FLOPS ON HOLOMORPHIC SYMPLECTIC FOURFOLDS AND DETERMINANTAL CUBIC HYPERSURFACES

BRENDAN HASSETT AND YURI TSCHINKEL

1. Introduction

Let Σ be a K3 surface. Any birational map Σ −−→ Σ extends to an automorphism; this follows from the uniqueness of minimal models for surfaces of non-negative Kodaira dimension. By the Torelli Theorem, the group of automorphisms of Σ is isomorphic to the group of automorphisms of H^{2}(Σ, ℤ) compatible with the intersection pairing ⟨, ⟩ and the Hodge structure on H^{2}(Σ, ℂ), and preserving the cone of nef (numerically eventually free) divisors. The nef cone admits an intrinsic combinatorial description (see, for example, [31]), once we specify a polarization g: A divisor h on Σ is nef if and only if ⟨h, D⟩ ≥ 0 for each divisor class D with ⟨D, D⟩ ≥ −2 and ⟨g, D⟩ > 0. This characterization of the automorphism group has many interesting applications to arithmetic and geometric questions.

In this paper, we study certain aspects of the birational geometry of higher-dimensional analogs of K3 surfaces, i.e., irreducible holomorphic symplectic varieties F. These share many geometric properties with K3 surfaces. For example, the group H^{2}(F, ℤ) carries a canonical integral quadratic form (,), the Beauville-Bogomolov form (see, for example, [26]). Its definition uses the symplectic form on F but it can be characterized by the fact that the self-intersection form on H^{2}(F, ℤ) is proportional to a power of the Beauville-Bogomolov form [26, 1.11] [18]

D^{dim(F)} = c_{F} (F, F)^{dim(F)/2} .

Moreover, these varieties satisfy local Torelli theorems [4] and surjectivity of the period map [26]. In contrast to the surface case, F may have numerous minimal models and may admit birational self-maps which are not regular. Furthermore, naive generalizations of the Torelli Theorem to higher dimensions are false; for counterexamples, consult [14], [35], and [32, Cor. 1.7, Thm. 4.5].

Date: July 14, 2009.

Perhaps the best-known examples of irreducible holomorphic symplectic varieties are punctual Hilbert schemes of K3 surfaces and their deformations [4]. Here we focus on the case of length-two subschemes, which are isomorphic to the symmetric square of the K3 surface blownup along the diagonal. These also arise as varieties of lines on cubic fourfolds [6]. By [23], given a polarization g on F, a divisor h on F is nef if (h, ρ) ≥ 0 for each divisor class ρ satisfying

(1) (g, ρ) > 0; and

(2) (ρ, ρ) ≥ −2, or (ρ, ρ) = −10 and (ρ, H^{2}(F, ℤ)) = 2ℤ. (These

are called (−10)-classes.) We have conjectured that these conditions are also necessary [22]. The main challenge in proving this is to show that the divisors ρ described above obstruct line bundles from being ample. For example, we expect extremal (−10)-classes ρ to be Poincaré dual to multiples of lines contained in planes P ⊂ F. The presence of such planes has implications for the birational geometry of F, as we can take the Mukai flop or elementary transformation along P [34, 0.7]

Bl_{P} F ≃ Bl_{P′}F′
↙  ↘
F  F′.

Indeed, since P is Lagrangian, N_{P/F} ≃ Ω_{P}^{1} so the exceptional divisor E ⊂ Bl_{P} F is isomorphic to ℙ(Ω_{P}^{1} ). This admits two ℙ^{1}-bundle structures over ℙ^{2}, so we can blow down E to obtain a nonsingular variety F′ birational to F. This is also an irreducible holomorphic symplectic variety, deformation equivalent to F [25, 3.4].

One especially interesting case is when there are no (0) or (−2)- classes (i.e., divisors ρ with (ρ, ρ) = 0, −2) but multiple (−10)-classes. Here the nef cones of birational models of F should be completely controlled by (−10)-classes. Not only are the integral extremal rays of F Poincaré dual to (−10)-classes, but this remains true for Mukai flops of F. In this situation, we expect F to admit infinite sequences of Mukai flops. However, Morrison [33] and Kawamata [29] have conjectured the following:

Conjecture (Finiteness of models). Let F be a (simply-connected) Calabi-Yau manifold. Then there are finitely many minimal models of F up to isomorphism.

How can this be reconciled with the existence of infinite sequences of flops? The only possibility is that after a finite sequence of flops of F, we arrive at a variety isomorphic to F. This gives rise to birational maps F −−→ F that are not automorphisms. These in turn act on H^{2}(F, ℤ), preserving the cone of moving divisors but not the nef cone.

More specifically, consider a general cubic fourfold X containing a cubic scroll, or equivalently, a hyperplane section with six double points (see Proposition 23). The Picard lattice Pic(F) of the corresponding variety of lines F = F (X) has rank two and the associated quadratic form represents −10 but not −2 or 0. For such fourfolds we

• compute the ample and moving cone in Pic(F);

• prove that F does not admit biregular automorphisms;

• exhibit a birational automorphism of infinite order explaining the chamber decomposition of the moving cone.

Our principal results are Theorems 24 and 31. The first exhibits explicit birational involutions on F and factors their indeterminacy. The second describes the action of the birational automorphism group on H^{2}(F, ℤ).

We list some previous results in a similar vein: Miles Reid [36, 6.8] has offered examples of elliptically-fibered threefolds with an infinite number of distinct minimal models. Michael Fryers [17] classified isomorphism classes of minimal models of the general Horrocks-Mumford quintic threefold. Morrison [33] and Kawamata [29] have proven finiteness results (up to isomorphism!) for Calabi-Yau fiber spaces F → B where 0 < dim(B) ≤ dim(F) ≤ 3. The case of Calabi-Yau manifolds of dimension ≥ 3 remains open.

The first half of the paper is devoted to classical results on cubic hypersurfaces. In Section 2 we analyze cubic threefolds Y with six ordinary double points in general position and their varieties of lines F (Y). Section 3 establishes a dictionary between determinantal cubic surfaces and determinantal cubic threefolds, which generally have six ordinary double points in linear general position. Section 4 develops this to explain the geometric properties of Y, e.g., a transparent description of the components of F (Y) and how they are glued together. Finally, Section 5 shows that cubic threefolds with six double points are determinantal.

The second half focuses on applications to the birational geometry of certain irreducible holomorphic symplectic varieties. Section 6 uses this information to construct birational involutions on the variety of lines F on a cubic fourfold containing Y. In Section 7 we explain the connection to our conjecture on nef cones. We close with an application to Zariski-density of rational points on F.

Throughout, the base field is algebraically closed of characteristic zero.

Acknowledgments: We are grateful to János Kollár and Nick Shepherd-Barron for useful conversations, and to Igor Dolgachev, Balazs Szendrői, and Claire Voisin for helpful comments and references. The first author was partially supported by National Science Foundation Grants 0554491 and 0134259. The second author was partially supported by National Science Foundation Grants 0554280 and 0602333.

2. Cubic threefolds with six double points

We assume that Y ⊂ ℙ^{4} is a cubic hypersurface with ordinary double points at p_{1}, …, p_{6}, which are in linear general position.

Lemma 1. The cubic hypersurface Y contains no planes and the variety of lines F (Y) has the expected dimension two.

Proof. Let Y′ denote a cubic threefold containing the plane

Π = {x_{0} = x_{1} = 0}.

Suppose G is a homogeneous cubic equation for Y′. Then we can write

G = x_{0}Q_{0} + x_{1}Q_{1}

for quadratic forms Q_{0} and Q_{1}. The singular locus of Y′ contains the subscheme defined by

x_{0} = x_{1} = Q_{0} = Q_{1} = 0

which consists of four coplanar points. Thus the singularities of Y′ are not in linear general position.

