Special Cubic Fourfolds

Brendan Hassett ^{∗} University of Chicago

September 18, 1998

Abstract

A cubic fourfold is a smooth cubic hypersurface of dimension four; it is special if it contains a surface not homologous to a complete intersection. Special cubic fourfolds form a countably infinite union of irreducible families C_{d}, each a divisor in the moduli space C of cubic fourfolds. For an infinite number of these families, the Hodge structure on the nonspecial cohomology of the cubic fourfold is essentially the Hodge structure on the primitive cohomology of a K3 surface. We say that this K3 surface is associated to the special cubic fourfold. In these cases, C_{d} is related to the moduli space N_{d} of degree d K3 surfaces. In particular, C contains infinitely many moduli spaces of polarized K3 surfaces as closed subvarieties. We can often construct a correspondence of rational curves on the special cubic fourfold parametrized by the K3 surface which induces the isomorphism of Hodge structures. For infinitely many values of d, the Fano variety of lines on the generic cubic fourfold of C_{d} is isomorphic to the Hilbert scheme of length-two subschemes of an associated K3 surface.

Keywords: K3 surfaces, Noether-Lefschetz loci, period domains AMS Subject Classifications: 14C30, 14J28, 14J35

∗This work was supported by Harvard University, a National Science Foundation Graduate Fellowship, and a Sloan Foundation Dissertation Fellowship. This paper was revised while the author was visiting the Institut Mittag-Leffler.

1 Introduction

Let X be a special cubic fourfold, h its hyperplane class, and T the class of an algebraic surface not homologous to any multiple of h^{2}. The discriminant d is defined as the discriminant of the saturated lattice spanned by h^{2} and T. Let C_{d} denote the special cubic fourfolds of discriminant d (§ 3.2).

Theorem 1.0.1 (Classification of Special Cubic Fourfolds) ( Theorems 3.1.2, 3.2.3, and 4.3.1) C_{d} ⊂ C is an irreducible divisor and is nonempty iff d > 6 and d ≡ 0, 2(mod 6).

In section four, we give concrete descriptions of special cubic fourfolds with small discriminants and explain how certain Hodge structures at the boundary of the period domain arise from singular cubic fourfolds.

The nonspecial cohomology of a special cubic fourfold consists of the middle cohomology orthogonal to the distinguished classes h^{2} and T. In many cases, it is essentially the primitive cohomology of a K3 surface of degree d, which is said to be associated to the special cubic fourfold. Furthermore, the varieties C_{d} are often closely related to moduli spaces of polarized K3 surfaces. Let C_{d}^{mar} denote the marked special cubic fourfolds of discriminant d (§ 5.2). This is the normalization of C_{d} if d ≡ 2 (mod 6) and is a double cover of the normalization otherwise.

Theorem 1.0.2 (Associated K3 Surfaces and Maps of Moduli Spaces) (Theorems 5.1.3 and 5.2.4) Special cubic fourfolds of discriminant d have associated K3 surfaces iff d is not divisible by four, nine, or any odd prime p ≡ −1(mod 3). In these cases, there is an open immersion of C_{d}^{mar} into the moduli space of polarized K3 surfaces of degree d.

In particular, an infinite number of moduli spaces of polarized K3 surfaces may be realized as moduli spaces of special cubic fourfolds.

We can explain geometrically the existence of certain associated K3 surfaces. The Fano variety of X parametrizes the lines contained in it. For certain special cubic fourfolds these Fano varieties are closely related to K3 surfaces:

Theorem 1.0.3 (Geometry of Fano Varieties) (Theorem 6.1.4) Assume that d = 2(n^{2}+n+1) where n is an integer ≥ 2, and let X be a generic special cubic fourfold of discriminant d. Then the Fano variety of X is isomorphic to the Hilbert scheme of length-two subschemes of a K3 surface associated to X.

We should point out that the hypothesis on d is stronger than necessary, but simplifies the proof considerably. Combining this with the results on maps of moduli spaces, we obtain examples of distinct K3 surfaces with isomorphic Hilbert schemes of length-two subschemes (Proposition 6.2.2.)

One motivation for this work is the rationality problem for cubic fourfolds. The Hodge structures on cubic fourfolds and their relevance to rationality questions have previously been studied by Zarhin [30]. Izadi [15] also studied Hodge structures on cubic hypersurfaces with a view toward rationality questions. All the examples of cubic fourfolds known to be rational ([10] [27] [5] [28]) are special and have associated K3 surfaces. Indeed, a birational model of the K3 surface is blown up in the birational map from ℙ^{4} to the cubic fourfold. Is this the case for all rational cubic fourfolds? In a subsequent paper [14], we shall apply the methods of this paper to give new examples of rational cubic fourfolds. We show there is a countably infinite union of divisors in C_{8} parametrizing rational cubic fourfolds (C_{8} corresponds to the cubic fourfolds containing a plane).

Throughout this paper we work over ℂ. We use the term ‘generic’ to mean ‘in the complement of some Zariski closed proper subset.’ The term ‘lattice’ will denote a free abelian group equipped with a nondegenerate symmetric bilinear form. I would like to acknowledge the help I received in the course of this project. I benefitted from conversations with David Eisenbud and Elham Izadi, and from suggestions by Walter Baily and Robert Friedman. Barry Mazur provided important insights and comments, and Johan de Jong made many helpful observations and pointed out some errors in an early version of this paper. Finally, Joe Harris introduced me to this beautiful subject; his inspiration and advice have been invaluable.

2 Hodge Theory of Cubic Fourfolds

2.1 Cohomology and the Abel-Jacobi Map

Let X be a smooth cubic fourfold. The Hodge diamond of X has the form:

1

0  0
0  1  0
0  0  0  0
0  1  21  1  0

Let L denote the cohomology group H^{4}(X, ℤ), L^{0} the primitive cohomology H^{4}(X, ℤ)^{0}, and ⟨, ⟩ the symmetric nondegenerate intersection form on L. If h is the hyperplane class then h^{2} ∈ L and L^{0} = (h^{2})^{⊥}.

Our best tool for understanding the middle cohomology of X is the Abel-Jacobi mapping. Let F be the Fano variety of lines of X, the subvariety of the Grassmannian 𝔾(1, 5) parametrizing lines contained in X. This is a smooth fourfold ([1] §1). Let Z ⊂ F × X be the ‘universal line’, with projections p and q. The Abel-Jacobi map is defined as the map

α = p_{∗}q^{∗} : H^{4}(X, ℤ) → H^{2}(F, ℤ).

Let M = H^{2}(F, ℤ), M^{0} the primitive cohomology, and g the class of the hyperplane on F (induced from the Grassmannian). Recall that α(h^{2}) corresponds to the lines meeting a codimension-two subspace of ℙ^{5}, so α(h^{2}) = g. Following [3] and [5], we define the Beauville canonical form (,) on M so that g and M^{0} are orthogonal, (g, g) = 6, and (x, y) = \frac{1}{6}g^{2}xy for x, y ∈ M^{0}. Extending by linearity we obtain an integral form on all of M.

Proposition 2.1.1 (Beauville-Donagi [5] Prop. 6) The Abel-Jacobi map induces an isomorphism between L^{0} and M^{0}; moreover, for x, y ∈ L^{0} we have (α(x), α(y)) = − ⟨x, y⟩.

Indeed, we may interpret α is an isomorphism of Hodge structures H^{4}(X, ℂ)^{0} → H^{2}(F, ℂ)^{0}(−1). The −1 means that the weight is shifted by two; this reverses the sign of the intersection form.

Proposition 2.1.2 The middle integral cohomology lattice of a cubic fourfold is

L ≅ (+1)^{⊕21} ⊕ (−1)^{⊕2}

i.e. the intersection form is diagonalizable over ℤ with entries ±1 along the diagonal. The primitive cohomology is

L^{0} ≅ B ⊕ H ⊕ H ⊕ E_{8} ⊕ E_{8}

where B = (\begin{matrix}2 & 1 \\ 1 & 2 \end{matrix}), H = (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) is the hyperbolic plane, and E_{8} is the posi-

tive definite quadratic form associated to the corresponding Dynkin diagram.

We first prove the statement on the full cohomology. L is unimodular by Poincaré duality and has signature (21, 2) by the Riemann bilinear relations. L is odd because ⟨h^{2}, h^{2}⟩ = h^{4} = 3. Using the theory of indefinite quadratic forms (e.g. [25] ch.5 §2.2) we conclude the result.

