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Rational2ams.Pdf SOME RATIONAL CUBIC FOURFOLDS BRENDAN HASSETT Introduction The purp ose of this pap er is to give new examples of rational cubic fourfolds Let C denote the mo duli space of cubic fourfolds a twentydimensional quasipro jective variety The cubic fourfolds containing a plane form a divisor C C We prove the following theorem Main Theorem Theorem There is a countably innite col lection of divisors in C which parametrize rational cubic fourfolds Each of these is a codimension two subvariety in the moduli space of cubic fourfolds C Exp erimental evidence strongly suggests that the general cubic fourfold containing a plane is not rational but no smo oth cubic fourfold has yet b een proven to b e irrational The key to our construction is the following observation a cubic fourfold contain ing a plane is birational to a smo oth quadric surface over k P Indeed pro jecting from the plane gives a rational map to P whose b ers are quadric surfaces The cubic fourfold is rational if the quadric surface over k P is rational Using Ho dge theory we prove that the rational quadric surfaces corresp ond to a countably in nite union of divisors in C This is quite natural from an arithmetic p oint of view Over Q the rational quadric surfaces form a countably innite union of divisors in the Hilb ert scheme of quadric surfaces in P We conclude this intro duction by listing the cubic fourfolds known to b e rational There is an irreducible divisor C C parametrizing rational cubic fourfolds This divisor has many geometric characterizations For example it is the closure of the cubic fourfolds containing rational normal scrolls of degree four Fa Tr Ha and of the lo cus of Pfaan cubic fourfolds BD All the examples of rational cubic fourfolds known to the author are contained in C or one of the subvarieties of C Furthermore the birational map from P involves blowing up a surface birational to a K surface see x for details We work over the complex numb ers C unless mentioned otherwise Here generic means in the complement of a Zariski closed prop er subset and general means in the complement of a countable union of Zariski closed prop er subsets A lattice is a nitely generated free Zmo dule equipp ed with a nondegenerate integral quadratic form Part of this work was done while the author was visiting the Institut MittagLeer This pap er was revised while the author was supp orted by a National Science Foundation Postdo ctoral Fellowship BRENDAN HASSETT C 14 C 8 C Figure Known Rational Cubic Fourfolds Geometry of quadric surface bundles For our purp oses a quadric surface bund le is a at pro jective morphism q Q B of regular connected schemes such that the generic b er is a smo oth quadric surface The relative Fano scheme F B of a quadric surface bundle parametrizes the lines contained in the b ers of q For a smo oth quadric surface over a eld this consists of two disjoint smo oth genus zero curves corresp onding to the rulings of the surface Prop osition Let q Q Sp eck b e a smo oth quadric surface over a eld k Then the following are equivalent Q is rational over k The Fano scheme F of Q has a divisor dened over k with degree one on each comp onent Q has a zerocycle of o dd degree dened over k Proof Let Z denote the universal line over F so that we have a corresp ondence p Z q Q F and an induced Ab elJacobi map q p Ch Q PicF where Ch Q denotes the Chow group of zerocycles on Q The quadric Q is rational if and only if it has a p oint over k This p oint is mapp ed by to a pair of p oints dened over k one on each comp onent of F Conversely given such a pair of p oints the intersection of the corresp onding lines gives a k p oint of Q This proves the equivalence of the rst two conditions Clearly either of the rst two conditions implies the third we prove the converse Let z b e a cycle of o dd degree n on Q and dened over k The cycle z has degree n on each comp onent of F The canonical class K is dened over k F and has degree on each comp onent of F Consequently nK z has degree F one on each comp onent of F Given a nonzero section s H F nK z the F lo cus s consists of a pair of p oints on F one on each comp onent The prop osition has the following consequence SOME RATIONAL CUBIC FOURFOLDS Corollary Let q Q B b e a quadric surface bundle and assume B is rational over the base eld Let Q denote the class of the generic b er of q and assume there is a cycle T Ch Q dened over the base eld such that hT Qi is o dd Then Q is rational over the base eld We apply the prop osition to k k B the function eld of B Note that h i denotes the intersection pro duct on Q In our analysis of cubic fourfolds we shall use a transcendental