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 technical result

SOME RATIONAL CUBIC FOURFOLDS

Prop osition Consider the Ho dge structures in the global p erio d domain

nD for which there exists some T L H such that hT Qi is o dd These

Ho dge structures form a countable union of divisors indexed by the discriminant

of the saturation of K ZT This discriminant may b e any p ositive integer

n mo d

Here the discriminant of a lattice means the determinant of its intersection form

Later we shall intro duce a more rened notion

We are interested in the restriction of these divisors to the Zariski op en subset

mar

C of the p erio d domain These are also irreducible divisors and only nitely

mar

many of them are contained in the complement to C By Theorem these

divisors parametrize rational cubic fourfolds Thus we obtain our main theorem

Theorem Main Theorem There is a countably innite collection of divisors

in C which parametrize rational cubic fourfolds Each of these is a co dimension

two subvariety in the mo duli space of cubic fourfolds C

Remark There are rational cubic fourfolds X containing a plane such that

the quadric surface bundle X P do es not have a rational section For example

certain comp onents of C C are of this typ e

It seems likely that the only discriminants corresp onding to divisors in the b ound

mar

ary nD nC are n and

The remainder of this section is the pro of of Prop osition We rst establish

the following lemma

Lemma Let x b e a Ho dge structure in D Then the following conditions are

equivalent

There exists a cohomology class T L H x with hT Qi o dd

H x such that the There exists a saturated rank one sublattice S K

has o dd discriminant orthogonal complement to S in K

Furthermore the discriminant n of the saturation of K ZT is congruent to

mo dulo This is equal to the discriminant of the orthogonal complement to S in

K

Proof The key ingredient of this lemma is the following fact let K b e a saturated

nondegenerate sublattice of a unimo dular lattice and let K b e its orthogonal

complement Then K and K have the same discriminant up to sign This is

proved in Ni x

Assume the rst condition holds We may nd an element T in the saturation

of K ZT so that K ZT is saturated and hQ T i The intersection form

on K ZT takes the form

h Q T

h a

Q

a b T

with discriminant n a a b which is congruent to mo dulo Let

S b e the intersection of K ZT with K By construction S is saturated and

contained in H x The orthogonal complement to S in K coincides with the

orthogonal complement to K ZT in the full cohomology lattice In particular

this lattice has discriminant n which is o dd

BRENDAN HASSETT

Now assume the second condition holds Again the saturation of K S has the

same discriminant as the orthogonal complement to S in K If the discriminant

of the saturation of K S is o dd then it contains a class T with hT Qi o dd

Otherwise the intersection form could b e written

h Q T

h a

Q

T a b

for some T in the saturation of K S But the discriminant would then b e even

a contradiction

We now analyze the geometry of these Ho dge structures in the global p erio d

domain Supp ose we are given a p ositive denite rank one saturated sublattice

such that the orthogonal complement has o dd discriminant These are S K

precisely the sublattices arising from the lemma since the Ho dgeRiemann bilinear

relations imply that the intersection form on L H x is p ositive denite We

determine the x D for which S H x Because x corresp onds to H it is

necessary and sucient that x b e orthogonal to S with resp ect to the intersection

with resp ect to its form The lo cus of such x forms a hyp erplane section of D

imb edding in PK C It is not dicult to see that this is a nonempty irreducible

divisor in the p erio d domain Like D it is a b ounded symmetric domain of typ e

IV see the app endix of Sa x for more details Its image in nD parametrizes

those p erio ds with algebraic classes of typ e S This subvariety is an algebraic

divisor in the global p erio d domain b ecause b oth the global p erio d domain and

the normalization of the subvariety are arithmetic quotients of b ounded symmetric

domains BB Bo

As we cho ose various S satisfying the conditions stipulated ab ove we obtain

divisors in the global p erio d domain Now S and S determine the same divisor if

and only if S S for some we then say that S and S are equivalent

mo dulo Hence we obtain a bijective corresp ondence b etween the following two

typ es of data

irreducible divisors in parametrizing Ho dge structures x satisfying the nD

conditions of Lemma

equivalence classes of rank one p ositive denite saturated sublattices S

K such that the orthogonal complement has o dd discriminant

We shall prove that these equivalence classes are classied by the discriminants of

their orthogonal complements Our argument relies heavily on ideas of Nikulin Ni

If K is a lattice then the bilinear form induces an inclusion K K HomK Z

The discriminant group of K is dened as the quotient

dK K K

The order of dK is equal to the absolute value of the discriminant of K so dK

is trivial i K is unimo dular For example

ZZ h QZh QZ dK

The bilinear form on K extends to a Q valued bilinear form on K which induces

a Q Zvalued bilinear form on dK Furthermore if K is even then the quadratic

SOME RATIONAL CUBIC FOURFOLDS

form induces a Q Zvalued quadratic form on dK denoted q We use to

K K

denote the automorphisms of K acting trivially on dK

Under certain conditions imb eddings of lattices with compatible signatures are

classied by homomorphisms of the corresp onding discriminant groups The fol

lowing sp ecial case of results from Ni x illustrates this principle

Assume K is an even lattice of signature and S is a rank one

p ositive denite lattice A saturated imb edding of S into K corresp onds

to the following data

A subgroup H dS

S

A subgroup H dK

K

An isomorphism h q jH q jH

S S K K

An even lattice S of signature and an isomorphism

dS of discriminant quadratic forms Here q

S

q j ? H where H is the graph of h in dS dK

K

H

Two such imb eddings denoted S and S are conjugate under the

action of if the following conditions are satised

K

H H

S S

2 1

h H h H

S S

2 1

and dK dK such S There exist isomorphisms S

that and dS Here dS are the

maps induced by and resp ectively

In the same spirit we can often construct isomorphisms b etween lattices by con

structing isomorphisms b etween their discriminant groups The following statement