Now we turn to the primitive cohomology L^{0}. By Proposition 2.1.1 it suffices to compute M^{0}; we first compute M. In [5] Prop. 6, it is shown that F is a deformation of a variety S^{[2]}, where S is a degree-fourteen K3 surface and S^{[2]} denotes the Hilbert scheme of length-two zero-dimensional subschemes of S (also called the blown-up symmetric square of S). By [3] §6 we have the canonical orthogonal decomposition

H^{2}(S^{[2]}, ℤ) = H^{2}(S, ℤ) ⊕_{⊥} ℤδ

where (δ, δ) = −2 and the restriction of (,) to H^{2}(S, ℤ) is the intersection form. Geometrically, the divisor 2δ corresponds to the nonreduced lengthtwo subschemes of S. The cohomology lattice of a K3 surface is well-known (cf. [16] Prop. 1.2)

H^{2}(S, ℤ) ≅ Λ := H^{⊕3} ⊕ (−E_{8})^{⊕2},

so M ≅ H^{⊕3} ⊕ (−E_{8})^{⊕2} ⊕ (−2). Furthermore, the polarization g = 2f − 5δ, where f ∈ H^{2}(S, ℤ) satisfies (f, f) = 14 [5]. The automorphisms of H^{2}(S, ℤ) act transitively on the primitive vectors of a given nonzero length ([16] Theorem 2.4). If v_{1} and w_{1} are elements of the first summand H with (v_{1}, v_{1}) = (w_{1}, w_{1}) = 0 and (v_{1}, w_{1}) = 1, then we may take f = v_{1} + 7w_{1} and g = 2v_{1} + 14w_{1} − 5δ. Using v_{1} + 3w_{1} − 2δ and δ − 5w_{1} as the first two elements of a basis of M^{0}, we obtain the result. □ Remark: Note that our computation shows that L^{0} is even.

2.2 Hodge Theory and the Torelli Map

We review Hodge theory in the context of cubic fourfolds; a general introduction to Hodge theory is [12]. Recall that a complete marking of a polarized cubic fourfold is an isomorphism ϕ : H^{4}(X, ℤ) → L mapping the square of the hyperplane class to h^{2} ∈ L. If we are given a complete marking, the complex structure on X determines a distinguished subspace F^{3}(X) = H^{3,1}(X, ℂ) ⊂ L_{ℂ}^{0} satisfying the following properties:

1. F^{3}(X) is isotropic with respect to the intersection form <, >;

2. the Hermitian form H(u, v) = − ⟨u, v̄⟩ on F^{3}(X) is positive.

Let Q ⊂ ℙ(L_{ℂ}^{0}) be the quadric hypersurface defined by (1), and let U ⊂ Q be the topologically open subset where (2) also holds. U is a homogeneous space for the real Lie group SO(L_{ℝ}^{0}) = SO(20, 2). This group has two components; one of them reverses the orientation on the negative definite part of L_{ℝ}^{0}, which coincides with (F^{3} ⊕ \overline{F}^{3}) ∩ L_{ℝ}^{0}. Changing the orientation corresponds to exchanging F^{3} and \overline{F}^{3} (see §6 of the appendix to [24] for details). Hence the two connected components of U parametrize the subspaces F^{3} and \overline{F}^{3} = H^{1,3}(X) respectively; we denote them D′ and \overline{D′}. The component D′ is a twenty-dimensional open complex manifold, called the local period domain for cubic fourfolds.

Let Γ denote the group of automorphisms of L preserving the intersection form and the distinguished class h^{2}, and Γ^{+} ⊂ Γ the subgroup stabilizing D′. This is the index-two subgroup of Γ which preserves the orientation on the negative definite part of L_{ℝ}^{0}. Γ^{+} acts holomorphically on D′ from the left; for a point in D′ corresponding to the marked cubic fourfold (X, ϕ) the action is γ(X, ϕ) = (X, γ ∘ ϕ). The orbit space D = Γ^{+}\D′ exists as an analytic space and is called the global period domain.

Two cubic fourfolds are isomorphic iff they are projectively equivalent. Let C denote the coarse moduli space for smooth cubic fourfolds, constructed as a Geometric Invariant Theory quotient [18] ch.4 §2. Each cubic fourfold determines a point in D, and the corresponding map τ : C ⟶ D is called the period map. By general results of Hodge theory, this is a holomorphic map of twenty-dimensional analytic spaces. For cubic fourfolds we can say much more. First, we have the following result due to Voisin:

Theorem 2.2.1 (Torelli Theorem for Cubic Fourfolds[29]) τ : C → D is an open immersion of analytic spaces.

In particular, if X_{1} and X_{2} are cubic fourfolds and there exists an isomorphism of Hodge structures ψ : H^{4}(X_{1}, ℂ) → H^{4}(X_{2}, ℂ), then X_{1} and X_{2} are isomorphic. Second, τ is not just an analytic map:

Proposition 2.2.2 D is a quasi-projective variety of dimension twenty and τ : C → D is an algebraic map.

In §6 of the appendix to [24], it is shown that the manifold D′ is a bounded symmetric domain of type IV. The group Γ^{+} is arithmetically defined and acts holomorphically on D′. In this situation we may introduce the Borel-Baily compactification ([2] §10): there exists a compactification of D′, compatible with the action of Γ^{+}, so that the quotient is projective. Moreover, Γ^{+}\D′ is a Zariski open subvariety of this quotient. To complete the proof, we use the following consequence of A. Borel’s extension theorem [6]:

Let D′ be a bounded symmetric domain, and G an arithmetically defined torsion-free group of automorphisms. Let D = G\D′ be the quasi-projective quotient space, and Z an algebraic variety. Then any holomorphic map Z → D is algebraically defined.

While Γ^{+} has torsion, some normal subgroup H of finite index is torsionfree ([24] IV Lemma 7.2). Let Γ^{+}(N) denote the subgroup of Γ^{+} acting trivially on L/NL. For some large N, Γ^{+}/H acts faithfully on L/NL so Γ^{+}(N) ⊂ H and is torsion-free. Let C(N) denote the moduli space of cubic fourfolds with marked ℤ/Nℤ cohomology. This is a finite (and perhaps disconnected) cover of C; we use C^{0}(N) to denote a connected component. Let D(N) = Γ^{+}(N)\D′, which is also finite over D. The period map lifts to a map τ_{N} : C^{0}(N) → D(N). By Borel’s theorem τ_{N} is algebraic, and a descent argument implies τ is also algebraic. □ Remark: It follows that C is a Zariski open subset of D and its complement is defined by algebraic equations.

3 Special Cubic Fourfolds

3.1 Basic Definitions

Definition 3.1.1 A cubic fourfold X is special if it contains an algebraic surface T not homologous to a complete intersection.

Let A(X) = H^{2,2}(X) ∩ H^{4}(X, ℤ), which is positive definite by the Riemann bilinear relations. The Hodge conjecture is true for cubic fourfolds [31], so A(X) is generated (over ℚ) by the classes of algebraic cycles; X is special if and only if the rank of A(X) is at least two. This is equivalent to saying that the rank of L ∩ F^{3}(X)^{⊥} is at least two, or that L^{0} ∩ F^{3}(X)^{⊥} ≠ 0. A Hodge structure x ∈ D′ is special if L^{0} ∩ F^{3}(x)^{⊥} is nonzero.

Theorem 3.1.2 (Structure of Special Cubic Fourfolds) Let K ⊂ L be a positive definite rank-two saturated sublattice containing h^{2}, [K] the Γ^{+} orbit of K, and C_{[K]} the cubic fourfolds X such that A(X) ⊃ K′ for some K′ ∈ [K]. Every special cubic fourfold is contained in some C_{[K]}, which is an irreducible algebraic divisor of C, and is nonempty for all but a finite number of [K].

