Rational curves on holomorphic symplectic fourfolds

Brendan Hassett and Yuri Tschinkel

March 2001

1 Introduction

One of the main problems in the theory of irreducible holomorphic symplectic manifolds is the description of the ample cone in the Picard group. The goal of this paper is to formulate explicit Hodge-theoretic criteria for the ampleness of line bundles on certain irreducible holomorphic symplectic manifolds. It is well known that for K3 surfaces the ample cone is governed by (−2)-curves. More generally, we expect that certain distinguished twodimensional homology classes of the symplectic manifold should correspond to explicit families of rational curves, and that these govern its ample cone.

The program for analyzing the ample cone of a symplectic manifold divides naturally into three parts. First, for each deformation type of irreducible holomorphic sympectic manifolds we identify distinguished Hodge classes in H_{2}(ℤ) that should be represented by rational curves. We consider H_{2}(ℤ) and H^{2}(ℤ) as quadratic lattices with respect to a natural quadratic form (the Beauville form discussed in Section 2) and distinguish orbits in H_{2}(ℤ) under the orthogonal group. These orbits are often characterized by the ‘squares’ of the corresponding elements, i.e., the value of the quadratic form. These distinguished classes should be in one-to-one correspondence with certain geometrically described rational curves on F. In many cases, one can use deformation arguments to show that, if a given distinguished class represents a rational curve of a certain type then this remains true under deformation (see Section 4). Second, one shows that rational curves with the geometry described govern the ample cone of F. This entails classifying possible contractions of symplectic manifolds - a very active topic of current research (see the discussion of the literature below) - and interpretting this classification in terms of the numerical properties of the contracted curves.

It also involves the classification of base loci for sections of line bundles on a symplectic manifold. The third part of the program is to show that a divisor class satisfying certain numerical conditions arises from a big line bundle and thus yields a birational transformation of the symplectic manifold. This aspect of the program is still largely conjectural; see the work of Huybrechts cited below.

We are mainly concerned with the first step of this program in a specific case. Let F be an irreducible holomorphic symplectic fourfold deformation equivalent to the punctual Hilbert scheme S^{[2]} for some K3 surface S. Given the Hodge structure on H^{2}(F), we describe explicitly (but conjecturally) the cone of effective curves on F and, by duality, the ample cone of F. As in the case of K3 surfaces, each divisor class of square −2 induces a reflection preserving the Hodge structure. The ‘birational ample cone’ is conjectured to be the interior of a fundamental domain for this reflection group. However, the ample cone may be strictly smaller than the birational ample cone, owing to the existence of elementary transformations along ℙ^{2}’s in F. The corresponding classes have square −10 with respect to the Beauville form.

Here we give a brief and incomplete overview of work on related problems. Wilson has studied Calabi-Yau threefolds from a similar point of view (see [29],[30],[31], and [32].) If F is a Calabi-Yau threefold then the Picard group of F is equipped with two integer-valued forms: a cubic form μ (the intersection form) and a linear form c_{2}(F) (obtained by intersecting with the second Chern class of the tangent bundle.) Wilson gives criteria for the existence of birational contractions and elliptic fibrations in terms of the number-theoretic properties of these forms.

Namikawa [22] and Shepherd-Barron [25], have proven structural results on the geometry of birational morphisms from holomorphic symplectic manifolds. There are also results in this direction by Burns, Hu, and Luo [7] and Wierzba [28]. Matsushita [19] [20] has proven a detailed description of fiber structures on irreducible holomorphic symplectic manifolds. Huybrechts has conjectured a projectivity criterion for irreducible holomorphic symplectic manifolds and elaborated consequences of his criterion. See [13] and the erratum in [14]; we will be careful to distinguish results fully proved in [13] from those which remain conjectural. Markman [17] has developed a theory of generalized elementary transformations for moduli spaces of sheaves on K3 surfaces.

Our study of rational curves on symplectic manifolds began with the detailed study of a particular example: the variety F parametrizing lines on a cubic fourfold X is an irreducible holomorphic symplectic manifold. The existence of rational curves on F coincides with the presence of rational ruled surfaces on X. Our conjectures therefore shed light on the effectivity of certain codimension-two cycles on X. Conversely, the projective geometry of cubic fourfolds provides a useful laboratory where we may test our claims. As an application of our conjectures, we find that the presence of distinguished Hodge classes on X often implies the existence of special unirational parametrizations of X.

This paper is organized as follows. In Section 2 we recall basic results and conjectures for irreducible holomorphic symplectic manifolds. In the next section, we introduce the notion of nodal classes and state our main conjectures. In Section 4 we give some deformation-theoretic evidence for our conjectures. The rest of the paper is devoted to examples supporting the conjectures. Section 5 is devoted to Hilbert schemes S^{[2]} for K3 surfaces of small degree. We describe examples of nonnodal rational curves and certain codimension-two behavior in Section 6. We turn to the projective geometry of cubic fourfolds in the last section. Questions of rationality and unirationality are addressed in Section 7.2.

Throughout, we work over ℂ. A primitive element of an abelian group A is one that cannot be written in the form nx for any x ∈ A or n ∈ ℤ, n > 1. An indecomposable element of a monoid is an element which cannot be written in the form a + b for some nonzero a and b in this monoid. Recall that an element v of a convex real cone C is an extremal ray if, for any u, w ∈ C with u + w = v we necessarily have u, w ∈ ℝ_{+}v.

Acknowledgements. The first author was partially supported by an NSF postdoctoral fellowship, NSF continuing research grant 0070537, and The Institute of Mathematical Sciences at the Chinese University of Hong Kong. The second author was partially supported by the NSA. We thank D. Matsushita and Y. Namikawa for sending us their preprints. Our treatment of rationality questions in Section 7.2 benefitted from discussions with D. Saltman and I. Dolgachev.

2 Generalities

Let F be an irreducible, holomorphic symplectic manifold of dimension 2n. This means that F is compact, Kähler, simply connected, and H^{0}(F, Ω_{F}^{2} ) is spanned by an everywhere-nondegenerate 2-form (cf. [2] and [6]). The second cohomology group H^{2}(F, ℤ) carries a nondegenerate integer-valued quadratic form (,), the Beauville form. It has signature (3, b_{2}(F) − 3) and its restriction to H^{1,1} ∩ H^{2}(F, ℤ) has signature (1, b_{2}(F) − 3), where b_{2}(F) is the second Betti number (see [3] and [13] §1.9 for more details).

Using the universal coefficient theorem, we can extend the Beauville form to a ℚ-valued form on H_{2}(F, ℤ). Concretely, given some primitive R ∈ H_{2}(F, ℤ), there exists a unique class w ∈ H^{2}(F, ℚ) such that Rv = (w, v) for all v ∈ H^{2}(F, ℤ). We set (R, R) = (w, w). Let ρ ∈ H^{2}(F, ℤ) denote the primitive class such that cρ = w for some c > 0. Note that R is of type (2n − 1, 2n − 1) iff ρ is of type (1, 1). Conversely, given a primitive ρ ∈ H^{2}(F, ℤ) with (ρ, H^{2}(F, ℤ)) = dℤ and d > 0, there exists a primitive class R ∈ H_{2}(F, ℤ) with dRv = (v, ρ) for all v ∈ H^{2}(F, ℤ).

Throughout this paper, the square of a divisor class means the square with respect to the Beauville form. In the sequel we will assume that F has a polarization g; note that (g, g) > 0 [13] §1.9. Denote by

Pic_{+}(F, g) = {v ∈ Pic(F) | (v, g) > 0}

the positive halfspace (with respect to g and the Beauville form). Let Λ_{+}(F, g) ⊂ Pic_{+}(F, g) be the vectors with positive square.

We denote by Λ_{amp}(F) and Λ_{nef}(F) the monoids of ample and nef divisor classes. Let N_{1}(F) be the group of classes of 1-cycles (up to numerical equivalence), NE(F) ⊂ N_{1}(F)_{ℝ} the cone of effective curves, and \overline{NE}(F) its closure. We denote by Λ_{+}^{∗}(F, g) the set of classes R such that the corresponding ρ is contained in Λ_{+}(F, g).

We next review properties of line bundles on polarized irreducible holomorphic symplectic manifolds which follow from standard results of the minimal model program.

Proposition 2.1 Let (F, g) be a polarized irreducible holomorphic symplectic manifold.

1. A class λ ∈ Pic(F) is ample iff λ ∈ Λ_{+}(F, g) and λC > 0 for each curve C ⊂ F (see [13] Cor. 6.4).

2. Any class λ ∈ Λ_{nef}(F) which is big has the property that the line bundle L(λ) has no cohomology and L(mλ) is globally generated for m ≫ 0; it therefore defines a birational morphism b : F →Y (see [15], Remark 3-1-2 and Theorem 1-2-3).

The following statements were conjectured by Huybrechts [14] (but stated as Theorem 3.11 and Corollary 3.10 in [13] with an incomplete proof):

Conjecture 2.2 Let F be an irreducible holomorphic symplectic manifold.

