y )] == L · Sym ( M Y ) + L · [ P d − ] · M Y ++ (cid:16) Sym [ P d ] − (1 + L d )[ P d ] (cid:17) + L d · [ Sing ( Y )] . Finally it is easy to see that in fact
Sym [ P d ] − (1 + L d )[ P d ] = L · Sym ([ P d − ]) − L d and we get the claim dividing (5.5) by L . (cid:3) Corollary 5.7.
There is the following relation in K ( V ar/k )[ L − ] : [ F ( Y )] = Sym ( M Y ) + [ P d − ] · M Y + d − X k =0 a k L k + L d − · [ Sing ( Y )] where a k = (cid:2) k +22 (cid:3) , k < d − (cid:2) d − (cid:3) , k = d − (cid:2) d − − k (cid:3) , k > d − F ( Y )] in termsof the rational defect M Y in two examples of rational cubics. Example 5.8.
Let Y is a cubic hypersurface with a single ordinary doublepoint P over an arbitrary field. Projecting from the point P one establishes anisomorphism Bl P ( Y ) ≃ Bl V ( P d ) , where V ⊂ P d is a smooth complete intersection of a cubic with a quadric.We find the rational defect of Y . Let E be the exceptional divisor of Bl P ( Y ) .We have: [ Y ] − [ P ] + [ E ] = [ Bl P ( Y )] = [ Bl V ( P d )] = [ P d ] + L [ V ] so that [ Y ] = [ P d ] + L [ V ] − ([ E ] − . The exceptional divisor E is a ( d − -dimensional quadric, so that [ E ] =[ P d − ] for d even and [ E ] = [ P d − ] + L d − for d odd. This leads to the followingformula: [ Y ] = (cid:26) [ P d ] + L ([ V ] − [ P d − ]) , d even [ P d ] + L ([ V ] − [ P d − ] − L d − ) , d oddand for the rational defect of Y we get M Y = (cid:26) [ V ] − [ P d − ] , d even [ V ] − [ P d − ] − L d − , d oddThe two varieties F ( Y ) and V (2) are known to be birational: V parametrizeslines passing through P and for two such lines there is a residual line in theplane spanned by the two lines [CG72] . Now we can find the class of the Fanovariety in K ( V ar/k )[ L − ] using Theorem 5.6: (5.6) [ F ( Y )] = (cid:26) Sym ([ V ]) , d even Sym ([ V ] − L d − ) , d oddIf d = 2 , and k is algebraically closed, then V consists of six isolated points,and there are Sym (6) = 21 lines on a cubic surface with one node in accor-dance with Example 5.3. ANO VARIETY AND RATIONALITY FOR A CUBIC 21 If d = 3 , C := V is a canonically embedded genus curve and we have [ F ( Y )] = Sym ([ C ] −
1) ==
Sym ([ C ]) − Sym (1) − ([ C ] −
1) ==
Sym ([ C ]) − [ C ] . It is known [CG72] that in this case the birational morphism
Sym ( C ) → F ( Y ) glues two disjoint copies of C together.If d = 4 , S := V is a K -surface and [ F ( Y )] = [ Sym ( S )] which agrees with [Has00, Lemma 6.3.1] . Example 5.9.