Suppose that F (Y) has dimension > 2. As the singularities of Y are ordinary double points, there is at most a one-parameter family of lines through each singularity. Thus the generic line ℓ of F (Y) is contained in a smooth hyperplane section H∩Y. Consider the incidence correspondence

Z = {(ℓ, H) : ℓ ⊂ Y, ℓ ⊂ H} ⊂ Gr(2, 5) × Gr(4, 5)

in the partial flag variety. Since Z has dimension five the fibers of projection onto the second factor have dimension one, which is impossible as smooth cubic surfaces have a finite number of lines. □

Proposition 2. Let Y be a cubic hypersurface with six ordinary double points p_{1}, …, p_{6} in linear general position. Projection from the point p_{6}

Y −−→ ℙ^{3}

factors

\xymatrix{Ỹ := Bl_{p_{6}}Y \ar[d]_{δ} \ar[r]^{γ} & ℙ^{3} \\ Y \ar@{-->}[ur]}

where γ is the blow up of a complete intersection C_{6} of a smooth quadric and a cubic in ℙ^{3}, consisting of two twisted cubic curves meeting in five nodes.

Proof. The morphism γ blows down all the lines in Y incident to p_{6}; since p_{6} is an ordinary double point, these are parametrized by a complete intersection C_{6} of a smooth quadric Q (the projectivized tangent cone of Y at p_{6}) and a cubic in ℙ^{3}. Furthermore, an easy computation using the Jacobian criterion shows that C_{6} is smooth except at the points n_{i} = γ(p_{i}), i = 1, …, 5. Note that n_{i} corresponds to the line ℓ(p_{i}, p_{6}) joining p_{i} to p_{6}.

We claim that C_{6} has two irreducible components E_{6} and E_{6}^{∨}, each smooth and rational of degree three, and n_{1}, …, n_{5} are nodes of C_{6}. Since the normalization of C_{6} has genus −1 it is necessarily reducible. Consider the alternatives for the combinatorics of components: If C_{6} were to contain a component of degree one then this would meet the rest of C_{6} in three nodes, say n_{1}, n_{2}, n_{3}. Then the ordinary double points {p_{6}, p_{1}, p_{2}, p_{3}} ∈ Y would all lie in a plane, contradicting our general position hypothesis. If C_{6} were to contain a component of degree two then this would meet the rest of C_{6} in four coplanar nodes, say n_{1}, n_{2}, n_{3}, n_{4}. Then {p_{6}, p_{1}, p_{2}, p_{3}, p_{4}} ∈ Y would span a threedimensional space, again contradicting our hypothesis. If C_{6} contained a component of degree three and arithmetic genus one (i.e., a nodal plane cubic) then the quadric Q would be degenerate. □

Remark 3. This analysis implies that

n_{1}, n_{2}, n_{3}, n_{4}, n_{5} ∈ Q ≃ ℙ^{1} × ℙ^{1}

satisfy the following genericity conditions:

• the n_{i} are distinct;

• no two of the n_{i} lie on a ruling of Q;

• no four of the n_{i} lie on a hyperplane section of Q ⊂ ℙ^{3}. Hence S = Bl_{n_{1},...,n_{5}}Q is isomorphic to a nonsingular cubic surface. While we will not prove this, S is isomorphic to the cubic surface constructed from Y in (7) of Section 5. In particular, S does not depend on which double point p_{i} ∈ Y we choose for our projection.

Corollary 4. The singular locus F (Y)^{sing} ⊂ F (Y) is equal to the lines meeting the singular points p_{1}, …, p_{6} ∈ Y. The irreducible components of F (Y)^{sing} consist of twelve smooth rational curves

E_{1}, E_{1}^{∨}, …, E_{6}, E_{6}^{∨},

where E_{j} ∪ E_{j}^{∨} parametrizes the lines through p_{j}. The singularities of F (Y)^{sing} are the 15 lines ℓ(p_{i}, p_{j}) joining singularities of Y, and

ℓ(p_{i}, p_{j}) = E_{i} ∩ E_{j} = E_{i} ∩ E_{j}^{∨} = E_{i}^{∨} ∩ E_{j} = E_{i}^{∨} ∩ E_{j}^{∨}.

Proof. It is a general fact [1, §1] that for any cubic hypersurface Y′, the variety of lines F (Y′) is smooth at lines avoiding the singularities of Y′. Moreover, F (Y′) is singular at lines passing through an ordinary double point of Y′ [11, 7.8]. The structure of the singular locus then follows from Proposition 2. □

Corollary 5. The pair (Y, p_{6}) is uniquely determined up to isomorphism by the isomorphism class of the nodal curve C_{6}.

Proof. The curve C_{6} is a stable curve of genus four and C_{6} ↪ ℙ^{3} is its canonical embedding. We can characterize Y as the image of ℙ^{3} under the linear series of cubics passing through C_{6}. □

3. Determinantal cubic surfaces and threefolds

We review determinantal representations of smooth cubic surfaces. The story begins with Grassmann [20] who showed that cubic surfaces arise as the common points of three nets of planes in ℙ^{3}, i.e., the locus where a 3 × 3 matrix of linear forms on ℙ^{3} has nontrivial kernel. Schröter [37] showed that a generic surface admits such a realization and Clebsch [10] tied these representations to the structure of the lines on the cubic surface. Dickson [15] addressed the problem of expressing arbitrary smooth cubic surfaces in determinantal form. See [5, 6.4] and [8] for modern accounts and [16] for further historical discussion.

Proposition 6. Let S ⊂ ℙ^{3} be a smooth cubic surface. Then there exists a 3 × 3 matrix M = (m_{ij}) with entries linear forms on ℙ^{3} such that

S = {det(M) = 0}.

Up to the left/right action of GL_{3} × GL_{3}, there are 72 such representations, corresponding to sextuples of disjoint lines on S.

This was extended by B. Segre [38] (cf. [5, 6.5]) to smooth cubic surfaces defined over arbitrary fields:

Proposition 7. Let S be a smooth cubic surface defined over an arbitrary field k. Then the following conditions are equivalent:

• There exists a 3 × 3 matrix of linear forms over k such that S = {det(M) = 0}.

• S contains a rational point and a sextuple of disjoint lines defined over k.

• S admits a birational morphism to ℙ^{2} defined over k.

We emphasize that each individual line in the sextuple need not be defined over k.

C. Segre [39, §12-14] analyzed determinantal representations of cubic threefolds:

Proposition 8. Let Y ⊂ ℙ^{4} be a generic cubic hypersurface realized as the determinant of a 3 × 3 matrix of linear forms. Then Y has six ordinary double points, in linear general position. Conversely, any cubic hypersurface with six ordinary double points in linear general position is determinantal.

For completeness, we will provide an argument in Propositions 10 and 19.

Our main goal is to explain how all these classical theorems are related. Here is the key geometric ingredient: Let W be a vector space with a nondegenerate bilinear form (,); taking orthogonal complements, we obtain a natural identification

(1)

Gr(n, W) = Gr(dim(W) − n, W)
Λ ↦ Λ^{⊥}.

Let G be a group acting linearly on W, with the natural induced action on Gr(n, W) and the action on Gr(dim(W) − n, W) induced by (1).

We are especially interested in the case where W = End(V) for some vector space V of dimension d, the bilinear form (,) is the trace pairing

(A, B) = tr(AB),

and GL(V) × GL(V) acts on End(V) by left-right multiplication

(g_{1}, g_{2}) · M = g_{1}Mg_{2}^{t}.

Here we are using the transpose operator

t : End(V) → End(V)

obtained by dualizing and then applying the trace pairing. Consider the semidirect product

(2)  1 → GL(V) × GL(V) → G → 𝔖_{2} → 1

where 𝔖_{2} acts by exchanging the factors. Since

(g_{1}, g_{2}) · M^{t} = g_{1}M^{t}g_{2}^{t} = ((g_{2}, g_{1}) · M)^{t}

G also acts naturally on End(V) and thus on the Grassmannians

Gr(n, End(V)) = Gr(d^{2} − n, End(V)).