version of this result Prop osition Let q Q B b e a quadric surface bundle over a rational pro jective variety Assume there is a class T H Q Z H Q such that hT Qi is o dd Then X is rational over C Proof By the previous prop osition it suces to construct a divisor on the relative Fano scheme F B intersecting the comp onents of the generic b er in h Q T i p oints We may discard any comp onents of F that fail to dominate B Cho ose a resolution of singularities F F and set Z F Z We again obtain a F corresp ondence of smo oth varieties p Z q Q F and an induced map on cohomology q p H Q Z H Q H F Z H F By the Lefschetz Theorem on classes T is a divisor on F The image of this divisor in F has the desired prop erties Cubic fourfolds containing a plane First we x some notation The Hilb ert scheme of cubic hyp ersurfaces in P is a pro jective space P The smo oth hyp ersurfaces form an op en subset U P and are called cubic fourfolds Two cubic fourfolds are isomorphic if and only if they are equivalent under the action of SL Consequently the isomorphism classes of cubic fourfolds corresp ond to elements of the orbit space C U SL Applying Geometric Invariant Theory GIT x one may prove that C has the structure of a twentydimensional quasipro jective variety C is called the moduli space of cubic fourfolds Now consider a cubic fourfold X containing a plane P A dimension count shows the isomorphism classes of such cubic fourfolds form a divisor C C We shall restrict our attention to these sp ecial cubic fourfolds for more details see V Ha or Ha Let h denote the hyp erplane class of X and let Q denote the class of a quadric surface residual to P in a threedimensional linear space so that h P Q Let X denote the blowup of X along P Pro jecting from the plane P we obtain a morphism q X P The b ers of this morphism corresp ond to quadric surfaces in the class Q In particular a cubic fourfold containing a plane is birational to a quadric surface BRENDAN HASSETT bundle over P Applying the results of the previous section we obtain the following theorem Theorem Let X b e a cubic fourfold containing a plane P and let Q b e the class of a quadric surface residual to P Assume there is a class T H X Z H X such that hQ T i is o dd Then X is rational over C Analysis of the periods Our next goal is to determine when the hyp otheses of the theorem are satised We retain the terminology of the previous section The metho ds we use are ex plained in more detail in Ha x and Ha V also contains a detailed discussion of the p erio ds of cubic fourfolds containing a plane Let K H X Z denote the sublattice spanned by h and Q The intersection form on H X Z restricts to h Q h Q denote the orthogonal complement to K in H X Z Let K We now recall some results ab out the p erio ds of cubic fourfolds containing a plane More general statements are proved in Ha x and Ha Let L b e a lattice isomorphic to the middle cohomology of a cubic fourfold L has signature Fix distinguished elements h and P in L corresp onding to the hyp erplane class squared and a plane contained in some cubic fourfold Let X b e a cubic fourfold containing a plane P and let H X Z L b e a complete marking of its cohomology preserving the classes h and P This induces a map H X C L C Now the Ho dge structure on the middle cohomology of X is entirely determined C which is isotropic with by the onedimensional subspace H X K resp ect to the intersection form Consequently each completely marked cubic four fold containing a plane yields a p oint on the quadric hyp ersurface of PK C where the intersection form is zero The local period domain for cubic fourfolds containing a plane is a top ologically op en subset of this hyp ersurface consisting of one of the connected comp onents of the op en set where the Hermitian form hu vi is p ositive This manifold has the structure of a nineteendimensional b ounded symmetric domain of typ e IV Sa x of the app endix it is denoted D Let denote the automorphisms of L which preserve the intersection form act trivially on K and resp ect the orientation on the negative denite part of K The group acts from the left on D PK C The quotient nD is called the global period domain for cubic fourfolds containing a plane This is the quotient of a b ounded symmetric domain by an arithmetic group and so is a normal quasi mar pro jective varietyBB Let C b e the variety parametrizing the pairs X P where X is a cubic fourfold and P is plane contained in X The p erio d map mar nD C is an algebraic op en immersion of quasipro jective varieties This follows from the Torelli theorem for cubic fourfolds V and the Borel extension theorem Bo We now state our main
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