1. F is projective iff there exists a class g ∈ Pic(F) with (g, g) > 0.

2. If g is a polarization for F then any class λ ∈ Λ_{+}(F, g) is big.

Now assume F = S^{[n]}, the Hilbert scheme of length n subschemes of a K3 surface S. Then we have an isomorphism

Pic(S^{[n]}) ≃ Pic(S) ⊕_{⊥} ℤe

compatible with the Beauville form (see [3]). Each divisor f on S determines a divisor on S^{[n]}, also denoted by f, and corresponding to the subschemes with some support in f. The locus of subschemes with support at fewer than n points has class 2e and (e, e) = −2(n − 1).

More generally, if F is deformation equivalent to S^{[n]} then the Beauville form on L := H^{2}(F, ℤ) is an even, integral form isomorphic to

U^{⊕3} ⊕_{⊥} (−E_{8})^{⊕2} ⊕_{⊥} ℤe,

where U is a hyperbolic plane, E_{8} the positive-definite quadratic form associated to the corresponding Dynkin diagram.

Proposition 2.3 Assume that n = 2 so that

L ≃ U^{⊕3} ⊕_{⊥} (−E_{8})^{⊕2} ⊕_{⊥} (−2).

The orbits of primitive elements v ∈ L under the action of Γ = Aut(L) are classified by (v, v) and the ideal (v, L) (which equals ℤ or 2ℤ.)

Proof. We shall classify primitive imbeddings of the lattice K := ℤv = (2d) (where (v, v) = 2d) into L. For simplicity, we first restrict to the case d ≠ 0. The basic technical tool is the discriminant group d(L) := L^{∗}/L and the associated ℚ/2ℤ-valued quadratic form q_{L} [23]. We have d(L) ≃ ℤ/2ℤ with q_{L} equal to −\frac{1}{2} (mod 2ℤ) on the generator. Let K^{⊥} denote the orthogonal complement to K in L, d(K) and d(K^{⊥}) the discriminant groups, and q_{K} and q_{K^{⊥}} the corresponding forms, so that d(K) ≃ ℤ/2dℤ with q_{K} equal to \frac{1}{2d} (mod 2ℤ) on the generator \frac{v}{2d}. We have the sequence of inclusions

K ⊕ K^{⊥} ⊂ L ⊂ L^{∗} ⊂ K^{∗} ⊕ (K^{⊥})^{∗};

L^{∗} consists of the elements of K^{∗} ⊕ (K^{⊥})^{∗} which are integral on L. Let H (resp. H^{∗}) be the image of L (resp. L^{∗}) in d(K) ⊕ d(K^{∗}), so that d(L) = H^{∗}/H. Note that H is isotropic and H^{∗} is the annihilator of H with respect to q_{K}⊕q_{K^{⊥}} (or the ℚ/ℤ-valued bilinear form associated to it). The projection of H into d(K) is injective and its image is a subgroup of index one or two, depending on whether (v, L) = ℤ or 2ℤ.

In the first case, d(K^{⊥}) contains the projection of H as an index two subgroup. Since H is isotropic for q_{K} ⊕ q_{K^{⊥}}, the restriction q_{K^{⊥}}|H takes value −\frac{1}{2d} (mod 2ℤ) on the generator. Let x ∈ H^{∗} be a nontrivial element projecting to 0 in d(K), which may be regarded as an element of d(K^{⊥}). We have q_{K^{⊥}}(x) = −\frac{1}{2} (mod 2ℤ) because x generates d(L). Since x annihilates H, it follows that d(K^{⊥}) = H + ℤx ≃ ℤ/2dℤ ⊕_{⊥} ℤ/2ℤ. This determines the discriminant form (and signature) of K^{⊥} completely, which determines it up to isomorphism [23] 1.14.2. The classification of the primitive imbeddings K ↪ L follows from [23] 1.15.1.

The proof in the second case is similar but easier, as d(K^{⊥}) is equal to H, so its discriminant form is easily computed.

If d = 0, we may produce an element e′ ∈ L with (v, e′) = 0, (e′, e′) = −2, and (e′, L) = 2ℤ. By our previous argument e′ ∈ Γe, so we may assume that w ∈ U^{⊕3} ⊕ (−E_{8})^{⊕2}. Then the result follows from [16] §2. □

When n = 2, we have an identity c_{2}(F) · v · v = 30 (v, v) (given in [13] 1.11 up to a multiplicative constant). Hence Riemann-Roch takes the form

χ(F, L(v)) = \frac{1}{8}((v, v) + 4)((v, v) + 6).

3 Conjectures

In this section (F, g) is a g-polarized irreducible holomorphic symplectic manifold, deformation equivalent to the Hilbert scheme of length-two subschemes of a K3 surface. Let E be the (possibly infinite!) set of classes ρ ∈ Pic_{+}(F, g) satisfying one of the following:

1. (ρ, ρ) = −2 and (ρ, L) = 2ℤ,

2. (ρ, ρ) = −2 and (ρ, L) = ℤ,

3. (ρ, ρ) = −10 and (ρ, L) = 2ℤ,

Let E^{∗} be the corresponding classes R ∈ H_{2}(F, ℤ); this means that for some ρ ∈ E we have

(v, ρ) = \begin{cases}Rv & where (ρ, L) = ℤ \\ 2Rv & where (ρ, L) = 2ℤ \end{cases}

for each v ∈ L. In particular, R satisfies one of the following

1. (R, R) = −\frac{1}{2},

2. (R, R) = −2,

3. (R, R) = −\frac{5}{2}.

Let N_{E}(F, g) ⊂ H_{2}(F, ℤ) be the smallest real cone containing E^{∗} and the elements R ∈ N_{1}(F) such that R · g > 0 and the corresonding ρ has nonnegative square. Note that the boundary of N_{E}(F, g) is polyhedral in a neighborhood of any boundary point with negative square. This follows from the fact that the Beauville form is negative definite on the orthogonal complement to g in Pic(F).

We now can state our main conjecture.

Conjecture 3.1 (Effective curves conjecture)

NE(F) = N_{E}(F, g).

The analogous theorem for K3 surfaces may be found in §1.6 and 1.7 of [16] (see also [4]).

The classes in E^{∗} that are extremal in (the closure of) N_{E}(F, g) will be called nodal classes, in analogy with the terminology for K3 surfaces (see [16], Section 1.4). The nodal classes are denoted E_{nod}^{∗} and the corresponding classes in E are denoted E_{nod}. Since the boundary of N_{E}(F, g) is polyhedral in a neighborhood of any nodal class, it follows that a class R ∈ E^{∗} is nodal iff no positive multiple of R is decomposable.

Remark 3.2 Consider the monoid N_{E}(F, g) ∩ H_{2}(F, ℤ). In contrast to the situation for K3 surfaces, it is possible for a (−2)-class in the monoid to be indecomposable but not nodal. Indeed, there exist examples with rk N_{1}(F) = 2.

Conjecture 3.1 and Proposition 2.1 yield a characterization of the ample monoid:

Λ_{amp}(F) = {λ ∈ Λ_{+}(F, g) : (λ, v) > 0 for each
v ∈ Pic_{+}(F, g) with (v, v) ≥ 0 or v ∈ E}.

The signature of the Beauville form implies that if (λ, λ) and (v, v) are both nonnegative then (λ, v) ≥ 0; indeed if λ and v are linearly independent then strict inequality follows. Furthermore, verifying the positivity condition against nonextremal classes is clearly redundant. We therefore obtain the following simplification:

Conjecture 3.3 (Consequence of Conjecture 3.1)

Λ_{amp}(F) = {λ ∈ Λ_{+}(F, g) : (λ, v) > 0 for each v ∈ E_{nod}}.

This generalizes Proposition 1.9 of [16].

We digress to consider the monoid of effective divisors. Its description is the same as for K3 surfaces.

Conjecture 3.4 (Effective divisors conjecture) The monoid of effective divisors is generated by the elements of Pic_{+}(F, g) with square ≥ −2.

Remark 3.5 A (−2)-class is extremal in the (conjectured) cone of effective divisors iff it is indecomposable in the (conjectured) monoid of effective divisors. We expect that each of these (−2)-classes is realized by a conic bundle over a K3 surface.

We next discuss classes λ ∈ Λ_{nef}(F), i.e., those in the boundary of the ample cone. Conjecture 2.2 implies that when (λ, λ) > 0 the sections of L(mλ) for m ≫ 0 give a birational morphism b : F →Y. Any curve represented by a nodal class R ∈ E_{nod}^{∗} orthogonal to λ is contracted by b.

Conjecture 3.6 (Nodal classes conjecture) Each nodal class R ∈ E_{nod}^{∗} represents a rational curve contracted by a birational morphism b given by sections of L(mλ), where λ is any class on the boundary of the ample cone satisfying λR = 0.