Consider the rank stratification on End(V)

0 ⊂ Σ_{1} ⊂ Σ_{2} ⊂ … ⊂ Σ_{d−1} ⊂ End(V),

which is invariant under the group actions. We have the incidence correspondence

{(W_{1}, W_{2}, A) : W_{1} ⊂ ker(A), im(A) ⊂ W_{2}}
⊂ Gr(d − k, V) × Gr(k, V) × End(V),

a vector bundle of rank k^{2} over Gr(d−k, V)×Gr(k, V). The projection onto End(V) induces a birational morphism to Σ_{k}, invertible away from Σ_{k−1}. In particular, Σ_{k} is ruled by k^{2}-dimensional vector spaces; through each A ∈ Σ_{k} \ Σ_{k−1} there passes a unique ruling

R_{A} := {M ∈ End(V) : ker(A) ⊂ ker(M), im(M) ⊂ im(A)}.

Recall the description of the tangent space of Σ_{k} (see, for instance, [3, pp. 68-69]): When A has rank < k then T_{A}Σ_{k} = End(V); furthermore, Σ_{k−1} is the singular locus of Σ_{k}. If A has rank k then

T_{A}Σ_{k} = {M ∈ End(V) : M(ker(A)) ⊂ im(A)};

note that T_{A} ⊃ R_{A}. We can express

(T_{A}Σ_{k})^{⊥} = {N ∈ End(V) : NA = AN = 0},

which is a linear subspace of dimension (d − k)^{2} in Σ_{d−k}. Thus for each matrix B satisfying

im(B) = ker(A) and ker(B) = im(A),

we have (T_{A}Σ_{k})^{⊥} = R_{B}.

Proposition 9. Assume V is a three-dimensional vector space and Λ ⊂ End(V) is a four-dimensional subspace. Let Λ^{⊥} ⊂ End(V) denote the orthogonal complement of Λ with respect to the trace pairing. Then the following conditions are equivalent:

• Λ is tangent to Σ_{2} at a smooth point or intersects Σ_{1} nontrivially;

• Λ^{⊥} is tangent to Σ_{1} at a nonzero point or is tangent to Σ_{2} at a smooth point.

In other words, Λ is transverse to the rank strata if and only if Λ^{⊥} is transverse to the rank strata.

Proof. Suppose that Λ is tangent to Σ_{2} at a rank-two matrix A_{0}, i.e.,

(3)  A_{0} ∈ Λ ⊂ T_{A_{0}}Σ_{2}.

Let B_{0} be a matrix with ker(B_{0}) = im(A_{0}) and im(B_{0}) = ker(A_{0}), which is unique up to scalars; it follows that

(T_{A_{0}}Σ_{2})^{⊥} = R_{B_{0}}, (T_{B_{0}}Σ_{1})^{⊥} = R_{A_{0}}.

Dualizing (3) we obtain

R_{B_{0}} = (T_{A_{0}}Σ_{2})^{⊥} ⊂ Λ^{⊥} ⊂ A_{0}^{⊥}.

We also have

R_{B_{0}} ⊂ T_{B_{0}}Σ_{1} = R_{A_{0}}^{⊥} ⊂ A_{0}^{⊥}.

Since

dim R_{B_{0}} = 1, dim T_{B_{0}}Σ_{1} = dim Λ^{⊥} = 5, dim A_{0}^{⊥} = 8,

we deduce

span(B_{0}) ⊊ Λ^{⊥} ∩ T_{B_{0}}Σ_{1}.

Thus Λ^{⊥} fails to intersect Σ_{1} transversely at B_{0}.

Now suppose Λ is incident to Σ_{1} at a rank-one matrix B_{0}, i.e.,

R_{B_{0}} ⊂ Λ.

Again, let A_{0} be a matrix with ker(A_{0}) = im(B_{0}) and im(A_{0}) = ker(B_{0}). Dualizing, we have Λ^{⊥} ⊂ B_{0}^{⊥} = T_{A_{0}}Σ_{2} and the intersection

Λ^{⊥} ∩ R_{A_{0}} ⊂ T_{A_{0}}Σ_{2}

is nonzero. Pick a nonzero matrix

C ∈ Λ^{⊥} ∩ R_{A_{0}}.

If C has rank two then im(C) = im(A_{0}) and ker(C) = ker(A_{0}), thus R_{C} = R_{A_{0}} and T_{C}Σ_{2} = T_{A_{0}}Σ_{2}. Here Λ^{⊥} fails to be transverse to Σ_{2} at C. If C has rank one then

T_{C}Σ_{1} = {M ∈ End(V) : M(ker(C)) ⊂ im(C)}
⊂ {M ∈ End(V) : M(ker(A_{0})) ⊂ im(A_{0})}
= T_{A_{0}}Σ_{2} = B_{0}^{⊥}.

Since Λ^{⊥} ⊂ B_{0}^{⊥} as well, the intersection

T_{C}Σ_{1} ∩ Λ^{⊥} ⊂ B_{0}^{⊥}

has dimension at least two, i.e., Λ^{⊥} meets Σ_{1} nontransversely at C. Conversely, suppose that Λ^{⊥} is tangent to Σ_{1} at B_{0}, i.e.,

span(B_{0}) ⊊ Λ^{⊥} ∩ T_{B_{0}}Σ_{1}.

Let A_{0} denote a matrix with im(A_{0}) = ker(B_{0}) and ker(A_{0}) = im(B_{0}). Dualizing, we find that

Λ + R_{A_{0}} ⊊ B_{0}^{⊥} and Λ ∩ R_{A_{0}} ≠ 0.

Let C be a nonzero matrix in this intersection. If C has rank one then Λ meets Σ_{1} nontrivially and we’re done. If C has rank two then it has the same image and kernel as A_{0}, whence T_{C}Σ_{2} = B_{0}^{⊥} and Λ ⊂ T_{C}Σ_{2}, i.e., Σ_{2} and Λ fail to be transverse at C.

Now suppose Λ^{⊥} is tangent to Σ_{2} at a matrix A_{0} of rank two, i.e.,

A_{0} ⊂ Λ^{⊥} ⊂ T_{A_{0}}Σ_{2}.

Again, choose B_{0} to be a matrix with im(B_{0}) = ker(A_{0}) and ker(B_{0}) = im(A_{0}) so that B_{0} ∈ R_{B_{0}} = T_{A_{0}}Σ_{2}^{⊥}. Dualizing yields B_{0} ∈ Λ, i.e., Λ intersects Σ_{1}. □

We use this to interpret our determinantal expressions for cubic hypersurfaces. Recall that dim(V) = 3. Tensor multiplication

V × V^{∨} → End(V)

gives the Segre embedding

ℙ(V) × ℙ(V^{∨}) ↪ ℙ(End(V)),

whose image has degree six and coincides with ℙ(Σ_{1}). Given a fourdimensional subspace

Λ ⊂ End(V)

the intersection

(4)  S := ℙ(Λ ∩ Σ_{2}) ⊂ ℙ(Λ) ≃ ℙ^{3}

is a determinantal cubic surface. It is smooth precisely when Λ meets Σ_{2} transversely at smooth points. Then we obtain an embedding

\begin{matrix}S & ↪ & ℙ(V) × ℙ(V^{∨}) \\ s & ↦ & (ker(s), im(s)) \end{matrix}

such that the projections induce the blow-up realizations of S (cf. Proposition 6)

β : S → ℙ(V), β^{∨} : S → ℙ(V^{∨}).

Let Λ^{⊥} be the orthogonal complement to Λ with respect to the trace pairing. Then

(5)  Y := ℙ(Λ^{⊥} ∩ Σ_{2}) ⊂ ℙ(Λ^{⊥}) ≃ ℙ^{4}

is a determinantal cubic threefold. It is necessarily singular along the points of ℙ(Λ^{⊥} ∩ Σ_{1}). If ℙ(Λ^{⊥}) intersects ℙ(Σ_{1}) and the smooth points of ℙ(Σ_{2}) transversely then the Bezout theorem implies that the singular locus of Y is

{p_{1}, …, p_{6}} := ℙ(Σ_{1} ∩ Λ^{⊥}).

Note that these give a sextuple of points in ℙ(V) × ℙ(V^{∨}) ≃ ℙ^{2} × ℙ^{2}; a straightforward cohomology computation shows these are in linear general position in ℙ(End(V)).