Let d be even and assume that Y is a smooth cubic d -foldcontaining two disjoint d/ -planes P , P . In this case Y is rational as thereis a birational map P × P → Y mapping ( a, b ) ∈ P × P to the third pointof intersection of the line L a,b through a and b with Y .Resolving indeterminacy locus of this map and its inverse we find as iso-morphism Bl P ,P ( Y ) ≃ Bl V ( P × P ) . Here V is a ( d − -dimensional variety consisting of points ( a, b ) such that L a,b ⊂ V . In fact V is a complete intersection of divisors (1 , and (2 , in P × P . V can also be considered as a subvariety in F ( Y ) parametrizing lines inter-secting both P and P .For the rational defect of Y we have M Y = M P × P + [ V ] − P d/ ][ P d/ − ] unless d = 2 in which case the third term disappears, V is a set of points and M Y = 6 . For d = 4 , S := V is a K surface and M Y = [ S ] + [ P ] − P ] .In this example F ( Y ) is again birationally equivalent to V (2) : two lines L , L on Y intersecting both P and P and which are generic with this property,determine a smooth cubic surface T in their span h L , L i . T is equipped withtwo more lines E = P ∩ T , E = P ∩ T . There is a unique line L on T which does not intersect the quadrilateral formed by L , L , E , E .One can see that the assignment { L , L } 7→ L defines a birational morphism V (2) → F ( Y ) . For the inverse map, starting with a generic line L ⊂ Y notintersecting P , P one finds the unique -plane containing L and intersecting P and P in some lines E , E . The intersection of this -plane with Y isa smooth cubic surface T , and one finds a unique pair of skew lines L , L intersecting both E , E and not intersecting L .Theorem 5.6 gives an expression of [ F ( Y )] in terms of [ V ] , which will be ofthe form [ F ( Y )] = [ V [2] ] + L · ( . . . ) . The term in brackets is a certain combination of classes L i and L j · [ V ] . For d = 4 this last term vanishes and we simply get [ F ( Y )] = [ S [2] ] ∈ K ( V ar/k )[ L − ] . However, as Hassett remarks in [Has00] for d = 4 these two varieties are notisomorphic (see [Has98, Section 6.1] for details). Hodge structure of the Fano variety F ( Y )In this section we assume Y to be a smooth complex cubic d -fold. Wecompute the Hodge structure of the Fano variety of lines F ( Y ) in terms of theHodge structure of Y .By the Weak Lefschetz theorem there is the following decomposition ofHodge structure of Y : H ∗ ( Y, Q ) = d M k =0 Q ( − k ) ⊕ H d ( Y, Q ) prim , where H d ( Y, Q ) prim is the primitive cohomology with respect to the hyperplanesection. We have H d ( Y, Q ) = (cid:26) H d ( Y, Q ) prim , d oddH d ( Y, Q ) prim ⊕ Q ( − d ) , d even The Hodge structure of F ( Y ) is expressed in terms of the weight ( d − H Y := H d ( Y, Q ) prim (1) . Theorem 6.1.
Let Y be a smooth complex cubic hypersurface of dimension d . There is the following decomposition for the Hodge structure of the Fanovariety of lines on Y : H ∗ ( F ( Y ) , Q ) ≃ Sym ( H Y ) ⊕ d − M k =0 H Y ( − k ) ⊕ d − M k =0 Q ( − k ) a k and a k = (cid:2) k +22 (cid:3) , k < d − (cid:2) d − (cid:3) , k = d − (cid:2) d − − k (cid:3) , k > d − In particular, if d is even, then all odd-dimensional cohomology of F ( Y ) vanish. Note that for d ≥ F ( Y ) is H d − ( F ( Y ) , Q ) = (cid:26) H Y , d odd H Y ⊕ Q ( − d − ) [ d +24 ] , d even ANO VARIETY AND RATIONALITY FOR A CUBIC 23
Proof.
Consider the Hodge realization homomorphism µ Hdg : K ( V ar/ C ) → K ( HS/ Q ) .µ Hdg maps the Tate class L p = [ A p ] to the class of the Hodge-Tate structure[ Q ( − p )] of weight 2 p , which is invertible; this implies that µ Hdg descends to awell-defined ring homomorphism K ( V ar/ C )[ L − ] → K ( HS )which we will also denote by µ Hdg .Definition 2.3 of the rational defect M Y is compatible with the definition of H Y : [ H Y ] = µ Hdg ( M Y ) ∈ K ( HS ) , see Example 2.4.Applying the realization µ Hdg to the decomposition of Corollary 5.7 we get[ H ∗ ( F ( Y ) , Q )] = [ Sym ( H Y )] + (cid:2) d − M k =0 H Y ( − k ) (cid:3) + [ d − M k =0 Q ( − k ) a k ] . It is well-known that the category of polarizable Hodge structures is semisim-ple [PS08, Corollary 2.12], in particular if two polarizable pure Hodge struc-tures H and H have equal classes in the Grothendieck ring, then H and H are isomorphic. Thus we obtain H ∗ ( F ( Y ) , Q ) ≃ Sym ( H Y ) ⊕ d − M k =0 H Y ( − k ) ⊕ d − M k =0 Q ( − k ) a k . (cid:3) In principle Theorem 6.1 allows to compute all the Hodge numbers of theFano variety F ( Y ) of a smooth cubic d -fold using the following Lemma: Lemma 6.2.
The primitive Hodge numbers h d − q,q of a smooth complex cubic d -fold Y are contained in the range d − ≤ q ≤ d +13 and for those q are givenas follows: h d − q,qprim ( Y ) = (cid:18) d + 23 q − d + 1 (cid:19) . Proof.