Given such a lattice K, we set K^{0} = K ∩ L^{0}. Let D′_{K} be the x ∈ D′ such that K^{0} ⊂ x^{⊥}; this is a hyperplane section of D′ ⊂ ℙ(L_{ℂ}^{0}). Each special Hodge structure is contained in some D′_{K}. Let K^{⊥} denote the orthogonal complement to K in L. We see that D′_{K} is a topologically open subset of a quadric hypersurface in ℙ(K_{ℂ}^{⊥}), has dimension nineteen, and classifies Hodge structures structures on the lattice K^{⊥}. As in the previous section, we can prove that D′_{K} is a bounded symmetric domain of type IV. Let Γ_{K}^{+} = {γ ∈ Γ^{+} : γ(K) ⊂ K}. As before, the quotient Γ_{K}^{+}\D′_{K} is quasi-projective. Furthermore, the induced holomorphic map Γ_{K}^{+}\D′_{K} → Γ^{+}\D′ = D is algebraically defined, so its image is an irreducible algebraic divisor.

We enumerate the divisors parametrizing special Hodge structures in D. Each one corresponds to Γ_{K}^{+}\D′_{K} for some K ⊂ L as above, but K is not uniquely determined. K_{1} and K_{2} give rise to the same divisor if and only if K_{1} = γ(K_{2}) for some γ ∈ Γ^{+}, i.e. Γ_{K_{1}}^{+} and Γ_{K_{2}}^{+} are conjugate in Γ^{+}. Let D_{[K]} denote the corresponding irreducible divisor in D. Since C is Zariski open in D (Proposition 2.2.2), C_{[K]} = C ∩ D_{[K]} is an irreducible algebraic divisor in C, and D_{[K]} ⊂ (D − C) for finitely many [K]. □

Definition 3.1.3 Let (K, ⟨, ⟩) be a positive definite rank-two lattice containing a distinguished element h^{2} with ⟨h^{2}, h^{2}⟩ = 3. A marked (resp. labelled) special cubic fourfold is a cubic fourfold X with the data of a primitive imbedding of lattices K ↪ A(X) preserving h^{2} (resp. the image of such an imbedding.) A special cubic fourfold is typical if it has a unique labelling.

We write D_{[K]}^{lab} for Γ_{K}^{+}\D′_{K}. The morphism D_{[K]}^{lab} → D_{[K]} is birational (indeed D_{[K]}^{lab} is the normalization of D_{[K]}), so a general point in D_{[K]} has a unique labelling. The fiber product D_{[K]}^{lab} ×_{D} C will be denoted C_{[K]}^{lab}.

3.2 Discriminants and Special Cubic Fourfolds

Definition 3.2.1 The discriminant of a labelled special cubic fourfold (X, K) is the determinant of the intersection matrix of K.

Proposition 3.2.2 Let (X, K) be a labelled special cubic fourfold of discriminant d and let v be a generator of K^{0}.

1. d > 0 and d ≡ 0, −1 (mod 3) ;

2. d′ := ⟨v, v⟩ = \begin{cases}3d & if d ≡ −1 (mod 3) \\ \frac{d}{3} & if d ≡ 0 (mod 3) \end{cases};

3. ⟨v, L^{0}⟩ = \begin{cases}3ℤ & if d ≡ −1 (mod 3) \\ ℤ & if d ≡ 0 (mod 3) \end{cases};

4. d is even.

The first three statements are straightforward computations, so we omit their proofs. The fourth follows from the remark after Proposition 2.1.2. □

We refine the results of the previous section by classifying the orbits of the rank-two sublattices under the action of Γ^{+}. The following theorem is a consequence of Theorem 3.1.2 and Proposition 3.2.4:

Theorem 3.2.3 (Irreducibility Theorem) The special cubic fourfolds possessing a labelling of discriminant d form an irreducible (possibly empty) algebraic divisor C_{d} ⊂ C.

Elements of C_{d} are called special cubic fourfolds of discriminant d; the corresponding rank-two lattice is denoted K_{d}. We write D_{d} for D_{[K_{d}]}, D_{d}^{lab} for D_{[K_{d}]}^{lab}, C_{d} for C_{[K_{d}]}, and C_{d}^{lab} for C_{[K_{d}]}^{lab}.

Proposition 3.2.4 Let K and K′ be saturated rank-two nondegenerate sublattices of L containing h^{2}. Then K = γ(K′) for some γ ∈ Γ^{+} if and only if K and K′ have the same discriminant.

We claim it suffices to prove the result for Γ. We find some g ∈ Γ − Γ^{+} stabilizing sublattices with every possible discriminant. Take g to be the identity except on the second hyperbolic plane in the orthogonal decomposition for L^{0}; on this component set g equal to multiplication by −1. (We refer to the computation of L^{0} in Proposition 2.1.2.)

Now we analyze the action of Γ on our rank-two sublattices, or equivalently, on saturated nondegenerate rank-one sublattices K^{0} ⊂ L^{0}. We apply the results of Nikulin on discriminant groups and quadratic forms; see [22] or [9] for basic definitions and proofs. The elements of Γ fix h^{2}, so they act trivially on the discriminant groups d(ℤh^{2}) and d(L^{0}) [22] §1.5. Conversely, any automorphism of L^{0} that acts trivially on d(L^{0}) extends to an element of Γ [22] 1.5.1.

Let K^{0} denote a lattice generated by an element v with ⟨v, v⟩ = d′, q_{K^{0}} the quadratic form on d(K^{0}), and q the quadratic form on d(L^{0}). The lattice L^{0} is the unique even lattice of signature (20, 2) with discriminant quadratic form q [22] 1.14.3. Any saturated codimension-one sublattice K^{⊥} ⊂ L^{0} is the orthogonal complement in L of a rank-two sublattice K, so there is an induced isomorphism d(K^{⊥}) ≅ d(K) [22] 1.6.1, and d(K^{⊥}) is generated by at most two elements. This implies the isomorphism class of K^{⊥} is determined by its signature and discriminant form, and any isomorphism of d(K^{⊥}) preserving the discriminant quadratic form is induced by an automorphism of K^{⊥} [22] 1.14.3.

Two primitive imbeddings of i : K^{0} → L^{0} differing only by an element of Γ are said to be congruent. Applying the results of [22] §1.15 in our situation, we find the primitive imbeddings i : K^{0} → L^{0} correspond to the following data:

1. a subgroup H_{q} ⊂ d(L^{0});

2. a subgroup H_{K^{0}} ⊂ d(K^{0});

3. an isomorphism ϕ : H_{K^{0}} → H_{q} preserving the restrictions of the quadratic forms to these subgroups, with graph Γ_{ϕ} = {(h, ϕ(h)) : h ∈ H_{K^{0}}} ⊂ d(K^{0}) ⊕ d(L^{0});

4. an even lattice K^{⊥} with complementary signature and discriminant form q_{K^{⊥}}, and an isomorphism ϕ_{K^{⊥}} : q_{K^{⊥}} → −δ, where δ = ((q_{K^{0}} ⊕ −q)|Γ_{ϕ}^{⊥})/Γ_{ϕ} (and Γ_{ϕ}^{⊥} is the orthogonal complement to Γ_{ϕ} with respect to q_{K^{0}} ⊕ q).

Another imbedding i′ with data (H′_{q}, H′_{K^{0}}, ϕ′, (K′)^{⊥}, ϕ_{(K′)^{⊥}}) is congruent to i if and only if H_{K^{0}} = H′_{K^{0}} and ϕ = ϕ′.

Our proof now divides into two cases. In the first case H_{q} = {0}, or equivalently, ⟨i(K^{0}), L^{0})⟩ = ℤ (i.e. 3|d). By the characterization above, all primitive imbeddings of K^{0} are congruent. In the second case H_{q} = d(L^{0}) ≅ ℤ/3ℤ, or equivalently, ⟨i(K^{0}), L^{0})⟩ = 3ℤ. In this case, d(K^{0}) has a subgroup H_{K^{0}} of order three and 3|d′. There are two possible isomorphisms between d(L^{0}) and H_{K^{0}}, thus two congruence classes of imbeddings of K^{0} into L^{0}. □

Using [22] §1.15 and Proposition 3.2.2, we can compute the discriminant quadratic forms of the lattices K_{d}^{⊥}:

Proposition 3.2.5 If d ≡ 0 (mod 6) then d(K_{d}^{⊥}) ≅ ℤ/\frac{d}{3}ℤ ⊕ ℤ/3ℤ, which is cyclic unless 9|d. We may choose this isomorphism so that q_{K_{d}^{⊥}}(0, 1) = \frac{2}{3} (mod 2ℤ) and q_{K_{d}^{⊥}}(1, 0) = −\frac{3}{d} (mod 2ℤ). If d ≡ 2 (mod 6) then d(K_{d}^{⊥}) ≅ ℤ/dℤ. We may choose a generator u so that q_{K_{d}^{⊥}}(u) = \frac{2d−1}{3d} (mod 2ℤ).