Proposition 10. Let S and Y be determinantal cubic hypersurfaces defined by Equations 4 and 5 above. Then Y is a cubic threefold with six ordinary double points in linear general position if and only if S is a smooth cubic surface. We thus obtain a identification

\{ \begin{matrix}determinantal cubic \\ threefolds with six \\ ordinary double points \\ in linear general position \end{matrix} \} ≃ \{ \begin{matrix}determinantal cubic surfaces \\ without singularities \end{matrix} \}

that is equivariant with respect to the action of G.

Indeed, Proposition 9 says we can identify the open subsets in

Gr(4, End(V)) = Gr(5, End(V))

where our transversality conditions hold.

4. Geometric applications of the determinantal description

In this section, we assume that S and Y satisfy the conclusions of Proposition 10. The determinantal description allows a transparent derivation of many of the key properties of Y.

Proposition 11. Let F (Y) denote the variety of lines on Y. We have a natural surjective morphism

ν : ℙ(V) ⊔ S ⊔ ℙ(V^{∨}) → F (Y)

that maps each component birationally onto its image.

Proof. For each point [v] ∈ ℙ(V), let

ℓ_{[v]} = {y = [ϕ] : ϕ(v) = 0} = {y = [ϕ] : v ∈ ker(ϕ)} ⊂ Y

where ϕ ∈ End(V) represents y ∈ Y ⊂ ℙ(Λ^{⊥}). This is a linear subspace of codimension at most three in ℙ(Λ^{⊥}). Indeed, elements of

Λ^{⊥} ∩ {M : v ⊂ ker(M)}

automatically have vanishing determinants. Lemma 1 guarantees Y does not contain any planes, so we conclude that ℓ_{[v]} is a line.

Similarly, for [v^{∨}] ∈ ℙ(V^{∨}) we also get lines

ℓ_{[v^{∨}]} = {y = [ϕ] : v^{∨} ∘ ϕ = 0} = {y = [ϕ] : v^{∨} ∈ ker(ϕ^{t})} ⊂ Y.

Given s = [σ] ∈ S with σ ∈ Λ, we have the locus

ℓ_{s} = {y = [ϕ] : σϕσ = 0} ⊂ Y.

Since σ has rank two, this condition translates into the vanishing of the 2 × 2 matrix of the induced map

im(σ) →^{ϕ} V/ker(σ).

However, the orthogonality assumption tr(σϕ) = 0 implies that there are only three independent linear conditions. In particular, ℓ_{s} is a line in Y.

Combining these three constructions, we obtain the morphism ν. We next show that ν is surjective. Lemma 1 implies that F (Y) is twodimensional. A standard intersection theory computation [19, 14.7.13] shows that deg F (Y) = 45 (with respect to the Plücker embedding of the Grassmannian). However, we can compute the pull back

ν^{∗}O_{F (Y)}(1) = (O_{ℙ(V)}(3), O_{S}(3), O_{ℙ(V^{∨})}(3))

which means that

deg(ℙ(V)) = deg(ℙ(V^{∨})) = 9, deg(S) = 27.

Thus all the components of F (Y) are in the image of ν; furthermore, ν maps each component birationally onto its image. □

Corollary 12. Retain the notation of Proposition 11 and let y ∈ Y be a nonsingular point. The components of F (Y) dominated by ℙ(V) and ℙ(V^{∨}) each admit a unique line passing through y. The component dominated by S admits four lines passing through y.

Proof. The first statement is easily verified using linear algebra. The second can be deduced from the fact that a generic y ∈ Y lies on six lines in Y. □

Proposition 13. The morphism

ν : ℙ(V) ⊔ S ⊔ ℙ(V^{∨}) → F (Y)

induces the following identifications: Consider the distinguished doublesix on S

{E_{1}, …, E_{6}; E_{1}^{∨}, …, E_{6}^{∨}},

with each 6-tuple blowing down to a collections of points

{q_{1}, …, q_{6}} ⊂ ℙ(V), {q_{1}^{∨}, …, q_{6}^{∨}} ⊂ ℙ(V^{∨}).

Let {ℓ_{1}, …, ℓ_{6}} and {ℓ_{1}^{∨}, …, ℓ_{6}^{∨}} be the lines in ℙ(V^{∨}) and ℙ(V) dual to these points. We have natural isomorphisms for each i:

\begin{matrix}ψ_{i} : ℓ_{i} →^{∼} E_{i}, & ψ_{i}(ℓ_{i} ∩ ℓ_{j}) = E_{i} ∩ E_{j}^{∨} ⊂ S, \\ ψ_{i}^{∨} : ℓ_{i}^{∨} →^{∼} E_{i}^{∨}, & ψ_{i}(ℓ_{i}^{∨} ∩ ℓ_{j}^{∨}) = E_{i}^{∨} ∩ E_{j} ⊂ S. \end{matrix}

Proof. We break up the argument into two lemmas:

Lemma 14. The morphism ν maps E_{i}, E_{i}^{∨} ⊂ S, ℓ_{i} ⊂ ℙ(V^{∨}) and ℓ_{i}^{∨} ⊂ ℙ(V) to the locus C_{i} of lines passing through p_{i}. Furthermore, E_{i} and ℓ_{i} parametrize y = [ϕ] ∈ Y such that im(ϕ) ⊃ im(p_{i}); E_{i}^{∨} and ℓ_{i}^{∨} parametrize y = [ϕ] such that ker(ϕ) ⊂ ker(p_{i}). Here we regard the singularity p_{i} ∈ Y as an element Λ^{⊥} ∩ Σ_{1}.

Proof. The determinantal description of S identifies

E_{i} = {s ∈ Λ : ker(s) = im(p_{i})}.

Similarly, we have

E_{i}^{∨} = {s ∈ Λ : im(s) = ker(p_{i})}.

On the other hand,

ℓ_{i} = {im(s) : s ∈ Λ with ker(s) = im(p_{i})} ⊂ ℙ(V^{∨})

and

ℓ_{i}^{∨} = {ker(s) : s ∈ Λ with im(s) = ker(p_{i})} ⊂ ℙ(V).

Thus for s_{i} = [σ] ∈ E_{i}

ν(s_{i}) = [{y = [ϕ] : ϕ(im(σ)) ⊂ ker(σ) = im(p_{i})}]

which is a line through p_{i}. On the other hand, for v_{i}^{∨} ∈ ℓ_{i} (where v_{i}^{∨} ∈ V^{∨} satisfies v_{i}^{∨}(im(p_{i})) = 0) we have

ν(v_{i}^{∨}) = [{y = [ϕ] : v_{i}^{∨}(im(ϕ)) = 0}]

which is also a line through p_{i}. As we vary s_{i} ∈ E_{i} and v_{i}^{∨} ∈ ℓ_{i}, we get the locus of y = [ϕ] such that im(ϕ) ⊃ im(p_{i}).

The analogous statements for E_{i}^{∨} and ℓ_{i}^{∨} are proven similarly. □

There is an obvious identification

(6)  \begin{matrix}ℓ_{i} = ℙ(q_{i}^{⊥}) = ℙ((V/q_{i})^{∨}) = ℙ(V/q_{i}) = ℙ(Hom(q_{i}, V/q_{i})) \\ = ℙ(T_{q_{i}}ℙ(V)) = E_{i}; \end{matrix}

note that if W is a two-dimensional vector space then the isomorphism W = W^{∨}⊗⋀^{2} W induces a natural isomorphism ℙ(W) = ℙ(W^{∨}). This is not the gluing inducing ν. However, note that this takes the points ℓ_{i} ∩ ℓ_{j} to the intersections E_{i} ∩ 𝔩_{ij}, where 𝔩_{ij} is the proper transform of the line joining q_{i} and q_{j}. Using (6), it suffices to express ψ_{i} and ψ_{i}^{∨} as automorphisms of E_{i} and E_{i}^{∨}.