1. If (R, R) = −\frac{1}{2}, −2 (i.e., the corresponding ρ is a (−2)-class) then ρ is represented by a family of rational curves parametrized by a K3 surface. This family can be blown down to rational double points.

2. If (R, R) = −\frac{5}{2} (i.e., the corresponding ρ is a (−10)-class) then ρ is represented by a family of lines contained in a plane ℙ^{2}. This plane can be contracted to a point.

The following theorem of Namikawa provides support for this conjecture (part of the theorem was proved first by Shepherd-Barron [25] and related results were obtained by Wierzba [28]).

Theorem 3.7 ([22] Props. 1.1 and 1.4, [25]) Let b : F →Y be a birational projective morphism from a projective holomorphic symplectic manifold to a normal variety.

1. There exists a subvariety Z ⊂ Y of codimension at least four, so that Y \ Z is singular along a smooth codimension-two subvariety S, which admits a nondegenerate holomorphic two-form. Furthermore, Y \Z has rational double points of fixed type along each connected component of S.

2. Assume that P ∈ Y is an isolated Gorenstein singular point and that at least one irreducible component of G := b^{−1}(P) is normal. Then G ≃ ℙ^{n} with normal bundle Ω_{ℙ^{n}}^{1}.

A key tool in the study of K3 surfaces is the Weyl group, the group generated by reflections with respect to the (−2)-classes in the Picard group (see [4] or [16]). Our conjectures imply that there is an analog in higher dimensions. For each ρ ∈ E with square −2, we obtain a reflection s_{ρ} given by the formula

s_{ρ}(v) = v + (v, ρ) ρ.

The group W generated by these reflections is called the generalized Weyl group. Let C_{+}(F, g) be the smallest real cone containing Λ_{+}(F, g) and D(F, g) the subcone of C_{+}(F, g) defined by (v, ρ) ≥ 0 for each ρ ∈ E with square −2. This is a fundamental domain for the action of W on C_{+}(F, g). The (−10)-classes in E are walls for a subdivision of D(F, g) into subchambers. The interior of each subchamber is the ample cone for a symplectic birational model for F; the subchamber containing g is the ample cone of F. In other words, D(F, g) is the closure of the birational ample cone of F.

One does not generally expect that the closure of the ample cone should be a fundamental domain for the action of a group W′ ⊋ W. Algebraically, (−10)-classes do not yield reflections. Geometrically, one does not expect to find a group relating the various birational models of F. For a concrete example, see the discussion below of the Fano variety of lines on a cubic fourfold of discriminant eight.

We now describe the square-zero classes on the boundary of the closure of the ample cone:

Conjecture 3.8 Let λ be a primitive square-zero class on the boundary of the closure of the ample cone. Then the corresponding line bundle L = L(λ) has no higher cohomology and its sections yield a morphism

a : F →ℙ^{2}

whose generic fiber is an abelian surface.

The following theorem of Matsushita describes fiber space structures on holomorphic symplectic manifolds.

Theorem 3.9 ([19]) Let F be a projective irreducible holomorphic symplectic manifold and a : F →B be a fiber space structure with normal base B and generic fiber F_{b}. Then we have:

1. a is equidimensional in codimension-two points of B.

2. K_{F_{b}} is trivial and there exists an abelian variety F̃_{b} and an étale morphism F̃_{b}→F_{b}.

3. dim(B) = n and B has ℚ-factorial log-terminal singularities.

4. −K_{B} is ample and B has Picard number 1.

5. The polarization on B pulls back to a square-zero divisor on F.

Furthermore, if F has dimension four then F_{b} is an abelian surface.

Matsushita has also proved that such fibrations are Lagrangian [20]. For convenience, we provide a proof in our special case:

Proposition 3.10 In addition to the hypotheses of Theorem 3.9 assume that the dimension of F is four. Then the fibers F_{b} are Lagrangian.

Proof. It suffices to prove this result for smooth fibers. Recall the formula for the Beauville form (see [13], 1.9)

(α, α) = ∫ (σ\overline{σ})α^{2} − (∫ σ\overline{σ}^{2}α) (∫ σ^{2}\overline{σ}α) ,

where σ is the generator of H^{0}(F, Ω_{F}^{2} ), normalized so that ∫ (σ\overline{σ})^{2} = 1. If α is the pullback of the polarization on B then the class of F_{b} is equal to some rational multiple cα^{2}. Since α is of type (1, 1), type considerations imply that the second term vanishes. On the other hand, the first term is equal to

\frac{1}{c} ∫_{F_{b}} σ|_{F_{b}} \overline{σ}|_{F_{b}}.

Since (α, α) is zero by the previous theorem it follows that σ restricted to F_{b} is also zero and therefore F_{b} is Lagrangian (here we use the Hodge-Riemann bilinear relations). □

Remark 3.11 It is instructive to compare Conjecture 3.8 with Wilson’s results. Let F be a Calabi-Yau threefold and D a nef divisor class corresponding to a nonsingular point of the cubic hypersurface μ_{F} = 0 and satisfying Dc_{2}(F) ≠ 0. Then for some n > 0 the linear series |nD| is free and induces an elliptic fiber space structure on F (see [29] §3 and [30] §1). The assumption c_{2}(F)D ≠ 0 can be weakened under further technical hypotheses (see [31] and [32]).

Remark 3.12 Let F be an irreducible holomorphic symplectic manifold and λ ∈ Λ_{nef}(F) a divisor with (λ, λ) = 0. Is λ necessarily semiample?

4 Deformation theory

4.1 Deformations of subvarieties

In this section, we work with arbitrary irreducible holomorphic symplectic manifolds.

Theorem 4.1 Let F be an irreducible holomorphic symplectic manifold of dimension 2n and Y a submanifold of dimension k. Assume either that Y is Lagrangian, or that all of the following hold: N_{Y/F} = Ω_{Y}^{1} ⊕ O_{Y}^{⊕2n−2k}, the restriction of the symplectic form to Y is zero, and H^{1}(O_{Y}) = 0. Then the deformation space of Y in F is smooth of dimension 2n − 2k.

The deformations of F arising from deformations of the pair (Y, F) are precisely those preserving the sub-Hodge structure

ker(H^{∗,∗}(F) → H^{∗,∗}(Y)).

Proof. This is very similar to the proof of Corollary 3.4 in [24] (see [27] for the Lagrangian case). By [24], Corollary 3.2, any obstructions to deforming Y in F lie in the kernel of the natural projection

π_{0,2} : H^{1}(N_{Y/F}) → H^{1}(Ω_{Y}^{1} ) ⊗ ker(H^{0}(Ω_{F}^{2} ) → H^{0}(Ω_{Y}^{2} ))^{∗}.

Under our hypotheses this map is an isomorphism. □

Corollary 4.2 Keep the hypotheses of Theorem 4.1. Assume furthermore that the cohomology of Y is generated by divisor classes. Then the deformations of F arising from a deformation of the pair (Y, F) are precisely those for which the image of

H_{2}(Y, ℤ) → H_{2}(F, ℤ)

remains algebraic.

4.2 Applications

In this section we will assume that F is deformation equivalent to S^{[2]} for a K3 surface S. We show that the locus where our conjectures hold is open (in the analytic topology) in the moduli space. We will find points in the moduli space where our conjectures hold in subsequent sections.

Theorem 4.3 Let F be as above and R a nodal class on F. Assume that Conjecture 3.6 holds for R and F. Let F′ be a small projective deformation of F such that R deforms to a class R′ of type (3, 3). Then the Conjecture remains true for F′ and R′.

Proof. Let π : F→Δ be a deformation of F over a disc and A a nef and big line bundle on F such that A has degree zero on R. By Corollary 4.2, R deforms to a family of rational curves R ⊂ F over Δ. In each fiber R_{t} itself deforms in a two parameter family. This family is parametrized by a deformation S_{t} of the K3 surface S if ρ is a (−2)-class. It is parametrized by a ℙ^{2} if ρ is a (−10)-class. Clearly, these are contracted by the sections of some power of A. □

Theorem 4.4 Let F be as above and λ be a nef, square-zero divisor class on F. Assume that Conjecture 3.8 holds for λ and F. Let F′ be a small projective deformation of F such that λ deforms to a divisor class λ′. Then the Conjecture remains true for F′ and λ′.

Proof. This is a consequence of Theorem 4.1 and the Lagrangian property proved in 3.10. □

4.3 Examples

In this section we give examples of submanifolds satisfying the conditions of Theorem 4.1. We also refer the reader to [27] for examples in the Lagrangian case. Assume that Y is a complete homogeneous space under a reductive algebraic group or a toric variety, and assume that the normal bundle to Y is of the form stated above. Theorem 4.1 shows that the deformation space of Y in F is smooth. Moreover, the locus in the deformation space of F corresponding to manifolds containing a deformation of Y has codimension equal to the rank of the Néron-Severi group of Y. Here are some specific examples:

Example 4.5 Given Y = ℙ^{n} ⊂ F, the deformations of F containing ℙ^{n} form a divisor in the deformation space. For instance, if S is a K3 surface containing a smooth rational curve C then C^{[n]} ≃ ℙ^{n} ⊂ S^{[n]}. Let R ∈ H_{2}(F, ℤ) be the class of a line in ℙ^{n} and ρ ∈ H^{2}(F, ℤ) the corresponding divisor (i.e. 2Rv = (ρ, v) for v ∈ H^{2}(F, ℤ)). Then ρ = 2C − e and (ρ, ρ) = −2(n + 3).