4 Examples

4.1 Special Cubic Fourfolds with Small Discriminants

The examples here are discussed in more detail in [13]. If T is a smooth surface contained in a cubic fourfold X then ⟨T, T ⟩ = c_{2}(N_{T/X}) = 6h_{T}^{2} + 3h_{T} K_{T} + K_{T}^{2} − χ_{T} where χ_{T} is the topological Euler characteristic and h_{T} is the hyperplane class restricted to T.

4.1.1 d=8: Cubic Fourfolds Containing a Plane (see [29])

X contains a plane P, so that ⟨P, P ⟩ = 3 and our marking is

K_{8} = \begin{aligned}& h^{2} & P \\ h^{2} & 3 & 1 \\ P & 1 & 3 \end{aligned} .

The cubic fourfolds in C_{8} generally contain other surfaces, like quadric surfaces and quartic del Pezzo surfaces.

4.1.2 d=12: Cubic Fourfolds Containing a Cubic Scroll

X contains a rational normal cubic scroll T, so that ⟨T, T ⟩ = 7 and our marking is

K_{12} = \begin{aligned}& h^{2} & T \\ h^{2} & 3 & 3 \\ T & 3 & 7 \end{aligned} .

4.1.3 d=14: Cubic Fourfolds Containing a Quartic Scroll/Pfaffian Cubic Fourfolds

X is a cubic fourfold containing a rational normal quartic scroll T, so that ⟨T, T ⟩ = 10 and our marking is

K_{14} = \begin{aligned}& h^{2} & T \\ h^{2} & 3 & 4 \\ T & 4 & 10 \end{aligned} .

Special cubic fourfolds of discriminant 14 generally also contain quintic del Pezzo surfaces and quintic rational scrolls. One can show that the quartic scrolls, quintic scrolls, and quintic del Pezzos on X form families of dimensions two, two, and five respectively. Note that Morin [17] uses a spurious parameter count to deduce that the quartic scrolls form a one dimensional family. From this, he concludes incorrectly that every cubic fourfold contains a quartic scroll.

Another description of an open subset of C_{14} is the Pfaffian construction of Beauville and Donagi [5]. The dimension counts above follow easily from their results. They also show that the Pfaffian cubic fourfolds are rational. Finally, we should point out that the cubic fourfolds containing two disjoint planes possess a marking with discriminant 14, and thus are also contained in C_{14}. (See [10] and [27] for more discussion of these examples.)

4.1.4 d=20: Cubic Fourfolds Containing a Veronese

X contains a Veronese surface V, so that ⟨V, V ⟩ = 12 and our marking is

K_{20} = \begin{aligned}& h^{2} & V \\ h^{2} & 3 & 4 \\ V & 4 & 12 \end{aligned} .

4.2 d=6: Cubic Fourfolds with Double Points

A double point is ordinary if its projectivized tangent cone is smooth. Cubic hypersurfaces in ℙ^{5} with an ordinary double point are stable in the sense of Geometric Invariant Theory. This is proved using Mumford’s numerical criterion for stability ([18] §2.1) and the methods of ([18] §4.2). Let C̃ denote the quasi-projective variety parametrizing cubic fourfolds with (at worst) a single ordinary double point.

Let X_{0} be a cubic fourfold with a single ordinary double point p. Projection from p gives a birational map π_{p} : X_{0} −−→ ℙ^{4} which can be factored

\overline{X_{0}} = Bl_{S}(ℙ^{4}) ⟶^{q_{1}} X_{0}
q_{2}↓
ℙ^{4}

where q_{1} is the blow-up of the double point p and q_{2} is the blow-down of the lines contained in X_{0} passing through p. These lines are parametrized by a surface S ⊂ ℙ^{4}, which is the complete intersection of a quadric and a cubic. The quadric is nonsingular because p is ordinary; the complete intersection is smooth because p is the only singularity of X_{0}. In particular, S is a sextic K3 surface. The inverse map π_{p}^{−1} is given by the linear system of cubic polynomials through this K3 surface. Conversely, given any sextic K3 surface contained in a smooth quadric, the image of ℙ^{4} under this linear system is a cubic fourfold with an ordinary double point. Note that the sextic K3 surfaces contained in a singular quadric hypersurface are precisely those containing a cubic plane curve.

This construction suggests that we associate a sextic K3 surface to any element of C̃ − C:

Proposition 4.2.1 The Torelli map extends to an open immersion τ̃ : C̃ → D. The closed set C̃_{6} := C̃ − C is mapped into D_{6}.

In § 5.2 we shall see that D_{6} coincides with the period domain for sextic K3 surfaces. A detailed proof of the proposition is given in §4 of [29], so we merely explain some details needed for our calculations. (It also follows from the delicate analysis of singular cubic fourfolds in § 6.3.) Let X_{0} be a cubic fourfold with an ordinary double point and let S be the associated K3 surface. Smoothings of ordinary double points of even codimension have monodromy satisfying T^{2} = I, so any smoothing of X_{0} yields a pure limiting mixed Hodge structure H_{lim}^{4}. The corresponding point of the period domain is denoted τ̃(X_{0}). The limiting Hodge structure may be computed with the Clemens-Schmid exact sequence [7], which implies there is a natural imbedding of the primitive cohomology H^{2}(S, ℂ)^{0}(−1) into H_{lim}^{4}. The orthogonal complement to the image consists of a rank-two lattice of integral (2, 2) classes

K_{6} = \begin{aligned}& h^{2} & T \\ h^{2} & 3 & 0 \\ T & 0 & 2 \end{aligned}

so τ̃(X_{0}) ∈ D_{6}.

4.3 Existence of Special Cubic Fourfolds

D_{d} ⊂ D is nonempty if and only if d is positive and congruent to 0, 2 (mod 6) (Proposition 3.2.2), so we restrict to these values of d.

Theorem 4.3.1 (Existence of Special Cubic Fourfolds) Let d > 6 be an integer with d ≡ 0, 2 (mod 6) . Then the divisor C_{d} is nonempty.

We saw in the last section why there are no smooth cubic fourfolds of discriminant six: D_{6} corresponds to the limiting Hodge structures arising from cubic fourfolds with double points. In the next section we shall explain why there are no cubic fourfolds of discriminant two: D_{2} corresponds to the limiting Hodge structures arising from another class of singular cubic fourfolds. Is the complement D − C equal to D_{2} ∪ D_{6}?

To prove the theorem, we need the following lemmas:

Lemma 4.3.2 Let P be an indefinite even rank-two lattice representing six. Assume that P is not isomorphic to any of the following:

(\begin{matrix}6 & 1 \\ 1 & 0 \end{matrix}) (\begin{matrix}6 & 2 \\ 2 & 0 \end{matrix}) (\begin{matrix}6 & 0 \\ 0 & −2 \end{matrix}) (\begin{matrix}6 & 3 \\ 3 & 0 \end{matrix}) .

Then there exists a smooth sextic K3 surface S lying on a smooth quadric with Pic(S) ≅ P.

Lemma 4.3.3 Let P be a rank-two indefinite even lattice, f ∈ P a primitive element with d := f^{2} > 0, and assume there is no E ∈ P with E^{2} = −2 and fE = 0. Then there exists a K3 surface S with Pic(S) = P and f a polarization on S. Moreover, f is very ample unless there exists an elliptic curve C on S with C^{2} = 0 and fC = 1 or 2.

Recall that Λ denotes the lattice isomorphic to the middle cohomology of a K3 surface. Using the results of §2 of [16], there exists an imbedding P ↪ Λ. So for some elements of the period domain P equals the lattice of (1, 1)-classes. The surjectivity of the period map for K3 surfaces implies the existence of a K3 surface S with Picard group P so that f contained in the Kähler cone of S (see pp. 127 of [4]). This implies f is a polarization of S.