The gluings ℓ_{i} ≃ E_{i} and ℓ_{i}^{∨} ≃ E_{i}^{∨} will be obtained from the following:

Lemma 15. There exists a projectivity ψ_{i} : E_{i} → E_{i} mapping E_{i} ∩ 𝔩_{ij} to E_{i} ∩ E_{j}^{∨} for each j ≠ i. The analogous statement holds for E_{i}^{∨}.

Proof. For notational simplicity we take i = 1. Consider the conic bundle S → ℙ^{1} given by the pencil of cubics on ℙ(V) double at q_{1} and containing q_{2}, …, q_{6}. The degenerate fibers are

𝔩_{1j} ∪ E_{j}^{∨}, j = 2, …, 6.

The curve E_{1} is a bi-section of this conic bundle, so there is a covering involution ψ_{1} : E_{1} → E_{1} taking 𝔩_{1j} ∩ E_{1} to E_{j}^{∨} ∩ E_{1}. □

It remains to check that this is in fact the identification induced by ν. However, we know from Corollary 4 that ν glues the points E_{i} ∩ E_{j}^{∨}, E_{i}^{∨} ∩ E_{j}, to [ℓ(p_{i}, p_{j})], the line in Y joining p_{i} and p_{j}. Now ℓ_{i} and ℓ_{j} meet in ℙ(V^{∨}), and ℓ_{i}^{∨} and ℓ_{j}^{∨} meet in ℙ(V); thus these points must also be mapped by ν to [ℓ(p_{i}, p_{j})]. In general, the isomorphisms ψ_{i} and ψ_{i}^{∨} are the unique ones identifying all these points. This finishes the proof of Proposition 13. □

Proposition 16. The morphism ν is obtained by gluing ℙ(V), S, and ℙ(V^{∨}) using the identifications described in Proposition 13.

Proof. Let F′ denote the surface obtained by gluing ℙ(V), S, and ℙ(V^{∨}) using the identifications ψ_{i} and ψ_{i}^{∨}. Again, F′ contains twelve distinguished rational curves

ℓ_{i} = E_{i}, ℓ_{i}^{∨} = E_{i}, i = 1, …, 6

and fifteen distinguished points

ℓ_{i} ∩ ℓ_{j} = E_{i} ∩ E_{j}^{∨} = ℓ_{i}^{∨} ∩ ℓ_{j}^{∨} = E_{i}^{∨} ∩ E_{j},

which map surjectively onto F (Y)^{sing} (by Corollary 4).

We have already seen that ν factors through F′; it only remains to prove that the induced morphism F′ → F (Y) is an isomorphism. Our analysis of the gluings over F (Y)^{sing} shows that ξ is a bijection over F (Y)^{sing}.

We first check that ν is the normalization of F (Y). Proposition 11 shows that ν maps each irreducible component birationally onto its image. It follows that the restrictions

ℙ(V) → ν(ℙ(V)), ℙ(V^{∨}) → ν(ℙ(V^{∨}))

are normalization maps. Consider the factorization of ν|S through the normalization of its image

S → ν(S)′ → ν(S);

this reverses the identifications induced by the ψ_{i} and ψ_{i}^{∨}. The images of the six lines E_{1}, …, E_{6}, (and E_{1}^{∨}, …, E_{6}^{∨}) in ν(S)′ are pairwise disjoint. Hence S → ν(S)′ contracts no curves and thus is an isomorphism; S is the normalization of ν(S).

This analysis implies F′ → F (Y) is bijective.

The Fano scheme F (Y) is defined by the degeneracy locus of a vector bundle over the Grassmannian Gr(2, 5), with the expected dimension (by Lemma 1). Thus F (Y) is a local complete intersection scheme and is Cohen-Macaulay; hence it has no embedded points and is seminormal. The universal property of seminormalization then implies that F′ → F (Y) is an isomorphism. □

The determinantal description offers a transparent construction for the cubic scrolls on Y. Each point [v] ∈ ℙ(V) determines a line in ℙ(V^{∨}), which may be interpreted as a ruled surface T_{v} ⊂ Y using the analysis of the components of F (Y) in Proposition 11:

Proposition 17. For each [v] ∈ ℙ(V), the locus

T_{v} = {y ∈ Y : v ∈ im(y)}

is a cubic scroll. The ruling arises from the morphism

\begin{matrix}T_{v} & → & ℙ(V/span(v)) \\ y & ↦ & im(y) \end{matrix}

with fibers ℓ_{[v^{∨}]}, where v^{∨} ≠ 0 ∈ V^{∨} with v^{∨}(v) = 0.

Similarly, for [v^{∨}] ∈ ℙ(V^{∨}) the locus

T_{v^{∨}} = {y ∈ Y : v^{∨}(ker(y)) = 0}

is a cubic scroll. Each union

T_{v} ∪ T_{v^{∨}} = Y ∩ Q

where Q is a quadric hypersurface.

If s ∈ S and ℓ_{s} denotes the corresponding line in Y then ℓ_{s} ⊂ T_{β(s)} (resp. T_{β^{∨}(s)}) is a section of the ruling.

Proof. Choose a basis v, v′, v′′ of V such that v^{∨}(v′) = v^{∨}(v′′) = 0. The matrices in the closure of the locus of rank-two matrices with image containing v can be written

B = (\begin{matrix}b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ b_{31} & b_{32} & b_{33} \end{matrix})

where the bottom two rows are linearly dependent. This defines a closed subset in ℙ(End(V)). Geometrically, this is a cone over the Segre embedding

ℙ^{1} × ℙ^{2} ⊂ ℙ^{5}

with a vertex a projective plane. Intersecting this with Λ^{⊥} yields a hyperplane section of ℙ^{1} × ℙ^{2}, which is a cubic scroll.

On the other hand, the closure of the locus of rank-two matrices with kernel annihilated by v^{∨} are those whose right two columns are linearly dependent. The union of these two loci is given by the intersection

{det(B) = 0} ∩ {b_{22}b_{33} − b_{23}b_{32} = 0} ⊂ ℙ(End(V)),

i.e., the intersection of ℙ(Σ_{2}) with a quadric hypersurface.

Recall that ℓ_{s} was defined in the proof of Proposition 11

ℓ_{s} = {y = [ϕ] : σϕσ = 0}.

Fix a ruling in T_{β(s)}: Regarding β(s) = ker(σ) as a line in V, we choose a two-dimensional subspace ker(σ) ⊂ U ⊂ V, and consider the matrices ϕ with image U. This imposes one additional linear constraint on the matrix entries of ϕ, so each ruling meets ℓ_{s} in one point.

□

Proposition 18. For each s ∈ S,

T_{β(s)} ∩ T_{β^{∨}(s)} = ℓ_{s} ∪ R_{s}

where R_{s} ⊂ Y is a twisted quartic curve passing through the singularities p_{1}, …, p_{6}.

Proof. Generically, the cubic scrolls are nonsingular and isomorphic to ℙ^{2} blown up at one point, in which case ℓ_{s} ⊂ T_{β(s)} is the exceptional curve. Let R_{s} denote the union of components of the intersection other than ℓ_{s}. We have shown that T_{βs} ∪T_{β^{∨}(s)} is a complete intersection of a quadric and cubic in ℙ^{4}, and thus is a singular K3 surface. Adjunction shows that

K_{T_{β(s)}} + R_{s} + ℓ_{s} ≡ 0;

hence R_{s} has degree four and genus zero. □

5. Cubic threefolds with six double points are determinantal

Here we complete C. Segre’s determinantal construction of cubic threefolds with six double points:

Proposition 19. Each cubic threefold with six ordinary double points in linear general position is determinantal.