Example 4.6 Again, let S be a K3 surface containing a smooth rational curve C and F = S^{[n]}. Consider the subschemes of S of length n with some support along C. The generic such subscheme is the union of a point of C and a length n − 1 subscheme disjoint from C. Thus we get a divisor D_{1} ⊂ S^{[n]} birational to a ℙ^{1} bundle over S^{[n−1]}. The normal bundle to a generic fiber Y of this bundle is Ω_{Y}^{1} ⊕ O_{Y}^{⊕2n−2}. Let R = [Y] ∈ H_{2}(F, ℤ) and ρ = [D_{1}] ∈ H^{2}(F, ℤ); note that R corresponds to ρ and (ρ, ρ) = (C · C)_{S} = −2. The deformations of S^{[n]} containing a deformation of D_{1} are those for which R (or ρ) remains algebraic. They have codimension one in the deformation space.

The relevance of these examples to Conjecture 3.6 is discussed in Section 5.4. We will give further examples in Section 6, where we consider cases where F is deformation equivalent to S^{[2]} and Y = 𝔽_{0}, 𝔽_{1}, or 𝔽_{4}. We digress to give one further example that is particularly interesting:

Example 4.7 We give a geometric realization of certain Mukai isogenies between K3 surfaces (cf. [21]). Let S_{8} be a generic degree 8 K3 surface. In particular, we assume S_{8} is realized as a complete intersection of 3 quadric hypersurfaces in ℙ^{5}, and the discriminant curve for these quadrics is a smooth sextic plane curve B. It follows that each such quadric Q has rank five or six, and the corresponding family of maximal isotropic subspaces in Q is parametrized by ℙ^{3} or a disjoint union two copies of ℙ^{3} respectively. As we vary Q, the families of maximal isotropic subspaces are parametrized by a K3 surface S_{2} of degree 2, the double cover of ℙ^{2} branched over B. Thus we obtain an étale ℙ^{3}-bundle E→S_{2}, mapping into S_{8}^{[4]}, with fibers satisfying the conditions of Proposition 4.1.

This yields an elegant universal construction of Brauer-Severi varieties representing certain 2-torsion elements of the Brauer group of a degree two K3 surface. Other 2-torsion elements are realized as étale ℙ^{1}-bundles E→S_{2} arising from families of nodal rational curves (see the discussion of cubic fourfolds of discriminant 8 in Examples 7.5 and 7.11). The relationship between Mukai isogenies and Brauer groups is explored more systematically in the upcoming thesis of Caldararu [8].

5 Symmetric squares of K3 surfaces

Let S_{2n} be a K3 surface with Picard group generated by a polarization f_{2n} of degree 2n. The Beauville form restricted to the Picard group takes the form

\begin{aligned}& f_{2n} & e \\ f_{2n} & 2n & 0 \\ e & 0 & −2 \end{aligned}

The effective divisor with class 2e is called the diagonal. It is isomorphic to a ℙ^{1}-bundle over S_{2n}; the fibers are nodal rational curves. It follows that an ample line bundle has class of the form xf_{2n} − ye with x, y > 0 and Proposition 2.1 implies that 2nx^{2} − 2y^{2} > 0. The conjectures in Section 3 give sufficient conditions on x and y for xf_{2n} − ye to be ample.

Proposition 5.1 Assume that S_{2n} is a K3 surface with a polarization f_{2n} which embeds S_{2n} as a subvariety of ℙ^{n+1}. The line bundle af_{2n} − e on S_{2n}^{[2]} is ample whenever a > 1 or a = 1 and S_{2n} does not contain a line. In particular, f_{2n} lies on the boundary of the closure of the ample cone.

Proof. Let S be a smooth surface embedded in projective space ℙ^{r} and not containing a line. Then there is a morphism from the Hilbert scheme S^{[2]} to the Grassmannian Gr(2, r). This morphism is finite onto its image. Therefore, the pullback of the polarization on the Grassmannian to S^{[2]} is ample. We apply this to the image of S_{2n} under the line bundle af_{2n}. □

Remark 5.2 In the event that S_{2n} does contain a line ℓ ⊂ ℙ^{n+1} the line bundle f_{2n} − e fails to be ample. However, it is nef and big and a sufficiently high multiple of it gives a birational morphism contracting the plane ℓ^{[2]} and inducing an isomorphism on the complement to this plane. In particular, there is a nodal (−10)-class 2[ℓ] − e orthogonal to f_{2n} − e.

5.1 Degree 2 K3 surfaces

A K3 surface S_{2} of degree two can be realized as a double cover of ℙ^{2} ramified in a curve of degree 6. The quadratic form 2x^{2} − 2y^{2} does represent −2 and −10. The corresponding nodal classes are e and 2f_{2} − 3e. The second class corresponds to the plane in S_{2}^{[2]} arising from the double cover. Our conjectures predict that the ample cone consists of classes xf_{2} − ye where x, y > 0 and 2x − 3y > 0. The quadratic form also represents 0, but the corresponding class f_{2} − e satisfies (2f_{2} − 3e, f_{2} − e) = −2. After flopping the plane the proper transform of f_{2} − e does yield an abelian surface fibration (the Jacobian fibration) - as expected.

Here is a sketch proof that the class 3f_{2} − 2e is nef and big. Indeed, 3f_{2} is very ample and embeds S_{2} into ℙ^{10}. The image is cut out by quadrics I(2). Each pair of points on S_{2} determines a line ℓ. The quadrics vanishing on that line form a hyperplane in I(2). This induces a morphism from S_{2}^{[2]} to ℙ^{27} = ℙ(I(2)^{∗}) given by the sections of the line bundle.

5.2 Degree 4 K3 surfaces

Let S_{4} be a K3 surface with Picard group generated by a polarization of degree 4. We take f_{4} − e as the polarization of S_{4}^{[2]}. Now we describe the (−2) and (−10)-lattice vectors in ℤf_{4} ⊕ ℤe and determine which are nodal classes. In fact, there are no (−10)-lattice vectors. The (−2)-vectors are of the form ±a_{m}f_{4} ∓ b_{m}e, where a_{m}\sqrt{2} + b_{m} = (2\sqrt{2} + 3)^{m}. The vectors in the positive halfspace Pic_{+}(S_{4}^{[2]}, f_{4} − e) satisfy 2x − y > 0. The nodal classes are 2f_{4} − 3e and e. It is easy to see that all the other (−2)-classes in the positive halfspace are decomposable. We therefore predict that the ample cone is the interior of the cone spanned by f_{4} and 3f_{4} − 4e. Indeed, S_{4}^{[2]} has an involution exchanging f_{4} and 3f_{4} − 4e (given p, q ∈ S_{4} the line spanned by p and q meets S_{4} in two more points).

5.3 Degree 8 K3 surfaces

Let S_{8} be a K3 surface with Picard group generated by a polarization of degree 8. This is the smallest degree case where there are no nodal classes besides the diagonal. Indeed, the quadratic form 8x^{2} − 2y^{2} represents −2 and −10 exactly when (x, y) = (0, ±1) and (±1, ±3). However, the parity condition for nodal classes of square −10 is not satisfied by f_{8} − 3e, i.e., (f_{8} − 3e, H^{2}(F, ℤ)) ≠ 2ℤ. Therefore, our conjectures imply that the ample cone is the interior of the cone spanned by f_{8} and f_{8} −2e, and the second line bundle yields an abelian surface fibration a : F →ℙ^{2}. We have already seen that the ample cone is contained in this cone. For an explicit construction of the abelian surface fibration, see [12] §7. There it is shown that the symmetric square of a generic K3 surface of degree 2n^{2} (n > 1) admits an abelian surface fibration.

Remark 5.3 This is a counterexample to the theorem in Section 2, p. 463 of [18]. There it is claimed that S^{[2]} of a K3 surface S admits a (Lagrangian) abelian surface fibration if and only if S is elliptic.

5.4 K3 surfaces containing a rational curve

Let S be a K3 surface containing a rational curve C and let T be the surface obtained by blowing down C. Of course, T has one rational double point. Consider the map b : S^{[2]}→Sym^{2}(T). This map contracts rational curves corresponding to both (−2) and (−10)-nodal classes.

The Hilbert scheme S^{[2]} contains a plane C^{[2]} and two distinguished divisors D_{1} and D_{2}, birational to ℙ^{1}-bundles over S. The divisor D_{1} is the locus of subschemes with some support in C and D_{2} is the diagonal. The map b contracts D_{1} and D_{2} to surfaces isomorphic to T and C^{[2]} to the point p where these surfaces intersect. The fiber b^{−1}(p) is the union of C^{[2]} and 𝔽_{4} (cf. Theorem 3.7).