is a sp ecial case of results from Ni x

Let M b e an even indenite lattice Let b e the minimal numb er of

generators of the discriminant group dM and assume that the rank

of M is greater than Let M b e another even lattice with the

dM dM same signature and assume there is an isomorphism

0

Then resp ecting the discriminant quadratic forms q and q is

M M

induced by an isomorphism M M

We use these results to prove the following lemma which completes the pro of of

Prop ostion

Lemma Let S and S b e rank one p ositive denite saturated sublattices of

K and assume that their orthogonal complements have the same o dd discriminant

n Then S and S are equivalent mo dulo Moreover such lattices exist for

each p ositive integer n mo d

Proof Set K K which is even by the results of Ha x or Ha For an

imb edding i S K of the typ e we are considering H dK is a cyclic

group of order eight and dS is cyclic of order n Let denote the group ?

K

8

which contains as an index two subgroup We rst claim that for any saturated

imb eddings i i S K there exists a such that i S i S This

is a consequence of the rst result of Nikulin quoted ab ove The existence of the

desired isomorphism follows from the second result

This result also implies that

A

K U E E

BRENDAN HASSETT

where U is the unique even unimo dular quadratic form of signature and

E is the p ositive denite quadratic form asso ciated to the corresp onding Dynkin

diagram For each p ositive integer a there exists a saturated Zv E with hv v i

a Se xVI I Consider elements of

A

E

of the form v resp ectively v let S denote the sublattice

generated by such an element The discriminant of S is a resp ectively

a and by the rst result of Nikulin the orthogonal complement to S

in K has discriminant a resp ectively a Thus for each p ositive

integer n mo d there is a rank one saturated sublattice of K such that the

orthogonal complement has discriminant n Moreover this sublattice is equivalent

mo dulo to one of the sublattices S

Now consider acting by multiplication by on U and trivially on the

other comp onents We nd that jS and so we conclude that S is

also equivalent to S In particular the arguments of the rst paragraph are

still valid if we replace by This completes the pro of of the lemma and the

prop osition

Geometry of the rational maps

Our rst goal is to prove the following result ab out the geometry of our rational

parametrizations

Prop osition Let X b e a cubic fourfold containing a plane P such that the

quadric surface bundle q X P has a rational section and let X K P

denote the birational map obtained by pro jecting from this section Then blows

down a family of lines parametrized by a surface birational to a degree two K

surface

Proof Recall that the discriminant curve C P is dened as the lo cus over which

q fails to b e smo oth In our case C is a sextic plane curve such that the double

branched over C is birational to a K surface V x cover of P

Let T X b e a closed integral subscheme meeting the generic b er of q in a

single reduced p oint The induced map T P is an isomorphism except over

a co dimension two subset of the base Let R denote the lines in the b ers of q

incident to T or the irreducible comp onent of this lo cus dominating the base

The induced map from R to P is generically nite of degree two and is ramied

over the discriminant curve In particular R is birational to a degree two K

surface

If Q is a smo oth quadric surface and t Q then pro jecting from t blows down the

two lines incident to t In particular the birational map from X to P constructed

from T blows down a family of lines birational to R

To conclude this section we suggest constructions of explicit linear series for

some of the rational maps constructed ab ove Consider a K surface S such that

SOME RATIONAL CUBIC FOURFOLDS

PicS is generated by two ample divisors h and h with intersection matrix

h h

h k

h k

where k Set n k k k which is p ositive and satises

n mo d The sections of O h give branched double covers s S P

S i i

The pair s s induces a map s S P P and the image S has k k

double p oints Fu x We assume that the double p oints of S are analytically

equivalent to the transverse intersection of two smo oth surfaces

Let H and H b e the divisors on P P such that h s H Let Y b e the

i i

blow up of P P along the surface S with exceptional divisor E Our assumption

on the singularities of S implies that Y is smo oth We set

L H k H E

and consider the linear series H Y O L We compute that L and a

Y

dimension count suggests that dim jLj We assume this linear series denes a

morphism

Y P

so that the image X Y is a smo oth cubic fourfold

Under these assumptions we can compute numerical invariants for some of the

surfaces contained in X Set F H k H E and F H k E

so that

L F L F

A dimension count suggests that F and F are eective and we assume they are

exceptional divisors for obtained by blowing up surfaces P and T in X Under

these assumptions we nd that h P T span a rank three sublattice of H X Z

with discriminant n k k and also that P has degree one and so is a

plane Since n mo d the intersection T h P is necessarily o dd

To lend some credibility to this numerology we should p oint out the geometry of

these examples is understo o d in the cases k The case k corresp onds to

the cubic fourfolds containing two disjoint planes The assumptions ab ove are easily

veried in this case The case k corresp onds to the cubic fourfolds containing

a plane P and Veronese surface T such that hP T i This example has b een

worked out by Tregub Tr

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BRENDAN HASSETT

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5

V C Voisin Theoreme de Torel li pour les cubiques de P Invent math

Current address Department of Mathematics University of Chicago University Avenue

Chicago IL

Email address hassettmathuchicago edu