We prove Proposition 19 using the geometry of the twisted quartic curves in a determinantal cubic threefold, following [12, 3.2-3.4]. One key tool is the Segre threefold 𝔖 ⊂ ℙ^{4}; we recall its basic properties:

• Given p_{1}, …, p_{6} ∈ ℙ^{4} in linear general position, the linear series of cubics double at these points induces a morphism

ϖ : Bl_{p_{1},...,p_{6}}ℙ^{4} → ℙ^{4}

with image 𝔖 and fibers twisted quartic curves containing the points p_{1}, …, p_{6}. If we choose p_{1} = [1, 0, 0, 0, 0],p_{2} = [0, 1, 0, 0, 0], p_{3} = [0, 0, 1, 0, 0], p_{4} = [0, 0, 0, 1, 0],p_{5} = [0, 0, 0, 0, 1], and p_{6} = [1, 1, 1, 1, 1] then the cubics double at these points are y_{0} = (x_{3} − x_{4})x_{0}(x_{1} − x_{2}), y_{1} = (x_{4} − x_{0})x_{1}(x_{2} − x_{3}), y_{2} = (x_{0} − x_{1})x_{2}(x_{3} − x_{4}), y_{3} = (x_{1} − x_{2})x_{3}(x_{4} − x_{1}),

y_{4} = (x_{2} − x_{3})x_{4}(x_{0} − x_{1})

which satisfy

y_{0}y_{1}y_{2} + y_{1}y_{2}y_{3} + y_{2}y_{3}y_{4} + y_{3}y_{4}y_{0} + y_{4}y_{0}y_{1} = 0.

• 𝔖 contains 10 ordinary double points and 15 planes.

• The nonsingular twisted quartic curves map to an open subset of 𝔖 that is isomorphic to M_{0,6}, the moduli space of genuszero curves with six marked points. The morphism ϖ is the universal family over M_{0,6}.

• The inclusion M_{0,6} ↪ 𝔖 extends to an isomorphism [24]

(ℙ^{1})^{6}//SL_{2} →^{∼} 𝔖

from the GIT quotient of six points in ℙ^{1} with the symmetric linearization.

Thus we have a morphism

(7) \{ \begin{matrix}cubic threefolds with \\ ordinary double points \\ at p_{1}, …, p_{6} \end{matrix} \} → \{ \begin{matrix}smooth cubic surfaces \\ arising as hyperplane \\ sections of 𝔖 ⊂ ℙ^{4} \end{matrix} \}

Remark 20. In general, M. Kapranov [28] [27, 4.3] has shown that M_{0,n−1} can be identified with the rational normal curves in ℙ^{n−3} passing through points p_{1}, …, p_{n−1} ∈ ℙ^{n−3} in linear general position. The rational normal curves are the universal curve, with p_{1}, …, p_{n−1} tracing out the marked points. Identifying the universal curve over \overline{M}_{0,n−1} with \overline{M}_{0,n}, there is a morphism

\begin{matrix}\overline{M}_{0,n} & → & ℙ^{n−3} \\ (C, p_{1}, …, p_{n}) & ↦ & p_{n}, \end{matrix}

factoring through Bl_{p_{1},...,p_{n−1}}ℙ^{n−3}.

Recall our previous notation: Let

{E_{1}, …, E_{6}; E_{1}^{∨}, …, E_{6}^{∨}}

denote the double-six on S, β : S → ℙ(V) and β^{∨} : S → ℙ(V^{∨}) the associated contractions, and

{q_{1}, …, q_{6}; q_{1}^{∨}, …, q_{6}^{∨}}

the images of the exceptional divisors. There is an involution of the Picard lattice taking E_{i} to E_{i}^{∨} for i = 1, …, 6. If the Picard lattice is presented

ℤL + ℤE_{1} + … + ℤE_{6}, E_{i}^{2} = −1, E_{i}E_{j} = δ_{ij}, L^{2} = 1, LE_{i} = 0,

the involution takes the form

(8)  E_{i} ↦ 2L − E_{j} − E_{k} − E_{a} − E_{b} − E_{c} = E_{i}^{∨},

where {i, j, k, a, b, c} is a permutation of the indices {1, …, 6}.

We shall need a version of Cremona’s hexahedral construction [13, 16]:

Proposition 21. Let S^{∘} ⊂ S denote the complement to the lines in S. For s ∈ S^{∘} consider the images of q_{1}, …, q_{6} and q_{1}^{∨}, …, q_{6}^{∨} under the projections

(9)  ℙ(V) −−→ ℙ(V/β(s)) ℙ(V^{∨}) −−→ ℙ(V^{∨}/β^{∨}(s)),

which determine elements j(s), j^{∨}(s) ∈ M_{0,6}. Then we have the following:

• j(s) = j^{∨}(s) for each s ∈ S^{∘};

• there exists an extension j : S → 𝔖;

• the image of j is a hyperplane section of 𝔖 ⊂ ℙ^{4};

• conversely, each smooth hyperplane section S ⊂ 𝔖 is a cubic

surface with a distinguished ordered double-six. Thus we obtain an identification

\{ \begin{matrix}smooth cubic surfaces \\ with a double-six of \\ ordered lines \end{matrix} \} ≃ \{ \begin{matrix}smooth hyperplane \\ sections of 𝔖 ⊂ ℙ^{4} \end{matrix} \} .

Proof. Fix s ∈ S^{∘} and consider the degree-two Del Pezzo surface S′ := Bl_{s}(S) = Bl_{q_{1},...,q_{7}}(ℙ(V)). The projections (9) induce conic bundles

φ : S′ → ℙ^{1}, φ^{∨} : S′ → ℙ^{1}

with degenerate fibers corresponding to the images of q_{1}, …, q_{6} and q_{1}^{∨}, …, q_{6}^{∨} respectively. However, each degree-two Del Pezzo admits a canonical involution, i.e. the covering involution of the anticanonical morphism S′ → ℙ^{2}. Moreover, φ and φ^{∨} are conjugate under this involution and thus have the same degenerate fibers. We conclude that j(s) = j^{∨}(s) in M_{0,6}.

We extend j to S: Assume first that β(s) ≠ q_{1}, …, q_{6}. We still have a conic bundle φ : S′ → ℙ^{1} but the images of q_{i} and q_{j} in ℙ^{1} coincide if β(s) ∈ 𝔩_{ij}, the line joining q_{i} and q_{j}. However, since no three of the q_{i} are collinear at most two points may coincide, so the image of (q_{1}, …, q_{6}) is a GIT-semistable point of (ℙ^{1})^{6}; this yields a well-defined point on 𝔖. If β(s) = q_{1} then we can identify E_{1} = ℙ(T_{q_{1}}ℙ(V)) = ℙ(V/q_{1}) and the images of the q_{j}, j = 2, …, 6 in ℙ(V/q_{1}) with the intersections of the proper transforms of the 𝔩_{1j} with E_{1}. The rule

j(s) = (s, 𝔩_{12} ∩ E_{1}, …, 𝔩_{16} ∩ E_{1})

extends the definition of j over E_{1} ⊂ S. (This argument is very similar to the proof of Lemma 15.)

A straightforward degree computation shows that j maps S to a hyperplane section of 𝔖.

For the final statement, the fifteen planes of 𝔖 cut out fifteen ordered lines of S. The remaining lines form a double-six. □

Let Y′ be a cubic threefold with ordinary double points at p_{1}, …, p_{6}; ϖ induces a rational map Y′ −−→ S contracting the twisted quartic curves in Y′ to points of a smooth hyperplane section i : S ↪ 𝔖 with a distinguished double-six. After ordering the two sextuples of disjoint lines, Proposition 10 yields a determinantal cubic hypersurface Y with ordinary double points at p_{1}, …, p_{6} corresponding to the marked cubic surface S. By Proposition 18, the image of Y under ϖ is a hyperplane section i_{2} : S ↪ 𝔖 with the planes of 𝔖 tracing out the corresponding 15 lines of S. Proposition 21 implies i_{1}(S) = i_{2}(S) and thus Y ≃ Y′; this yields an inverse to the morphism (7).

Remark 22. The natural map

\{ \begin{matrix}determinantal cubic threefolds \\ with ordinary double points \\ at p_{1}, …, p_{6} \end{matrix} \} → \{ \begin{matrix}cubic threefolds with \\ ordinary double points \\ at p_{1}, …, p_{6} \end{matrix} \}

is not an isomorphism. Under our identifications, these correspond to

\{ \begin{matrix}determinantal cubic surfaces \\ with a sextuple of ordered \\ lines \end{matrix} \} → \{ \begin{matrix}cubic surfaces with a \\ double-six of ordered \\ lines \end{matrix} \}

which has degree two. Indeed, this reflects the involution (8) interchanging the sextuples of our double-six.