The divisors D_{1} and D_{2} have classes C and 2e respectively. If R is the class of a line in C^{[2]} ≃ ℙ^{2} then the corresponding divisor class ρ = 2C − e. The (−2)-class e and the (−10)-class ρ are nodal; the class C is not nodal. Of course, it becomes nodal upon flopping C^{[2]} ⊂ S^{[2]}.

6 Nonnodal smooth rational curves

It is well known that for K3 surfaces all smooth rational curves are nodal and correspond to indecomposable (-2)-classes. In Section 5.4 we gave examples of nonnodal smooth rational curves; these curves were parametrized by a K3 surface. Here we discuss further examples of nonnodal smooth rational curves. As we shall see, these curves need not be parametrized by a K3 surface or a ℙ^{2}.

We first consider three examples where smooth rational curves do not correspond to nodal classes, but still correspond to classes with negative square. Let F = S^{[2]}, where S is a K3 surface which is a double cover of a rational surface Σ with Picard group of rank 2. Then F contains a surface isomorphic to Σ. We emphasize that the results of Subsection 4.1 apply in this case. This suggests certain refinements to Conjecture 3.6, which we formulate in each example.

Example 6.1 Let S→Σ = 𝔽_{0} be branched over a general curve of type (4, 4). Hence the rulings induce two elliptic fibrations E_{1} and E_{2} which generate the Picard group and intersect as follows:

\begin{aligned}& E_{1} & E_{2} \\ E_{1} & 0 & 2 \\ E_{2} & 2 & 0 \end{aligned} .

Let R_{1} and R_{2} denote the rulings of Σ ⊂ S^{[2]}, with ρ_{1} and ρ_{2} their Poincaré duals in Pic(S^{[2]}). We have ρ_{1} = E_{1} −e and ρ_{2} = E_{2} −e so that the Beauville form may be written

\begin{aligned}& ρ_{1} & ρ_{2} \\ ρ_{1} & −2 & 0 \\ ρ_{2} & 0 & −2 \end{aligned} .

Moreover, the ρ_{i} generate a saturated sublattice of the Picard group and

(ρ_{i}, H^{2}(F, ℤ)) = ℤ.

The smooth curves in the class R_{1} + R_{2} move in a 3-parameter family on Σ ⊂ F. However, ρ_{1} + ρ_{2} is not a nodal class. We conjecture that any holomorphic symplectic fourfold deformation equivalent to a symmetric square of a K3 surface with 2 nodal classes ρ_{1} and ρ_{2} as above should contain a surface Σ = 𝔽_{0}.

Example 6.2 Let S→Σ = 𝔽_{1} be branched over a general curve of type 6R_{0} + 4R_{−1}, where R_{0} is the class of the ruling and R_{−1} is the class of the exceptional curve. The ruling induces an elliptic fibration E on S and the exceptional curve yields a rational curve C ⊂ S; these generate the Picard group and intersect as follows:

\begin{aligned}& E & C \\ E & 0 & 2 \\ C & 2 & −2 \end{aligned} .

Let ρ_{0} (resp. ρ_{−1}) be the Poincaré dual to R_{0} (resp. 2R_{−1}). We have ρ_{0} = E − e and ρ_{−1} = 2C − e, so that ρ_{0} and ρ_{−1} generate a saturated sublattice on which the Beauville form may be written

\begin{aligned}& ρ_{0} & ρ_{−1} \\ ρ_{0} & −2 & 2 \\ ρ_{−1} & 2 & −10 \end{aligned} .

Moreover, (ρ_{0}, H^{2}(F, ℤ)) = ℤ and (ρ_{−1}, H^{2}(F, ℤ)) = 2ℤ.

The smooth curves in the class 2R_{0}+R_{−1} move in a 4-parameter family on Σ ⊂ F. However, 4ρ_{0}+ρ_{−1} is not a nodal class. In this case we conjecture that any F whose cohomology contains 2 nodal classes ρ_{0} and ρ_{−1} as above should contain a surface Σ = 𝔽_{1}. Furthermore, we expect that F is a specialization of a variety containing a plane Π which corresponds to a (−10)-nodal class. This class is equal to 2ρ_{0} + ρ_{−1} and Π specializes to a union of a ℙ^{2} and the 𝔽_{1} in F.

Example 6.3 Let S→Σ = 𝔽_{4} be branched over the union of a general curve of type 12R_{0} + 3R_{−4} and R_{−4}, where R_{0} is the class of the ruling and R_{−4} is the class of the exceptional curve. Again, the Picard group of S is generated by an elliptic fibration S and a rational curve C which intersect as follows

\begin{aligned}& E & C \\ E & 0 & 1 \\ C & 1 & −2 \end{aligned} .

Let ρ_{0} (resp. ρ_{−4}) be the Poincaré dual to R_{0} (resp. R_{−4}). Then we have ρ_{0} = E − e and ρ_{−4} = 2C + e, so ρ_{0} and ρ_{−4} generate a saturated sublattice with Beauville form

\begin{aligned}& ρ_{0} & ρ_{−4} \\ ρ_{0} & −2 & 4 \\ ρ_{−4} & 4 & −10 \end{aligned} .

Moreover, (ρ_{0}, H^{2}(F, ℤ)) = ℤ and (ρ_{−4}, H^{2}(F, ℤ)) = 2ℤ.

The smooth curves in the class 5R_{0} + R_{−4} move in a 7-parameter family on Σ ⊂ F. However, 5ρ_{0} +ρ_{−4} is not a nodal class. In this case we conjecture that any F with cohomology containing 2 nodal classes ρ_{0} and ρ_{−4} as above should contain a surface Σ = 𝔽_{4}. We also expect that F is a specialization of a variety containing a plane Π which corresponds to a (−10)-nodal class. This class is equal to 4ρ_{0} + ρ_{−4} and Π specializes to a union of a ℙ^{2} and the 𝔽_{4} in F.

Next we consider examples of smooth rational curves in F where the corresponding class ρ is of positive square.

Example 6.4 Let S_{2} be a general K3 surface of degree 2 with polarization f_{2}. Let C ⊂ S_{2} be a rational curve with two ordinary double points contained in the linear series |f_{2}|. Let F →ℙ^{2} be the compactified Jacobian for |f_{2}|. The fibers corresponding to C are isomorphic to a product of nodal curves with normalization ℙ^{1} × ℙ^{1}. Smooth curves of type (1, 1) in ℙ^{1} × ℙ^{1} yield smooth rational curves on F, deforming in a 3-parameter family. The homology class of these rational curves is double the class of the curve of type (1, 0), and is therefore not primitive.

Example 6.5 Let S_{4} ⊂ ℙ^{3} be a general K3 surface of degree 4 and f_{4} its polarization. Let C ⊂ S_{4} be an elliptic curve in |f_{4}| with two ordinary double points. Note that C^{[2]} is a nonnormal ruled surface. Its fibers are smooth rational curves such that the corresponding class ρ has square 2. Thus we get smooth rational curves in primitive homology classes such that the corresponding class ρ has positive square as well.

As the rank of the Picard group of F increases we expect more and more examples of nonnodal smooth rational curves parametrized by varieties of dimension > 2.

Remark 6.6 Let R ⊂ F be a smooth rational curve with primitive homology class. Then the Hilbert scheme of flat deformations of R need not be irreducible and may have arbitrarily large dimension. Take R ⊂ 𝔽_{0} ⊂ F of bidegree (1, n).

Question 6.7 Assume that rk Pic(F) = 1. Does there exist a smooth rational curve on F? Can we take its class to be primitive?

7 Cubic fourfolds

In this section, a cubic fourfold generally denotes a smooth cubic hypersurface X ⊂ ℙ^{5}. The variety F parametrizing lines on X is sometimes called the ‘Fano variety of lines’ - not to be confused with a variety with ample anticanonical class. It is known that F is an irreducible holomorphic symplectic fourfold deformation equivalent to the Hilbert scheme of length-two subschemes of a K3 surface [1] [5]. Consequently, the conjectures of Section 3 apply. The existence of smooth rational curves R ⊂ F translates into the existence of scrolls T ⊂ X. By definition, a scroll is the union of the lines parametrized by a smooth rational curve in the Grassmannian; it may have singularities. Our conjectures yield simple and verifiable predictions for the existence and nonexistence of scrolls in various homology classes of X. The presence of these scrolls yields unirational parametrizations of X of various degrees.