To complete the proof, we apply Saint Donat’s results for linear systems on K3 surfaces [23]. Specifically, we use Theorems 3.1, 5.2, and 6.1, along with the analysis of fixed components in §2.7. □ To prove Lemma 4.3.2, we note that the image under |f| is not contained

in a singular quadric because P ≇ (\begin{matrix}6 & 3 \\ 3 & 0 \end{matrix}) (i.e. S does not contain a plane

cubic). □

Now we prove the theorem. Let S be one of the K3 surfaces constructed in Lemma 4.3.2 and X_{0} the corresponding singular cubic fourfold. Let v ∈ P be primitive with respect to the sextic polarization. Recall that H^{2}(S, ℂ)^{0}(−1) is naturally imbedded into the limiting Hodge structure H_{lim}^{4} arising from X_{0}. The image of v is an integral class of type (2, 2) in H_{lim}^{4}, denoted v′. Relabel H_{lim}^{4} by letting K_{d} denote the saturation of the lattice ℤh^{2} + ℤv′. By Proposition 3.2.2, d = −\frac{1}{2} disc(P). For each d ≡ 0, 2 (mod 6) , d > 6, there exist lattices P satisfying the hypotheses of Lemma 4.3.2 with discriminant −2d. If d = 6n (resp. d = 6n + 2) we may take

P = (\begin{matrix}6 & 0 \\ 0 & −2n \end{matrix}) (resp. (\begin{matrix}6 & 2 \\ 2 & −2n \end{matrix})).

Set x_{0} = τ̃(X_{0}) so that x_{0} ∈ D_{6} ∩ D_{d}. We construct a smoothing ϕ : X → Δ′ where X_{t} is smooth for t ≠ 0, and τ(X_{t}) ∈ D_{d}. Let γ : Δ → D be a holomorphic map such that γ(0) = x_{0} and γ(u) ∈ D_{d} − D_{6} for u ≠ 0. The existence of such a curve follows from the construction of D as the quotient Γ^{+}\D′. Because τ̃ is an open immersion, we may shrink Δ so that γ lifts through τ̃, giving a map μ : Δ → C̃. Consequently, there exists a ramified base change b : Δ′ → Δ and a family X → Δ′ so that X_{t} = μ(b(t)). By construction we have X_{t} ∈ C ∩ τ^{−1}(D_{d}) = C_{d} for t ≠ 0, so C_{d} ≠ ∅. □

4.4 d=2: The Determinantal Cubic Fourfold

The determinantal cubic fourfold X_{0} is defined by the homogeneous equation:

R := |\begin{matrix}a & b & c \\ b & d & e \\ c & e & f \end{matrix}| = 0.

It is singular where the 2 × 2 minors of the determinant are simultaneously zero, i.e. along a Veronese surface V. We shall consider deformations X → Δ of X_{0} with equations R + tG, where G is the equation of a smooth cubic fourfold, and the curve C ⊂ V defined by the equation G|_{V} = 0 is also smooth. Let S be the double cover of V branched over C, a degree-two K3 surface.

Theorem 4.4.1 The limiting mixed Hodge structure arising from X → Δ is pure and special of discriminant two. The orthogonal complement to K_{2} is isomorphic to the primitive Hodge structure H^{2}(S, ℂ)^{0}(−1).

This result will not be used elsewhere in this paper. Its proof is essentially a calculation on the semistable reduction for X using the Clemens-Schmid exact sequence [7] (see [13] for details). Geometrically, X_{0} is contained in the indeterminacy locus of the Torelli map, but after blowing up the map is well-defined at the generic point of the exceptional divisor. Moreover, this exceptional divisor maps birationally to D_{2} ⊂ D.

5 Associated K3 Surfaces

5.1 Nonspecial Cohomology

Definition 5.1.1 The nonspecial cohomology lattice of a labelled special cubic fourfold (X, K_{d}) is defined as the orthogonal complement K_{d}^{⊥}. The nonspecial cohomology, denoted W_{X,K_{d}}, is the polarized Hodge structure induced on K_{d}^{⊥} by the Hodge structure on H^{4}(X, ℂ)^{0}.

Proposition 5.1.2 Let (X, K_{14}) be a generic Pfaffian cubic fourfold. Then there exists a degree-fourteen K3 surface S and an isomorphism of Hodge structures W_{X,K_{14}} = H^{2}(S, ℂ)^{0}(−1).

This is a consequence of [5] Prop. 6 (cf. § 2.1) and Proposition 6.1.1. However, it is best explained by observing that the birational map ℙ^{4} −−→ X blows up a surface birational to S, which therefore parametrizes a correspondence of rational curves on X.

Motivated by this example, we determine the special cubic fourfolds whose nonspecial cohomology is isomorphic to the primitive cohomology of a polarized K3 surface:

Theorem 5.1.3 (Existence of Associated K3 Surfaces) Let (X, K_{d}) be a labelled special cubic fourfold of discriminant d, with nonspecial cohomology W_{X,K_{d}}. There exists a polarized K3 surface (S, f) such that W_{X,K_{d}} ≅ H^{2}(S, ℂ)^{0}(−1) if and only if the following conditions are satisfied:

1. 4 |̸ d and 9 |̸ d;

2. p |̸ d if p is an odd prime, p ≡ −1(mod 3).

We say that the pair (S, f) is associated to (X, K_{d}).

We first show the theorem boils down to a computation of lattices (Proposition 5.1.4). Recall that a pseudo-polarization is a divisor f contained in the closure of the Kähler cone with (f, f) > 0; the primitive cohomology of a pseudo-polarized K3 surface (S, f) is the orthogonal complement to f in H^{2}(S, ℤ). Let Λ_{d}^{0} be a lattice isomorphic to the primitive middle cohomology of a degree d K3 surface. The isomorphism asserted in the theorem implies an isomorphism of lattices K_{d}^{⊥} ≅ −Λ_{d}^{0}. On the other hand, given a labelled special cubic fourfold (X, K_{d}) and an isomorphism of lattices K_{d}^{⊥} ≅ −Λ_{d}^{0}, W_{X,K_{d}}(+1) has the form of the primitive cohomology of a pseudo-polarized K3 surface. Indeed, since the Torelli map for K3 surfaces is surjective [4] [26], there exists a pseudo-polarized K3 surface (S, f) such that H^{2}(S, ℂ)^{0}(−1) ≅ W_{X,K_{d}}. Moreover, X is smooth so H^{4}(X, ℤ)^{0} ∩ H^{2,2}(X) does not contain any classes with self-intersection +2 ([29] §4 Prop. 1). Therefore there are no (−2)-curves on S orthogonal to f, and f is actually a polarization.

Proposition 5.1.4 Retain the notation above. K_{d}^{⊥} ≅ −Λ_{d}^{0} if and only if the conditions of Theorem 5.1.3 are satisfied.

The automorphisms of Λ = H^{2}(S, ℤ) act transitively on the primitive vectors with (v, v) = d ≠ 0 ([16] Theorem 2.4), so

Λ_{d}^{0} ≅ (−d) ⊕ H^{⊕2} ⊕ (−E_{8})^{⊕2};

let y denote the distinguished element with (y, y) = −d. The discriminant group d(Λ_{d}^{0}) and quadratic form q_{Λ_{d}^{0}} are equal to ℤ(\frac{y}{d})/ℤy, with q_{Λ_{d}^{0}}(\frac{y}{d}) = \frac{−1}{d}( mod 2ℤ).

We determine when d(K_{d}^{⊥}) and d(−Λ_{d}^{0}) are isomorphic as groups with a ℚ/2ℤ-valued quadratic form. We first consider the case d ≡ 2 (mod 6). Here both discriminant groups are isomorphic to ℤ/dℤ, so we just need to check when the quadratic forms are conjugate by an automorphism of ℤ/dℤ. Let u and w be generators of d(K_{d}^{⊥}) and d(−Λ_{d}^{0}) such that q_{K_{d}^{⊥}}(u) = \frac{2d−1}{3d} (mod 2ℤ) and q_{−Λ_{d}^{0}}(w) = \frac{1}{d} (mod 2ℤ) (see Proposition 3.2.5). The quadratic forms are conjugate if and only if the integer \frac{2d−1}{3} is a square modulo 2d, or equivalently, −3 is a square modulo 2d. By quadratic reciprocity this is the case if and only if d is not divisible by four and any odd prime p|d satisfies p ≢ −1 (mod 3). A similar argument holds in the case d ≡ 0 (mod 6).