6. Constructing flops

Let X be a smooth cubic fourfold with hyperplane class h.

Proposition 23. Assume that X admits a hyperplane section Y ⊂ X with six ordinary double points in linear general position. Then X contains two families of cubic scrolls T and T^{∨}, whose cycle classes satisfy

[T] + [T^{∨}] = 2h^{2}.

Conversely, if X is a smooth cubic fourfold containing a smooth cubic scroll T then the hyperplane section

Y = X ∩ span(T)

has at least six double points, counted with multiplicities.

Proof. Assume that X admits a hyperplane section Y as above. We may assume that Y is determinantal by Proposition 8. Proposition 17 guarantees that X contains two families of cubic scrolls, each parametrized by ℙ^{2}. Given T and T^{∨} from different families, we have

T ∪ T^{∨} = Y ∩ Q

for some quadric hypersurface in ℙ^{3}. The equation on cycle classes follows.

Now suppose that X contains a smooth cubic scroll T spanning the hyperplane section Y. In suitable coordinates,

T = \{[x_{0}, …, x_{4}] : rank ( \begin{matrix}x_{0} & x_{1} & x_{2} \\ x_{2} & x_{3} & x_{4} \end{matrix} ) = 1\}

and thus there exist linear forms y_{0}, y_{1}, y_{2} in x_{0}, …, x_{4} such that

Y = \{[x_{0}, …, x_{4}] : det ( \begin{matrix}x_{0} & x_{1} & x_{2} \\ x_{2} & x_{3} & x_{4} \\ y_{0} & y_{1} & y_{2} \end{matrix} ) = 0\},

i.e., Y is determinantal. The double points correspond to the matrices of rank one (cf. Proposition 9). □

Theorem 24. Let X be a smooth cubic fourfold not containing a plane, and F its variety of lines. Assume that X admits a hyperplane section Y with six ordinary double points in linear general position. Write

F (Y) = P ∪ S′ ∪ P^{∨}

with normalization ℙ^{2} ⊔ S ⊔ ℙ^{2}. Then there exist birational involutions

ι, ι^{∨} : F −−→ F

which are regular away from P ∪ S′ and P^{∨} ∪ S′ respectively. Precisely, ι is factored as follows:

(1) Flop P to get a new holomorphic symplectic fourfold F_{1}; the proper transform S_{1} of S is a plane in F_{1}.

(2) Flop the S_{1} in F_{1} to get F_{2}, which is isomorphic to F.

Proof. We construct ι: Let [m] ∈ F be a line not contained in F (Y). Then m ∩ Y = {y}, a nonsingular point of Y. By Corollary 12, there exists a unique line ℓ^{∨} ∈ P^{∨} containing y. Let Π denote the plane spanned by ℓ^{∨} and m; by assumption, Π ⊄ X. Thus we have

(10)  Π ∩ X = m ∪ ℓ^{∨} ∪ m̄

for some line m̄ ∈ F. Setting ι(m) = m̄, we get a morphism

ι : F \ F (Y) → F.

Since (10) is symmetric in m and m̄, ι is an involution.

As constructed, ι is not well-defined along F (Y). It remains to show that it extends to m ∈ P^{∨} \ (P ∪ S). Proposition 13 implies that m does not contain any singularities of Y. The normal bundle to m in X is one of the following [11, Proposition 6.19]

N_{m/X} ≃ O^{⊕2} ⊕ O(1), O(−1) ⊕ O(1)^{⊕2}.

Since m does not contain any of the singularities of Y, Corollary 12 implies that we have the first case. But then there exists a distinguished plane Π with

Π ∩ X = 2m ∪ m̄,

i.e., Π corresponds to the directions associated with the O(1)-summand. Consider the correspondence

Z = {(m, ℓ^{∨}, Π) : Π ∩ X ⊃ m ∩ ℓ^{∨}} ⊂ F × P^{∨} × Gr(2, 5);

the normal bundle computation guarantees that the projection

Z → F

is an isomorphism along P^{∨} \ (P ∪ S). By definition, ι is regular on Z and thus at the generic point of P^{∨}.

We will use the following notation for our factorization

F_{01}  F_{12}
↙^{β_{10}}  ↘^{β_{01}}  ↙^{β_{21}}  ↘^{β_{12}}
F = F_{0}  F_{1}  F_{2}
↘^{γ_{10}}  ↙^{γ_{01}}  ↘^{γ_{21}}  ↙^{γ_{12}}
F̄_{01}  F̄_{12}

where β_{10} blows up P, β_{01} blows down the exceptional divisor of β_{10}, β_{21} blows up S_{1} (the proper transform of S), and β_{12} blows down the exceptional divisor of β_{12}. Here F̄_{01} and F̄_{12} denote the singular varieties obtained by contracting P and S_{1} to a point. In other words, F_{2} is obtained from F_{0} by two Mukai flops. Moreover, we will show that ι : F −−→ F is resolved on passage to F_{2}, so the induced

(11)  ι_{2} : F_{2} → F

is necessarily an isomorphism.

Let P_{1} ⊂ F_{1} denote the plane that results from flopping P.

Lemma 25. S_{1} is isomorphic to ℙ^{2} and meets P_{1} transversely at six points.

Proof. Proposition 13 describes how S′ and P intersect: S′ has two smooth branches meeting transversely in F, each of which meets P ≃ ℙ^{2} in a line. If S^{∘} is the smooth locus of S′ then

P ∩ S^{∘} ⊂ S^{∘}

is Cartier, hence β_{10}^{−1}(S^{∘}) ≃ S^{∘}. However, S′ fails to be Cohen-Macaulay at the points of S′ \ S^{∘}, so any Cartier divisor through these points would necessarily have an embedded point. In particular,

P ∩ S′ ⊂ S′

is not Cartier at singular points of S′ and β_{10} necessarily modifies S′ at these points.

We claim that the proper transform S_{01} of S′ in F_{01} is just S. The easiest way to see this is through a local computation. At each singularity of S′ choose local coordinates {x_{1}, x_{2}, x_{3}, x_{4}} such that

S′ = {x_{1} = x_{2} = 0} ∪ {x_{3} = x_{4} = 0}, P = {x_{2} = x_{3} = 0}.

The blow-up of P has homogeneous equation

Ax_{3} = Bx_{2}

and thus the proper transforms of the components of S′ are disjoint and mapped isomorphically onto their images.

We next show that β_{01} contracts the double-six in S corresponding to the intersection of P with S′. The key ingredient is the local description of proper-transforms of Lagrangian submanifolds under Mukai flops given in [30, §4.2]: Locally, a holomorphic-symplectic fourfold containing a plane looks like the total space of the cotangent-bundle of ℙ^{2}. In the cotangent bundle, a complex Lagrangian submanifold is modelled locally as the conormal sheaf N_{V}^{∗} of a complex submanifold V ⊂ ℙ^{2}. The Mukai flop is realized as the cotangent bundle of the dual plane ℙ̌^{2}; the proper transform of the Lagrangian submanifold looks locally like the conormal sheaf N ∗ ; V̌ of the projective dual V̌ ⊂ ℙ̌^{2}. Since each branch of S′ looks locally like N_{ℓ}^{∗} for a line ℓ ⊂ ℙ^{2}, its proper transform looks locally like N_{[ℓ]}^{∗}, where [ℓ] ∈ ℙ̌^{2} classifies ℓ. In particular, the proper transform S_{1} meets P_{1} in six points and β_{01} : S_{01} → S_{1} blows down the double-six (corresponding to S′ ∩ P) to these points.

The analysis in Proposition 13 implies S_{1} ≃ ℙ^{2}, and thus is a Lagrangian plane in F_{1}. □

We define F_{2} as the Mukai flop of this plane and let S_{2} denote the resulting plane, P_{2} the proper transform of P_{1}, and

ϕ : F_{2} −−→ F

the composition of Mukai flops. Now P_{2} is isomorphic to a cubic surface, meeting S_{2} along a double-six as described before.