7.1 Lattices, Nodal Curves, and Scrolls

We recall standard facts about cubic fourfolds. We say that a cubic fourfold is special if it contains an algebraic surface not homologous to any multiple of the square of the hyperplane class h^{2}. Note that the intersection form ⟨, ⟩ on the primitive cohomology takes the form

(h^{2})^{⊥} ≃ ( \begin{matrix}2 & 1 \\ 1 & 2 \end{matrix} ) ⊕_{⊥} U^{⊕2} ⊕_{⊥} E_{8}^{⊕2}

(see [10], [9], [5]). Let K = ℤh^{2} + ℤT be a saturated sublattice of algebraic classes in the middle cohomology of X. Then the discriminant d = d(X, K) is the discriminant of K. It is a positive integer, congruent to 0 or 2 modulo 6. The special cubic fourfolds of discriminant d form an irreducible divisor C_{d} in the moduli space C of cubic fourfolds; C_{d} is nonempty iff d > 6. For instance, C_{8} corresponds to the cubic fourfolds containing a plane T_{1} and C_{14} corresponds to the cubic fourfolds containing a smooth quartic scroll T_{4}.

The cohomology of a cubic fourfold and its Fano variety are closely related (see [5] for most of what follows). The incidence correspondence between X and F induces the Abel-Jacobi map

α : H^{4}(X, ℤ)→H^{2}(F, ℤ),

respecting the Hodge structures. We have that (α(h^{2}), α(h^{2})) = 2 ⟨h^{2}, h^{2}⟩ and

(α(v), α(w)) = − ⟨v, w⟩

for v, w primitive. Note that g := α(h^{2}) is the polarization on F induced from the Grassmannian. The incidence correspondence induces a second map

β : H^{6}(F, ℤ)→H^{4}(X, ℤ)

respecting the Hodge structures. We can compose to obtain

ψ : H_{2}(F, ℤ)→H^{6}(F, ℤ) →^{β} H^{4}(X, ℤ) →^{α} H^{2}(F, ℤ)→H_{2}(F, ℤ),

where the first map is Poincaré duality and the last map is induced by the Beauville form. We have ψ(g) = 2g and ψ(v) = −v for v orthogonal to g.

Suppose that F contains a smooth rational curve R of degree n. Let T̃ be the universal line restricted to R and T ⊂ X the corresponding scroll sweeped out by R, which also has degree n. Note that the formula ⟨T, Σ⟩ = R · α(Σ) (for Σ ∈ H^{4}(X, ℤ)) follows from the incidence correspondence. Combining this with our computation of ψ, we obtain

⟨T, T ⟩ = R · α(T) = (R, ψ(R)) = \frac{n^{2}}{2} − (R, R) .

We use T_{n,Δ} to denote a scroll T of degree n for which the map

T̃→T ⊂ X

has singularities equivalent to Δ ordinary double points (by definition, Δ is the number given by the double point formula). A Chern class computation gives

⟨T_{n,Δ}, T_{n,Δ}⟩ = 3n − 2 + 2Δ  (1)

and we obtain the formula

Δ = \frac{1}{4}(n^{2} − 6n + 4 − 2 (R, R)).  (2)

The lattice generated by h^{2} and T_{n,Δ} has discriminant

d(n, Δ) = 3(3n − 2 + 2Δ) − n^{2} = 6Δ − (n^{2} − 9n + 6)
= \frac{n^{2}}{2} − 3 (R, R) .

This lattice has discriminant > 6, so we obtain the lower bound

Δ ≥ Δ_{min}(n) := ⌈\frac{1}{6}(n^{2} − 9n + 6) + 1⌉.

In particular, a cubic fourfold cannot contain smooth scrolls of degree > 7.

Remark 7.1 The lattice ℤh^{2} + ℤT_{n,Δ} need not be saturated. For instance, if n = 8 and Δ = 5 then d(8, 5) = 32. However, the lattice generated by h^{2} and T_{8,5} has index 2 in its saturation.

Proposition 7.2 Let X be a cubic fourfold, with Fano variety F. Let R ⊂ F be a nodal rational curve and T_{n,Δ} the corresponding scroll. Then Δ takes the following values:

Δ = \begin{cases}(m − 2)(m − 1) if n = 2m; \\ (m − 1)^{2} and m(m − 2) if n = 2m + 1. \end{cases}

Proof. This is a consequence of Equations 1 and 2 above. We observe that n is even when (R, R) = −2 and n is odd when (R, R) = −\frac{1}{2} or −\frac{5}{2}. □

We summarize the numerical predictions for nodal scrolls of small degree in the following table:

\begin{matrix}n & 2 & 3 & 4 & 5 & 5 & 6 & 7 & 7 & 8 & 9 & 9 & 10 & 11 & 11 \\ Δ & 0 & 0 & 0 & 0 & 1 & 2 & 3 & 4 & 6 & 8 & 9 & 12 & 15 & 16 \\ d(n, Δ) & 8 & 12 & 14 & 14 & 20 & 24 & 26 & 32 & 38 & 42 & 48 & 56 & 62 & 68 \end{matrix}

Remark 7.3 We can obtain cubic fourfolds containing scrolls T_{n,Δ} with more double points by exploiting nonnodal smooth rational curves on the corresponding Fano variety (see Examples 7.14 and 7.21).

7.2 Unirational parametrizations

We start with a classical example: if X is a cubic fourfold containing a smooth quartic scroll T_{4,0} then X is rational. One would like to generalize this construction to other special cubic fourfolds.

Proposition 7.4 Let X be a cubic fourfold with Fano variety F. Assume that F contains a smooth rational curve R of degree n, with corresponding scroll T_{n,Δ}. Assume that this corresponding scroll T is not a cone. Then there exists a rational map

ϕ : ℙ^{4} −−→ X

with

deg(ϕ) = \binom{n − 2}{2} − Δ = \frac{(n − 2)^{2}}{4} + \frac{(R, R)}{2} + 1.

Proof. Our assumptions imply that R parametrizes pairwise disjoint lines in X. Given generic ℓ_{1}, ℓ_{2}, the cubic surface

Span(ℓ_{1}, ℓ_{2}) ∩ X

contains two disjoint lines and thus is rational. We therefore obtain a cubic surface bundle

Y →^{σ} Sym^{2}(R) ≃ ℙ^{2}
ϕ↓
X

so that the fiber over the generic point contains two disjoint lines. Consequently, Y is rational over ℙ^{2} and thus is a rational variety.

To compute the degree of ϕ, it suffices to compute the number of double points arising from a generic projection of the scroll T into ℙ^{4}. The map T̃→ℙ^{4} has singularities equivalent to \binom{n−2}{2} double points; Δ of these are from the singularities of T. We obtain the second formula for deg(ϕ) by applying the Equation 2 of Section 7.1. □

This demonstrates that the existence of rational curves in certain homology classes of F implies that X is rational. Unfortunately, our conjectures indicate that such rational curves are quite rare. If R is nodal then (R, R) ≥ −\frac{5}{2}, so deg(ϕ) = 1 only when n = 4 (see also Examples 7.9 and 7.17).

However, we do obtain some interesting new unirational parametrizations of cubic fourfolds. Recall that in [10], the Fano variety of lines on the generic cubic fourfold of discriminant 2(N^{2} + N + 1) (N > 1) was shown to be isomorphic to S^{[2]} of a K3 surface S. In particular, it contains nodal rational curves R of degree 2N + 1 with (R, R) = −\frac{1}{2}. One can show that the scroll corresponding to a generic such curve is not a cone, hence Proposition 7.4 applies. We obtain

deg(ϕ) = N^{2} − N + 1,

which is always odd. In particular, the cubic fourfolds with odd degree unirational parametrizations are dense in the moduli space.

Cubic fourfolds are known to admit unirational parametrizations of degree two. Thus the cubic fourfolds described above admit unirational parametrizations of relatively prime degrees. There are few examples of irrational varieties with this property. Many common invariants used to detect irrationality (like the unramified cohomology of the function field) vanish in this situation.

7.3 Cubic fourfolds of small discriminant

In this section we specialize our conjectures to Fano varieties of lines on general special cubic fourfolds of discriminant d. We obtain predictions on the existence and nonexistence of scrolls T_{n,Δ} on X_{d} ∈ C_{d}. We verify these predictions in Section 7.4. Throughout we write g = α(h^{2}) and τ = α(T).

Example 7.5 (d = 8) For X_{8} ∈ C_{8} (resp. F_{8}) we have intersection pairing (resp. Beauville form):

\begin{aligned}& h^{2} & T \\ h^{2} & 3 & 1 \\ T & 1 & 3 \end{aligned} \begin{aligned}& g & τ \\ g & 6 & 2 \\ τ & 2 & −2 \end{aligned} ,

so τ is a (−2)-class (note that (τ, H^{2}(F_{8}, ℤ)) = ℤ.) There is also a (−10)- class: ρ = g − 2τ. One can check that these classes are nodal. Therefore our conjectures predict a plane in Π ⊂ F_{8} whose lines have degree one in the Grassmannian. This corresponds to a plane in X_{8}. They also predict a family of rational curves in F_{8} parametrized by a K3 surface which correspond to quadric cones in X_{8} (see Example 7.11).