We have seen that the conditions on d are necessary for K_{d}^{⊥} to be isomorphic to −Λ_{d}^{0}. On the other hand, K_{d}^{⊥} is the unique even lattice of signature (19, 2) with discriminant form (d(K_{d}^{⊥}), q_{K_{d}^{⊥}}) [22] 1.14.3. Hence if the discriminant forms of K_{d}^{⊥} and −Λ_{d}^{0} agree then K_{d}^{⊥} ≅ −Λ_{d}^{0}. □

5.2 Isomorphisms of Period Domains

We retain the notation of § 2.1 and § 5.1. Let Σ denote the automorphisms of Λ, and Σ_{d} the automorphisms fixing some primitive v ∈ Λ with (v, v) = d, which yield automorphisms of Λ_{d}^{0} = v^{⊥}. As in § 2.2, let N′_{d} be the local period domain for degree d K3 surfaces, an open nineteen-dimensional complex manifold. Let Σ_{d}^{+} ⊂ Σ_{d} denote the subgroup stabilizing N′_{d}. As before, N′_{d} is a bounded symmetric domain of type IV, Σ_{d}^{+} is an arithmetic group acting holomorphically on N′_{d}, and the quotient N_{d} := Σ_{d}^{+}/N′_{d} is therefore a quasi-projective variety, the global period domain for degree d K3 surfaces.

We introduce a bit more notation for special cubic fourfolds as well. Let G_{d}^{+} ⊂ Γ_{d}^{+} be the subgroup acting trivially on K_{d} and let D_{d}^{mar} denote the marked special Hodge structures of discriminant d, modulo the action of G_{d}^{+}. The fiber product D_{d}^{mar} ×_{D} C is written C_{d}^{mar}, the marked special cubic fourfolds of discriminant d. We have natural forgetting maps D_{d}^{mar} → D_{d}^{lab} and C_{d}^{mar} → C_{d}^{lab}.

Proposition 5.2.1 G_{d}^{+} = Γ_{d}^{+} if d ≡ 2 (mod 6) and G_{d}^{+} ⊂ Γ_{d}^{+} is an index-two subgroup if d ≡ 0 (mod 6). The natural map D_{d}^{mar} → D_{d}^{lab} is an isomorphism if d ≡ 2 (mod 6) and a double cover if d ≡ 0 (mod 6). Furthermore, D_{d}^{mar} = G_{d}^{+}\D′_{d} and thus is connected for all d ≠ 6.

We begin with the first statement. The lattice K_{d} has no automorphisms preserving h^{2} if d ≡ 2 (mod 6), so G_{d}^{+} = Γ_{d}^{+}. If d ≡ 0 (mod 6) then K_{d} has an involution, which acts on K_{d}^{0} as multiplication by −1. We claim it extends to an element γ ∈ Γ_{d}^{+}. By Proposition 3.2.4 we may assume K_{d}^{0} = ℤ(v_{1} + \frac{d}{6}w_{1}). We use the notation of § 2.1, so v_{1} and w_{1} form a basis for a hyperbolic summand H ⊂ L^{0}. Choose γ equal to multiplication by −1 on both hyperbolic summands of L^{0} and equal to the identity elsewhere. We have that γ ∈ Γ_{d}^{+} but γ ∉ G_{d}^{+}, so G_{d}^{+} is a proper subgroup of Γ_{d}^{+}.

The second statement follows immediately from the first. As for the third statement, recall that D_{d}^{lab} = Γ_{d}^{+}\D′_{d}. Hence for d ≡ 2 (mod 6) the result is immediate. For d ≡ 0 (mod 6), we must check that any γ ∈ Γ_{d}^{+} acting nontrivially on K_{d} also acts nontrivially on D′_{d}. For d ≠ 6, if γ acts nontrivially on K_{d} then the induced action on d(K_{d}) is not equal to ±1. However, the groups d(K_{d}) and d(K_{d}^{⊥}) are isomorphic, so the induced action on d(K_{d}^{⊥}) is not ±1. Now D′_{d} is a topologically open subset of a quadric hypersurface in ℙ(K_{d}^{⊥} ⊗ℂ), so only scalar multiplications act trivially on D′_{d}. In particular, γ necessarily acts nontrivially. □

Remark: There exists an element γ ∈ Γ_{6}^{+} − G_{6}^{+} acting trivially on K_{6}^{⊥}. It follows that D_{6}^{mar} ≠ G_{6}^{+}\D′_{6} but rather that D_{6}^{lab} = G_{6}^{+}\D′_{6}.

Theorem 5.2.2 Let d be a positive integer such that there exists an isomorphism j_{d} : K_{d}^{⊥} → −Λ_{d}^{0} (see Proposition 5.1.4.) Choose orientations on the negative definite parts of K_{d}^{⊥} and −Λ_{d}^{0} compatible with j_{d}, so there is an induced isomorphism of local period domains D′_{d} and N′_{d}. If d ≠ 6 then there is an induced isomorphism i_{d} : D_{d}^{mar} → N_{d}; we also have D_{6}^{lab} ≅ N_{6}.

The isomorphism of period domains depends on the choice of j_{d}. Each j_{d} induces an isomorphism of discriminant groups j′_{d} : d(K_{d}^{⊥}) → d(−Λ_{d}^{0}) preserving the ℚ/2ℤ-valued quadratic forms on these groups [22] §1.3. We denote the set of such isomorphisms Isom(d(K_{d}^{⊥}), d(−Λ_{d}^{0})); the group {n ∈ ℤ/dℤ : n^{2} = 1} acts faithfully and transitively on this set.

Theorem 5.2.3 For d ≠ 6, the various isomorphisms i_{d} : D_{d}^{mar} → N_{d} correspond to elements of Isom(d(K_{d}^{⊥}), d(−Λ_{d}^{0}))/(±1). The isomorphism i_{6} : D_{6}^{lab} → N_{6} is unique.

These two theorems have the following corollary:

Corollary 5.2.4 (Immersions into Moduli Spaces of K3 Surfaces) Let d ≠ 6 be a positive integer such that there exists an isomorphism j_{d} : K_{d}^{⊥} → −Λ_{d}^{0}. Then there is an imbedding i_{d} : C_{d}^{mar} ↪ N_{d}, unique up to the choice of an element of Isom(d(K_{d}^{⊥}), d(−Λ_{d}^{0}))/(±1). Moreover, there is a unique imbedding i_{6} : C̃_{6}^{lab} ↪ N_{6}.

As we shall see in §6, geometrical considerations will sometimes mandate specific choices of i_{d} (e.g. in the case d = 14).

We prove the first theorem. First, we compare the action of Σ_{d}^{+} on Λ_{d}^{0} to the action of G_{d}^{+} on K_{d}^{⊥}. We claim that Σ_{d}^{+} is the group of automorphisms of Λ_{d}^{0} preserving the orientation on the positive definite part of Λ_{d}^{0} ⊗ ℝ and acting trivially on the discriminant group d(Λ_{d}^{0}). This follows from the results of [22] §1.4, which imply that any such automorphism extends uniquely to an element of Σ_{d}^{+}. Similarly, G_{d}^{+} is the group of automorphisms of K_{d}^{⊥} preserving the orientation on the negative definite part of K_{d}^{⊥} ⊗ ℝ and acting trivially on the discriminant group d(K_{d}^{⊥}).

Now suppose we are given an isomorphism j_{d} : K_{d}^{⊥} → −Λ_{d}^{0}. This induces isomorphisms D′_{d} → N′_{d}, G_{d}^{+} → Σ_{d}^{+}, and i_{d} : G_{d}^{+}\D′_{d} → Σ_{d}^{+}\N′_{d}. Applying Proposition 5.2.1, we obtain an isomorphism i_{d} : D_{d}^{mar} → N_{d} for d ≠ 6. The remark after the proposition also yields an isomorphism i_{6} : D_{6}^{lab} → N_{6}. □

We turn to the proof of the second theorem. We must determine when two different isomorphisms j_{d}^{1} : K_{d}^{⊥} → −Λ_{d}^{0} and j_{d}^{2} : K_{d}^{⊥} → −Λ_{d}^{0} induce the same isomorphism i_{d} : G_{d}^{+}\D′_{d} → Σ_{d}^{+}\N′_{d}. If j_{d}^{2} = σ∘j_{d}^{1} for some σ ∈ Σ_{d}^{+} then j_{d}^{1} and j_{d}^{2} induce the same isomorphisms of period domains. Also, if j_{d}^{1} = −j_{d}^{2} then j_{d}^{1} and j_{d}^{2} induce the same isomorphism between D′_{d} and N′_{d}, because these manifolds lie in the projective spaces ℙ(K_{d}^{⊥} ⊗ ℂ) and ℙ(Λ_{d}^{0} ⊗ ℂ).