The proof is a standard computation of Hodge numbers of a smoothhypersurface based on the work of Griffiths [Grif69]. (cid:3)
Example 6.3. If d = 3 , H ( Y, Q ) has weight and Hodge numbers (0 , , , ;thus H Y has weight one with h , = h , = 5 and Theorem 6.1 gives a decom-position of the Hodge structure of the Fano surface: H Q ( − H H Y ( − H
10 25 10
Sym ( H Y ) H H Y H Q This result has been known since the work of Clemens and Griffiths [CG72] . Example 6.4. If d = 4 , H ( Y, Q ) has weight four and Hodge numbers (0 , , , , , H Y has weight two with h , = h , = 1 and h , = 20 (primitive classes areof codimension one in H , ( Y ) ) and we get a decomposition for the Hodgestructure of the hyperk¨ahler fourfold F ( Y ) : H Q ( − H H Y ( − ⊕ Q ( − H Sym ( H Y ) ⊕ H Y ( − ⊕ Q ( − H H Y ⊕ Q ( − H Q This has been deduced by Beauville and Donagi [BD85, Proposition 2] fromthe fact that F ( Y ) is deformation equivalent to the Hilbert scheme of two pointson a K surface. Rational cubic hypersurfaces
In this section k is a field of characteristic zero. In addition we assume thatConjecture 2.7 is true for k . Theorem 7.1.
Let Y be a smooth cubic hypersurface of dimension d ≥ over a field satisfying Conjecture 2.7. If Y is rational, then F ( Y ) is stablydecomposable in the sense of Definition 4.2.Proof. By Corollary 2.2 we have[ Y ] = [ P d ] + L · M Y , where M Y ∈ K ( V ar/k ) is a combination of classes of smooth projectivevarieties of dimension equal to d − M Y = m X i =1 [ V i ] − n X j =1 [ W j ] . Since we assume that L is not a zero-divisor, the formula in Theorem 5.6 isvalid in K ( V ar/k ): [ F ( Y )] = Sym ( M Y + P d − ) − L d − . ANO VARIETY AND RATIONALITY FOR A CUBIC 25
We set V m +1 := P d − and compute using (2.1): Sym ( M Y + P d − ) = Sym (cid:16) m +1 X i =1 [ V i ] − n X j =1 [ W j ] (cid:17) == Sym (cid:16) m +1 X i =1 [ V i ] (cid:17) − Sym (cid:16) n X j =1 [ W j ] (cid:17) −− (cid:16) m +1 X i =1 [ V i ] − n X j =1 [ W j ] (cid:17)(cid:16) n X j =1 [ W j ] (cid:17) == m +1 X i =1 Sym [ V i ] + X ≤ i
It can be seen from the proof of Theorem 5.6, that for Theorem7.1 to hold in dimension d , instead of relying on the general Cancellation Con-jecture 2.7 it is sufficient to assume that L does not annihilate combinationsof classes of varieties of dimension ≤ d − .It would even suffice if we knew for combinations α ∈ K ( V ar/k ) of classesof varieties of dimension ≤ d − : L · α = 0 = ⇒ α ∈ L · K ( V ar/k ) . Remark 7.3. If k is algebraically closed, then for the rational defect (7.1) wehave n = m as h , ( Y ) = h , ( P d ) + h , ( M Y ) = 1 + m − n. This makes the number of the product terms in the right-hand-side of (7.2)balanced, and leaves an extra V i [2] term. It is then very likely that in thedecomposition (7.2) the class of the Fano variety will in fact match one of the V i [2] , and not one of the products V i × V i ′ , W j × W j ′ . Using the results on indecomposability of the Fano variety obtained in Sec-tions 4.2, 4.3 we can make Theorem 7.1 very useful in dimensions d = 3 , Theorem 7.4.
Let k be a field satisfying Conjecture 2.7. Any smooth cubicthreefold Y /k is irrational.Proof.
Follows from Theorem 7.1 and Proposition 4.3. (cid:3)
Theorem 7.5.
1) Let
Y /k be a smooth cubic fourfold over a field k satisfyingConjecture 2.7. If Y is rational, then the Fano variety F ( Y ) is birational to S [2] for a K surface S/k .2) If Conjecture 2.7 is true for k = C , then a very general smooth complexcubic fourfold is irrational.Proof. Follows from Theorem 7.1, Proposition 4.5 and Colorollary 4.10. (cid:3)
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Sergey Galkin
National Research University Higher School of Economics (HSE)Faculty of Mathematics and Laboratory of Algebraic Geometry7 Vavilova Str.117312, Moscow, Russiae-mail:
Evgeny Shinder