This example illustrates our previous discussion concerning the action of the Weyl group. Here we have

C_{+}(F_{8}, g) = {ag − bτ : 3a + b > 0, a − b > 0}

and the fundamental domain for the action of the Weyl group is

D(F_{8}, g) = {ag − bτ : a + b ≥ 0, a − b ≥ 0}.

The conjectures predict that the ample cone should be

Λ_{amp}(F_{8}) = {ag − bτ : a + b > 0, a − 3b > 0};

the nef cone is bounded by clases of square 0 and 64. If F′ denotes the elementary transformation of F_{8} along the plane Π, we expect

Λ_{amp}(F′) = {ag − bτ : a − b > 0, −a + 3b > 0};

the nef cone is bounded by classes of square 8 and 64. In particular, the two subchambers of D(F_{8}, g) are not conjugate.

Example 7.6 (d = 12) For X_{12} and F_{12} we have pairings:

\begin{aligned}& h^{2} & T \\ h^{2} & 3 & 3 \\ T & 3 & 7 \end{aligned} \begin{aligned}& g & τ \\ g & 6 & 6 \\ τ & 6 & 2 \end{aligned} .

The (−10)-classes are given by 2τ − g and 3g − 2τ. Our conjectures predict that F_{12} contains two projective planes. The lines on these planes correspond to families of cubic scrolls on X_{12} (see Example 7.12).

Example 7.7 (d = 14) For X_{14} and F_{14} we have pairings:

\begin{aligned}& h^{2} & T \\ h^{2} & 3 & 4 \\ T & 4 & 10 \end{aligned} \begin{aligned}& g & τ \\ g & 6 & 8 \\ τ & 8 & 6 \end{aligned} .

The nodal classes classes are given by 2g − τ and 2τ − g. Note that (2g − τ, H^{2}(F_{14}, ℤ)) = ℤ and (2τ − g, H^{2}(F_{14}, ℤ)) = 2ℤ. The first corresponds to a family of rational curves of degree 4 on F_{14} parametrized by a K3 surface. The second corresponds to a family of rational curves of degree 5 also parametrized by a K3 surface (see Examples 7.13 and 7.15).

Example 7.8 (d = 20) For X_{20} and F_{20} we have pairings

\begin{aligned}& h^{2} & V \\ h^{2} & 3 & 4 \\ V & 4 & 12 \end{aligned} \begin{aligned}& g & v \\ g & 6 & 8 \\ v & 8 & 4 \end{aligned} ,

where v = α(V). There are no (−2)-classes but there are two nodal (−10)- classes: e_{1} = 2v − g, e_{2} = 19g − 8v. The corresponding rational curves on F_{20} have degrees 5 and 25, respectively (cf. Example 7.16). There is an involution interchanging e_{1} and e_{2} given by:

\begin{matrix}g & ↦ & 5g − 2v \\ v & ↦ & 12g − 5v \end{matrix} .

Example 7.9 (d = 26) For X_{26} and F_{26} we have pairings

\begin{aligned}& h^{2} & T \\ h^{2} & 3 & 5 \\ T & 5 & 17 \end{aligned} \begin{aligned}& g & τ \\ g & 6 & 10 \\ τ & 10 & 8 \end{aligned} .

This lattice does not represent −10. The nodal (−2)-classes are 2τ − g and 109g − 38τ. Note that

(2τ − g, H^{2}(F_{26}, ℤ)) = (109g − 38τ, H^{2}(F_{26}, ℤ)) = 2ℤ.

Our conjecture predicts two families of rational curves parametrized by K3 surfaces, with degrees 7 and 137 respectively.

We next apply our conjecture on effective classes to derive the nonexistence of a T_{5,2} on X_{26}. By Proposition 7.4, the existence of such a surface would imply the rationality of X_{26}. Let us assume that T_{5,2} ⊂ X_{26} with ruling R. We may take T for the class of T_{5,2}. We compute the class ρ corresponding to R. Since

\frac{1}{2}ρ · g = 5 \frac{1}{2}ρ · τ = 17

we get ρ = 5g − 2τ. If we write

ρ = a(2τ − g) + b(109g − 38τ)

then we find that a = −7/45 and b = 2/45. This implies that R is not contained in the (conjectured) monoid of effective classes. We shall show in Example 7.17 that a quintic scroll in ℙ^{5} cannot have two double points.

Question 7.10 How can one systematize the argument for the nonexistence of T_{5,2}’s on a (general) X_{26}? More precisely, let X be a cubic fourfold containing a scroll T_{n,Δ} and assume that the lattice containing h^{2} and T_{n,Δ} generates the lattice of algebraic classes in H^{4}(X, ℤ). Do the values obtained in Proposition 7.2 give upper bounds for Δ in terms of n?

7.4 Data

In this section we present data from projective geometry concerning the existence of scrolls on cubic fourfolds. We organize the information by the degree of the scroll.

First of all, let us observe that a scroll of degree n with ordinary double points can be obtained by projecting a smooth nondegenerate scroll of degree n in ℙ^{n+1} from a suitable linear subspace.

Example 7.11 (T_{2,0}) Observe that a scroll of degree two cannot have ordinary double points at all. It is easy to see that the general cubic fourfold of discriminant 8 contains such a scroll. Furthermore, these scrolls are parametrized by a K3 surface of degree 2 (see [26], [10], [11]).

Example 7.12 (T_{3,0}) A scroll of degree 3 also does not have any ordinary double points and it is contained in a general cubic fourfold of discriminant 12. On a fixed cubic fourfold these scrolls are parametrized by two disjoint ℙ^{2}’s; given one scroll T, there is a residual scroll T′ obtained by intersecting a linear and a quadratic hypersurface containing T (see [9]). These correspond to two distinct (−10)-classes.

Example 7.13 (T_{4,0}) A nondegenerate scroll of degree 4 in ℙ^{5} does not have any ordinary double points. A general cubic fourfold of discriminant 14 contains a family of such scrolls, parametrized by a smooth K3 surface of degree 14. The corresponding class is a nodal (−2)-class (see, for example, [5], [9]).

Example 7.14 (T_{4,1}) This example is closely related to Example 6.1. We will explain why the locus of cubic fourfolds containing a quartic scroll with one ordinary double point has codimension 2 in moduli. Consider a cubic fourfold X containing such a scroll T_{4,1}. Note that T_{4,1} is degenerate and is contained in a singular cubic threefold Y. We specialize first to the case where the quartic scroll degenerates to the union of two quadric scrolls. Each of these quadric scrolls is residual to a plane, and these planes intersect at a single point. What can we say about Y in this case? A cubic threefold containing two such planes is obtained as follows. Let C be a genus 4 stable curve obtained by taking a curve C_{1} of type (1, 3) on a quadric surface Q, along with the union of two rulings C_{2} and C_{3} of type (1, 0). Note that C is canonically imbedded in ℙ^{3}. Then Y is the image of ℙ^{3} under the linear series |L| of cubics cutting out C. The planes in Y are the total transforms of the lines C_{2} and C_{3}. We claim that Y contains a family of scrolls T_{4,1}, parametrized by ℙ^{3}. In particular, a cubic fourfold containing two planes meeting at a point also contains a three parameter family of T_{4,1}’s. These are obtained by taking the proper transforms of the quadric surfaces Z in ℙ^{3} containing the lines C_{2} and C_{3}. These form a linear series with projective dimension three. The restriction of |L| to Z is a linear series of type (1, 3) with two base points (i.e., the points of Z ∩ C_{1} not lying on C_{2} or C_{3}). The image of Z is a T_{4,1}.

This corresponds to a situation where F contains a surface isomorphic to ℙ^{1} × ℙ^{1}. The hyperplane sections give a 3-parameter family of rational curves on F. As we have seen, such fourfolds should lie in codimension two (see 4.1).

Example 7.15 (T_{5,0}) The cubic fourfolds of discriminant 14 also contain a family of quintic scrolls, parametrized by the same K3 surface which parametrizes the quartic scrolls. The corresponding class is a second nodal (−2)-class (see [5] or [9]).

Example 7.16 (T_{5,1}) A general cubic fourfold X_{20} of discriminant 20 contains a family of T_{5,1}’s parametrized by a ℙ^{2}. It is known that X_{20} contains a Veronese surface V. This also follows from Theorem 4.1 once we obtain a ℙ^{2} ⊂ F_{20}. The conic curves in X_{20} lying in V are parametrized by ℙ^{2} as well. For each such curve C, let H be the plane spanned by C so that

X_{20} ∩ H = C ∪ ℓ

where ℓ is a line. This yields a subvariety of F_{20} isomorphic to ℙ^{2}; the lines R ⊂ ℙ^{2} trace out T_{5,1}’s on X_{20}. The corresponding ρ ∈ Pic(F_{20}) is a (−10)- class.