On the other hand, assume that j_{d}^{1} and j_{d}^{2} induce the same isomorphism between G_{d}^{+}\D′_{d} and Σ_{d}^{+}\N′_{d}. Then there exist γ ∈ G_{d}^{+} and σ ∈ Σ_{d}^{+} such that j_{d}^{1} ∘ γ and σ ∘ j_{d}^{2} induce the same isomorphism between D′_{d} and N′_{d}, so j_{d}^{1} ∘ γ = ±σ ∘ j_{d}^{2}. We conclude that the isomorphisms between G_{d}^{+}\D′_{d} and Σ_{d}^{+}\N′_{d} correspond to certain elements of Isom(d(K_{d}^{⊥}), d(−Λ_{d}^{0}))/(±1).

It remains to check that each element of Isom(d(K_{d}^{⊥}), d(−Λ_{d}^{0}))/(±1) actually arises from an isomorphism between K_{d}^{⊥} and −Λ_{d}^{0} respecting the orientations on the negative definite parts. Now K_{d}^{⊥} has an automorphism g reversing the orientation on the negative part and acting trivially on d(K_{d}). Take g to be the identity except on a hyperbolic summand of the orthogonal decomposition for K_{d}^{⊥}; on the hyperbolic summand set g equal to multiplication by −1. Hence it suffices to show that the automorphisms of K_{d}^{⊥} induce all the automorphisms of d(K_{d}^{⊥}), which is proved in [22], Theorem 1.14.2 and Remark 1.14.3. □

6 Fano Varieties of Special Cubic Fourfolds

6.1 Introduction and Necessary Conditions

Here we provide a geometric explanation for the K3 surfaces associated to some special cubic fourfolds. The general philosophy underlying our approach is due to Mukai [19],[20],[21]. Let S be a polarized K3 surface and let M_{S} be a moduli space of simple sheaves on S. Quite generally, M_{S} is smooth and possesses a natural nondegenerate holomorphic two-form ([19] Theorem 0.1). Furthermore, the Chern classes of the ‘quasi-universal sheaf’ on S×M_{S} induce correspondences between S and M_{S}. If M_{S} is compact of dimension two then it is a K3 surface isogenous to S; the Hodge structure of M_{S} can be read off from the Hodge structure of S and the numerical invariants of the sheaves ([20] Theorem 1.5). Conversely, given a variety F with a nondegenerate holomorphic two-form and an isogeny H^{2}(S, ℚ) → H^{2}(F, ℚ), one can try to interpret F as a moduli space of sheaves on S. In the case where F is a K3 surface, we often have such interpretations ([20] Theorem 1.9). Note that F ≅ S^{[n]} can be interpretted as the moduli space of ideal sheaves on S of colength n; such sheaves are simple.

Proposition 6.1.1 Let X be a cubic fourfold with Fano variety F. Assume there is an isomorphism between F and S^{[2]} for some K3 surface S. Then X has a labelling K_{d} such that S is associated to (X, K_{d}); i_{d} : C_{d}^{mar} ↪ N_{d} may be chosen so that i_{d}(X, K_{d}) = S. If (X_{1}, K_{d}) is a generic element of C_{d}^{mar} and S_{1} = i_{d}(X_{1}, K_{d}), then the Fano variety F_{1} is isomorphic to S_{1}^{[2]}.

For nongeneric X_{1} the isomorphism between F_{1} and S_{1}^{[2]} can break down. Let X_{1} contain two disjoint planes π_{1} and π_{2}, so that X_{1} ∈ C_{14}. The proposition holds for d = 14, but the (birational) map between F_{1} and S_{1}^{[2]} acquires indeterminacy at the lines supported in the π_{i} (see [13] for details).

We prove the proposition. As in §2.1, there is an isomorphism H^{2}(F, ℤ) ≅ H^{2}(S, ℤ) ⊕_{⊥} ℤδ and the hyperplane class g = af − bδ where f is some polarization of S with d := (f, f). Let K_{d}^{⊥} equal α^{−1}(H^{2}(S, ℤ)^{0}(−1)) where α is the Abel-Jacobi map, and set K_{d} = (K_{d}^{⊥})^{⊥}. Applying Theorem 5.2.2 with j_{d} = α|K_{d}^{⊥}, we obtain a map i_{d} with the desired properties. To explain i_{d} geometrically, we need the following result:

Theorem 6.1.2 (Deformation Spaces of S^{[2]} [3]) Let S be a K3 surface and 2δ ⊂ S^{[2]} be the elements supported at a single point. The deformation space of S^{[2]} is smooth and has dimension twenty-one. Deformations of the form S_{1}^{[2]} correspond to a divisor in this space which may be characterized as the deformations for which δ remains a divisor.

We define C_{d} as the deformations of F for which δ remains algebraic. Applying Theorem 6.1.2, there is some small analytic neighborhood in C_{d} where the deformations are isomorphic to S_{1}^{[2]} for some deformation S_{1} of S. This isomorphism holds in an open étale neighborhood of X in C_{d}, so a generic cubic fourfold in C_{d} has Fano variety of the form S_{1}^{[2]}. □

For which values of d are the conclusions of Proposition 6.1.1 valid? Theorem 5.1.3 gives sufficient conditions for the existence of a K3 surface associated to (X, K_{d}), but these do not guarantee that F ≅ S^{[2]}:

Proposition 6.1.3 Assume that the Fano variety of a generic special cubic fourfold of discriminant d is isomorphic to S^{[2]} for some K3 surface S. Then there exist positive integers n and a such that d = 2\frac{n^{2}+n+1}{a^{2}}.

This is equivalent to the existence of a line bundle on S^{[2]} of degree 108, the degree of the Fano variety. For instance, Fano varieties of special cubic fourfolds of discriminant 74 are not generally of the form S^{[2]}, because 74a^{2} = 2(n^{2} + n + 1) has no integral solutions (see [11]).

We can produce infinitely many examples of special cubic fourfolds with Fano variety isomorphic to the symmetric square of a K3 surface:

Theorem 6.1.4 Assume that d = 2(n^{2} + n + 1) where n is an integer ≥ 2. Then the Fano variety of a generic special cubic fourfold X of discriminant d is isomorphic to S^{[2]}, where S is a K3 surface associated to (X, K_{d}).

This is proved in the next two sections. The condition on d corresponds to setting a = 1 in Proposition 6.1.3. The proof of the theorem suggests that the condition of the proposition is the correct sufficient condition.

6.2 Ambiguous Symplectic Varieties

Definition 6.2.1 Let F be an irreducible symplectic Kähler manifold, and assume that there exist K3 surfaces S_{1} and S_{2} and isomorphisms r_{1} : F → S_{1}^{[2]} and r_{2} : F → S_{2}^{[2]} such that r_{1}^{∗}δ_{1} ≠ r_{2}^{∗}δ_{2}. Then we say that F is ambiguous.

Our first example is a special case of a construction of Beauville and Debarre [8]. Let S be a smooth quartic surface in ℙ^{3}, p_{1} + p_{2} a generic point in S^{[2]}, and ℓ(p_{1} + p_{2}) the line containing p_{1} and p_{2}. By Bezout’s theorem ℓ(p_{1} + p_{2}) ∩ S = p_{1} + p_{2} + q_{1} + q_{2}. Setting j(p_{1} + p_{2}) = q_{1} + q_{2} for each p_{1} + p_{2}, we obtain a birational involution j : S^{[2]} −−→ S^{[2]}. If S contains no lines then j extends to a biregular morphism. Let f_{4} be the degree-four polarization on S and the corresponding class on S^{[2]}. Following [8], one may compute j^{∗}(x) = −x + (x, f_{4} − δ) (f_{4} − δ) on H^{2}(S^{[2]}, ℤ). Setting r_{2} = j ∘ r_{1}, we find that F = S^{[2]} is ambiguous.

We digress to give another beautiful example of ambiguous varieties:

Proposition 6.2.2 Assume that 3|d and that the Fano variety F of a generic cubic fourfold in C_{d} is isomorphic to S_{1}^{[2]} for some K3 surface S_{1}. Then F is ambiguous.