Example 7.17 (T_{5,2}) There are no quintic scrolls with two ordinary double points in ℙ^{5}. (This is highly unfortunate because a cubic fourfold containing such a scroll would be rational by Proposition 7.4.) Let T̃_{5,2} ⊂ ℙ^{6} be the normalization and p ∈ ℙ^{6} a point such that the projection of T̃_{5,2} from p is T_{5,2}. It follows that T̃_{5,2} contains four coplanar points. However, these points necessarily lie on a conic curve C ⊂ T̃_{5,2}. This forces T_{5,2} to be singular along the image of C.

Example 7.18 (T_{6,0}) This remains to be explored - the corresponding ρ is not nodal! The discriminant of the lattice ℤh^{2} + ℤT_{6,0} is 12.

Example 7.19 (T_{6,1}) In this discriminant (d = 18) the Fano variety has two square-zero classes (bounding the ample cone, by our conjectures). In particular, there are no nodal classes in this case.

Example 7.20 (T_{6,2}) The scroll T_{6,2} is contained in a general cubic fourfold X_{24} of discriminant 24. The family of such scrolls in a given cubic fourfold is parametrized by a K3 surface of degree 6. The corresponding class ρ is a nodal (−2)-class.

Example 7.21 (T_{6,3}) The cubic fourfolds X_{30} containing a sextic scroll with three ordinary double points are codimension 2 in moduli. The normalization T̃_{6,3} ⊂ ℙ^{7} has 6 points lying in a 4-dimensional linear subspace, containing the line ℓ from which we project. These points necessarily are contained in a rational normal curve C ⊂ T̃_{6,3} of degree 4. The image of C under projection is a quartic plane curve. This plane is necessarily contained in X_{30} by Bezout’s theorem, so H^{2,2}(X_{30}, ℤ) has rank at least 3. Let us remark that the Fano variety F_{30} contains a surface isomorphic to ℙ^{1} × ℙ^{1} and the rulings of the scrolls are given by type (1, 1)-curves of this surface (cf. the discussion of T_{4,1} and Example 6.1).

Example 7.22 (Further examples) Essentially the same argument shows that there are no scrolls T_{6,4} (or T_{6,5} or T_{7,5}): we look at the 8 points on the normalization T̃_{6,4} spanning a 5-dimensional linear subspace containing the line of projection ℓ. These points are necessarily contained on a rational normal curve of degree 5 on T̃_{6,4}. It projects to a quintic curve in ℙ^{3} with 4 ordinary double points. This violates Bezout.

References

[1] A. Altman, S. Kleiman, Foundations of the theory of Fano schemes, Compositio Math. 34 (1977), no. 1, 3–47.

[2] A. Beauville, Some remarks on Kähler manifolds with c_{1} = 0, Classification of algebraic and analytic manifolds (Katata, 1982), 1–26, Progr. Math., 39, Birkhäuser Boston, Boston, Mass., (1983).

[3] A. Beauville, Variétés Kähleriennes dont la premiére classe de Chern est nulle, J. Differential Geom. 18 (1983), no. 4, 755–782.

[4] A. Beauville, Préliminaires sur les périodes des surfaces K3, Géométrie des surfaces K3: modules et périodes (Exposé VII) (Palaiseau, 1981/1982), Astérisque No. 126 (1985), 91–97.

[5] A. Beauville, R. Donagi, La variété des droites d’une hypersurface cubique de dimension 4, C. R. Acad. Sci. Paris Sèr. I Math. 301 (1985), no. 14, 703–706.

[6] F. A. Bogomolov, The decomposition of Kähler manifolds with a trivial canonical class, (Russian) Mat. Sb. (N.S.) 93(135) (1974), 573–575.

[7] D. Burns, Y. Hu, T. Luo, Hyper-Kähler Manifolds and Birational Transformations in dimension 4, preprint (math.AG 0004154).

[8] A. Caldararu, Derived categories on varieties with trivial canonical bundle, Ph.D. Thesis, Cornell University (2000).

[9] B. Hassett, Special cubic hypersurfaces of dimension 4, Ph.D. Thesis, Harvard University, (1996).

[10] B. Hassett, Special cubic fourfolds, Compositio Math. 120 (2000), no. 1, 1-23.

[11] B. Hassett, Some rational cubic fourfolds, J. Algebraic Geom. 8 (1999), no. 1, 103–114.

[12] B. Hassett, Y. Tschinkel, Abelian fibrations and rational points on symmetric products, Internat. J. Math. 11 (2000), no. 9, 1163-1176.

[13] D. Huybrechts, Compact hyper-Kähler manifolds: basic results, Invent. Math. 135 (1999), no. 1, 63–113.

[14] D. Huybrechts, The Kähler cone of a compact hyperkähler manifold, preprint (math.AG 9909109).

[15] Y. Kawamata, K. Matsuda, K. Matsuki, Introduction to the minimal model problem, Algebraic geometry, Sendai, 1985, 283–360, Adv. Stud. Pure Math., 10, North-Holland, Amsterdam-New York, 1987.

[16] E. Looijenga, C. Peters, Torelli theorems for Kähler K3 surfaces, Compositio Math. 42 (1981), no. 2, 145–186.

[17] E. Markman, Brill-Noether duality for moduli spaces of sheaves on K3 surfaces, preprint (math.AG 9901072).

[18] D. G. Markushevich, Integrable symplectic structures on compact complex manifolds, Mat. Sb. (N.S.) 131(173) (1986), no. 4, 465-476; translation in Math. USSR-Sb. 59 (1988), no. 2, 459–469.

[19] D. Matsushita, On fibre space structures of a projective irreducible symplectic manifold, Topology 38 (1999), no. 1, 79–83.

[20] D. Matsushita, Addendum to: On fibre space structures of a projective irreducible symplectic manifold, preprint (math.AG 9903045) to appear in Topology.

[21] S. Mukai, Symplectic structure of the moduli space of sheaves on an abelian or K3 surface, Invent. Math. 77 (1984), 101-116.

[22] Y. Namikawa, Deformation theory of singular symplectic n-folds, preprint (math.AG 0010113) to appear in Math. Ann.

[23] V. Nikulin, Integral symmetric bilinear forms and some of their applications, Math. USSR Izvestija, 14 (1980), no. 1, 103–167.

[24] Z. Ran, Hodge theory and deformations of maps, Compositio Math. 97 (1995), no. 3, 309–328.

[25] N. Shepherd-Barron, Long extremal rays and symplectic singularities, preprint.

[26] C. Voisin, Théoréme de Torelli pour les cubiques de ℙ^{5}, Invent. Math. 86 (1986), no. 3, 577–601.

[27] C. Voisin, Sur la stabilité des sous-variétés lagrangiennes des variétés symplectiques holomorphes, Complex projective geometry (Trieste, 1989/Bergen, 1989), 294–303, London Math. Soc. Lecture Note Ser., 179, Cambridge Univ. Press, Cambridge, 1992.

[28] J. Wierzba, Contractions of symplectic varieties, preprint (math.AG 9910130).

[29] P.M.H. Wilson, Calabi-Yau manifolds with large Picard number, Invent. Math. 98 (1989), no.1, 139–155.

[30] P.M.H. Wilson, The Kähler cone on Calabi-Yau threefolds, Invent. Math. 107 (1992), no. 3, 561–583. Erratum: Invent. Math. 114 (1993), no. 1, 231–233.

[31] P.M.H. Wilson, The existence of elliptic fibre space structures on Calabi-Yau threefolds, Math. Ann. 300 (1994), no. 4, 693–703.

[32] P.M.H. Wilson, The existence of elliptic fibre space structures on Calabi-Yau threefolds II, Math. Proc. Cambridge Philos. Soc. 123 (1998), no. 2, 259–262.

Erratum, added March 5, 2010

In this Erratum we correct two mistakes in the published version of this paper:

Rational curves on holomorphic symplectic fourfolds, Geometric and Functional Analysis 11 (2001), no. 6, 1201-1228

We are grateful to Antoine Chambert-Loir for pointing out that the first paragraph of Theorem 4.1 should read as follows:

Let F be an irreducible holomorphic symplectic manifold of dimension 2n and Y a submanifold of dimension k. Assume either that Y is Lagrangian, or that all of the following hold: N_{Y/F} = Ω_{Y}^{1} ⊕ O_{Y}^{⊕2n−2k}, the restriction of the symplectic form to Y is zero, and H^{1}(O_{Y}) = 0. Then the deformation space of Y in F is smooth, of dimension 2n − 2k if the last three conditions above hold.

We are grateful to Claire Voisin for pointing out that Proposition 7.4 requires an additional hypothesis, and should read as follows:

Let X be a cubic fourfold with Fano variety F. Assume that F contains a smooth rational curve R of degree n, with corresponding scroll T_{n,Δ}. Assume that this corresponding scroll T is not a cone and has isolated singularities. Then there exists a rational map

ϕ : ℙ^{4} −−→ X

with

deg(ϕ) = \binom{n − 2}{2} − Δ = \frac{(n − 2)^{2}}{4} + \frac{(R, R)}{2} + 1.

Without this hypothesis the double point computation fails, as is shown by the example

T = {x^{2}z + y^{2}t = 0}.