This follows immediately from Proposition 6.1.1 and the results of §5.2, which imply that C_{d}^{lab} imbeds into a ℤ/2ℤ-quotient of N_{d} if 3|d.

6.3 Construction of the Examples

Let X_{0} ∈ C̃_{6}, F_{0} its Fano variety of lines, and S the sextic K3 surface associated to X_{0} (see § 4.2). Let ϕ : X → Δ be a family in C̃ with central fiber X_{0} and X_{t} smooth for t ≠ 0. Let F → Δ be the corresponding family of Fano varieties and X′ → Δ′ a semistable reduction of X → Δ. For simplicity, we assume that the central fiber of the semistable family is of the form X′_{0} = \overline{X_{0}} ∪ Q where \overline{X_{0}} = Bl_{S}(ℙ^{4}) is the desingularization of X_{0}, Q is a smooth quadric fourfold, and Q_{0} = \overline{X_{0}} ∩ Q is the smooth quadric in ℙ^{4} containing S. This is the case if ϕ is a sufficiently generic smoothing of X_{0}.

Lemma 6.3.1 F_{0} is singular along the lines through the double point, which are parametrized by S. These singularities are ordinary codimension-two double points and the blow-up σ : Bl_{S} F_{0} → F_{0} desingularizes F_{0}. If S_{0} does not contain a line then Bl_{S} F_{0} ≅ S^{[2]}.

The first part follows from § 4.2 and [1] 1.10. For the second part, we realize σ by blowing up the Grassmannian 𝔾(1, 5) along the locus L(p) of lines containing p. The fiber square

S → F_{0}
↓  ↓
L(p) → 𝔾(1, 5)

gives a natural closed imbedding of normal cones C_{S}F_{0} ↪ C_{L(p)}𝔾(1, 5)|S. The projectivization ℙ(C_{L(p)}𝔾(1, 5)) corresponds to ℙ(ℂ^{6}/S), where S is the restriction of the universal subbundle. Note L(p) ≅ ℙ^{4} and C_{L(p)}𝔾(1, 5)_{ℓ} corresponds to the lines λ such that ℓ ∈ λ ⊂ ℙ^{4}. For ℓ ∈ Sing(F_{0}) the fiber of ℙ(C_{S}F_{0})_{ℓ} corresponds to those lines λ such that ℓ ∈ λ ⊂ Q_{0}. These are parametrized by a smooth conic curve, hence F_{0} has codimension-two ordinary double points along S and Bl_{S} F_{0} is smooth.

This description implies that we can regard Bl_{S} F_{0} as a parameter space for certain curves on \overline{X_{0}}. These curves are of the following types:

1. lines on X_{0} disjoint from p;

2. unions of proper transforms of lines through p and lines contained in Q_{0} ⊂ \overline{X_{0}}.

These in turn may be identified with:

1. two-secants λ to S ⊂ ℙ^{4};

2. three-secants λ with a distinguished point s ∈ λ ∩ S.

We emphasize that each line meeting S in more than two points is contained in Q_{0} but not in S, and thus is a three-secant to S. We claim elements of S^{[2]} naturally correspond to curves of this type. For each ideal sheaf I of colength two there is a unique line λ containing the corresponding subscheme. Either λ is a two-secant, or λ is a three-secant and s is the support of I/I_{λ∩S}. □

Lemma 6.3.2 Retain the notation and assumptions introduced above. The family of Fano varieties F×_{Δ}Δ′ has ordinary codimension-three double points along the surface S. The variety F′ = Bl_{S}(F ×_{Δ} Δ′) is smooth, and the exceptional divisor E ⊂ F′_{0} is a smooth quadric surface bundle over S. The component of F′_{0} dominating F_{0} is isomorphic to S^{[2]}.

The proof is essentially the same as the first lemma. Our nest result is:

Proposition 6.3.3 Retain the notation and assumptions introduced above. Then there is a smooth family \overline{F} → Δ′, birational to F × Δ′, such that \overline{F}_{u} = F_{u} and \overline{F}_{0} = S^{[2]}.

We start with the family F′ described in the previous lemma. The fibers of E → S are all smooth quadric surfaces, so the variety parametrizing rulings of E is an étale double cover of S. Since S has no nontrivial étale coverings we may choose a ruling of E. Blowing down E in the direction of this ruling, we obtain a smooth family \overline{F}. This map induces an isomorphism from the proper transform of F_{0} in F′_{0} to the central fiber of \overline{F}. The proper transform to F_{0} in F′_{0} is isomorphic to S^{[2]}, so \overline{F} satisfies the conditions of the proposition. □

We now prove Theorem 6.1.4. Let S be an algebraic K3 surface with Picard group

P = \begin{aligned}& f_{6} & f_{4} \\ f_{6} & 6 & n + 5 \\ f_{4} & n + 5 & 4 \end{aligned}

and n ≥ 2. By Lemma 4.3.2, such a surface exists and we may assume that |f_{6}| imbeds it as a smooth sextic surface. The divisor f_{4} is effective because it has positive degree with respect to f_{6}. We claim that f_{4} is very ample. If f_{4} were not ample, then there would exist a (−2)-curve E with f_{4}E ≤ 0. This follows from the structure of the Kähler cone of S ([16] §1,§10). Note that f_{4}E ≠ 0 because P does not contain a rank-two sublattice of discriminant −8. Recall that the Picard-Lefschetz reflection associated to E is given by the equation r_{E}(x) = x + (E, x)E. Applying this to the class f_{4}, we find that r_{E}(f_{4})^{2} = 4 and (f_{6}, r_{E}(f_{4})) < (f_{6}, f_{4}). Hence that f_{6} and r(f_{4}) span a sublattice with discriminant smaller than that of P, which is impossible. Finally, applying Lemma 4.3.3 we see that the linear system |f_{4}| imbeds S as a smooth quartic surface.

Our hypothesis on P implies that the image of S under |f_{6}| lies on a smooth quadric hypersurface and does not contain a line, and that the image of S under |f_{4}| also does not contain a line. In particular, S corresponds to a singular cubic fourfold X_{0} ∈ C̃_{6}. Furthermore S^{[2]} is ambiguous, with an involution j : S^{[2]} → S^{[2]} so that δ_{2} := j^{∗}δ = 2f_{4} −3δ. Using Proposition 6.3.3 and the arguments of § 4.3, X_{0} has a smoothing ϕ : X → Δ such that (after base change) the corresponding family of smooth symplectic varieties \overline{F} → Δ′ is a deformation of S^{[2]} for which δ_{2} remains algebraic. By Theorem 6.1.2 the Fano variety \overline{F_{u}} of X′_{u} is isomorphic to S_{u}^{[2]}.

If we choose ϕ generally, we may assume that the X′_{u} are typical and that Pic(S_{u}) is generated by the polarization f′. Let Π = Pic(\overline{F_{u}}), a lattice (with respect to the canonical form) of discriminant −2deg (S_{u}). On the other hand, Π is the saturation of ℤg + ℤδ_{2}. Specializing to S^{[2]} we obtain Π = ℤ(2f_{6} − 3δ) + ℤ(f_{6} − f_{4}) with discriminant −4(n^{2} + n + 1). In particular, the S_{u} have degree d(n) = 2(n^{2}+n+1) and the X_{u} are special of discriminant d(n). □

We have shown that the pure limiting Hodge structures parametrized by D_{6} actually arise from smooth symplectic varieties. This may be interpretted as a weak surjectivity result for the corresponding Torelli map. It also explains the computation of the limiting mixed Hodge structure H_{lim}^{4} in § 4.2.

There are a number of ways Theorem 6.1.4 might be generalized. We need not assume that the polarizations f_{6} and f_{4} actually generate the Picard lattice of S. Another approach is to replace C̃_{6} by some other divisor C_{d} parametrizing special cubic fourfolds whose Fano varieties are of the form S^{[2]}. To make precise statements one requires explicit descriptions of two complicated closed sets: the complement D_{d} − C_{d} and the locus in C_{d} where the isomorphism between the Fano varieties and the blown-up symmetric squares breaks down. Finally, Mukai’s philosophy suggests that whenever we have an associated K3 surface S, the Fano variety F might be interpretted as a suitable moduli space of simple sheaves on S. It would be interesting to find such interpretations when F cannot be a blown-up symmetric square.

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