It remains to show that ι is resolved on F_{2}. Let g denote the polarization on F ⊂ Gr(2, 5) induced by the hyperplane class on the Grassmannian. The pull back ι^{∗}g is globally generated away S ∪P, the indeterminacy of ι. We shall use the following result of Boucksom [7]: Let Z be an irreducible holomorphic symplectic variety and f a divisor class on Z; then f is ample if and only if f meets each rational curve on Z positively. In particular, ι^{∗}g necessarily meets some rational curves supported in S ∪P negatively. Consider the map on one-cycles modulo homological equivalence induced by normalization and inclusion

j_{∗} : N_{1}(S̃, ℤ) ⊕ N_{1}(P, ℤ) → N_{1}(F, ℤ).

This has a nontrivial kernel: The description in Proposition 13 implies that

j_{∗}(E_{i}, 0) = j_{∗}(0, [line]), i = 1, …, 6,

hence j_{∗} has rank two, and the image of the effective curve classes is spanned by λ_{1} = j_{∗}(0, [line]) and λ_{2} = j_{∗}(β_{01}^{∗}[line], 0). In particular, all the effective curve classes where ι^{∗}g is negative can be expressed as linear combinations of λ_{1} and λ_{2} with nonnegative coefficients.

We now analyze the pull-back f = ϕ^{∗}ι^{∗}g. Since ϕ is an isomorphism away from P_{2} ∪ S_{2}, any curve of F_{2} not contained in P_{2} ∪ S_{2} meets f positively. Just as before, the image of the effective curve classes in P_{2} ∪ S_{2} are a cone generated by two classes λ′_{1} and λ′_{2}. However, using the identifications

H_{2}(F_{2}, ℚ) ⟶^{∼} H^{2}(F_{2}, ℚ) ⟶^{ι^{∗}} H^{2}(F, ℚ) ⟶^{∼} H_{2}(F, ℚ)

we see that λ′_{1} = −λ_{1} and λ′_{2} = −λ_{2}. It follows that f is positive along all curves classes in F_{2} and thus is ample.

To reiterate, the composition (cf. Equation 11)

ι_{2} = ι ∘ ϕ : F_{2} −−→ F

is a birational map of smooth projective varieties, and takes the ample divisor g to an ample divisor f. It follows that ϕ is an isomorphism. □

Remark 26. The strategy of our argument is due to Burns-Hu-Luo [9], who prove that any birational morphism of irreducible holomorphic symplectic varieties with normal exceptional loci is a sequence of Mukai flops. The normality assumption can be eliminated (cf. [40, 1.2]).

Proposition 27. Retain the notation of Theorem 24. Let [v] ∈ ℙ(V) and T^{∨} := T_{v} ⊂ Y denote one of the cubic scrolls described in Proposition 17. The divisor

τ^{∨} = {[m] ∈ F : m ∩ T^{∨} ≠ ∅} ⊂ F

is invariant under ι.

Proof. By Proposition 17, the ruling of T^{∨} is the rational curve

λ_{1}^{∨} = {[v^{∨}] : v^{∨}(v) = 0} ⊂ P^{∨} ⊂ F (Y) ⊂ F.

If m is incident to T^{∨} then m meets some ruling ℓ_{[v^{∨}]} ⊂ T^{∨} where [v^{∨}] ∈ λ_{1}^{∨}. Thus ℓ_{[v^{∨}]} coincides with the line ℓ^{∨} used in the construction of ι. Since the lines {m, ℓ_{[v^{∨}]}, ι(m)} are coplanar, we have ι(m) ∩ ℓ_{[v^{∨}]} ≠ ∅ and ι(m) ∈ τ^{∨}. □

7. Cones of moving and ample divisors

Let X be a smooth cubic fourfold and F its variety of lines. The incidence correspondence

Z
↙^{π}  ↘^{ψ}
X  F

induces the Abel Jacobi map of integral Hodge structures [6]

α = ψ_{∗}ϕ^{∗} : H^{4}(X) → H^{2}(F).

This is compatible with quadratic forms: Writing g = α(h^{2}) we have

(g, g) = 2 ⟨h^{2}, h^{2}⟩ = 6

and

(α(z_{1}), α(z_{2})) = − ⟨z_{1}, z_{2}⟩ for all z_{1}, z_{2} ∈ (h^{2})^{⊥}.

For general cubic fourfolds we have

H^{4}(X, ℤ) ∩ H^{2,2}(X, ℂ) = ℤh^{2},

but special cubic fourfolds admit additional algebraic cycles [21]:

Proposition 28. Let C denote the moduli space of smooth cubic fourfolds and C_{d} ⊂ C the cubic fourfolds X admitting a rank-two saturated lattice of Hodge cycles

h^{2} ∈ K_{d} ⊂ H^{4}(X, ℤ) ∩ H^{2,2}(X, ℂ)

of discriminant d. Then C_{d} is nonempty if and only if

d ≡ 0, 2 (mod 6), d > 6.

In this case, C_{d} is an irreducible divisor in C.

Assume that X contains a smooth cubic scroll T and let Y denote the hyperplane section of X containing T. The intersection form ⟨, ⟩ on the middle cohomology of X restricts to

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

a lattice of discriminant 12. Moreover, C_{12} is the closure of the locus of cubic fourfolds containing a cubic scroll or a hyperplane section with six double points in general position.

Let τ = α(T) so that the Beauville-Bogomolov form restricts to

J_{12} := \begin{aligned}& g & τ \\ g & 6 & 6 \\ τ & 6 & 2 \end{aligned}

a lattice of discriminant −24. We summarize elementary properties of this lattice:

(1) The elements of J_{12} ⊗_{ℤ} ℝ with nonnegative Beauville form are a union of convex cones P ∪ −P where

P = Cone (g − (3 − \sqrt{6})τ, (3 + \sqrt{6})τ − g) .

(2) J_{12} does not represent −2 or 0.

(3) The automorphism group of J_{12} is isomorphic to the direct product of ⟨±1⟩ and the infinite dihedral group

Γ := ⟨R_{1}, R_{2} : R_{1}^{2} = R_{2}^{2} = 1⟩

where the reflections R_{1}, R_{2} can be written

\begin{aligned}R_{1}(g) & = & g & R_{1}(τ) & = & 2g − τ \\ R_{2}(g) & = & −g + 6τ & R_{2}(τ) & = & τ. \end{aligned}

Γ consists of the automorphisms taking P to itself.

(4) J_{12} represents −10. We list (−10)-classes with positive intersection with g:

class  intersection with g
⋮  ⋮
ρ_{3}^{∨} = 16τ − 3g  (ρ_{3}^{∨}, g) = 78
ρ_{2}^{∨} = 4τ − g  (ρ_{2}^{∨}, g) = 18
ρ_{1}^{∨} = 2τ − g  (ρ_{1}^{∨}, g) = 6
ρ_{1} = 3g − 2τ  (ρ_{1}, g) = 6
ρ_{2} = 7g − 4τ  (ρ_{2}, g) = 18
ρ_{3} = 29g − 16τ (ρ_{3}, g) = 78
⋮  ⋮

For j ≥ 3 we define recursively

ρ_{j} = (R_{1}R_{2})ρ_{j−2} and ρ_{j}^{∨} = (R_{2}R_{1})ρ_{j−2}^{∨}.

(5) The element R_{1}R_{2} has infinite order and acts on the (−10)- classes with orbits:

\begin{matrix}\{…, ρ_{3}^{∨}, ρ_{1}^{∨}, −ρ_{2}, …\} \{…, −ρ_{3}^{∨}, −ρ_{1}^{∨}, ρ_{2}, …\} \\ \{…, −ρ_{2}^{∨}, ρ_{1}, ρ_{3} …\} \{…, ρ_{2}^{∨}, −ρ_{1}, −ρ_{3} …\}. \end{matrix}

The element R_{1} has order two and acts via

R_{1}(ρ_{i}) = ρ_{i}^{∨}

for each i.

Proposition 29. Let X be a smooth cubic fourfold with variety of lines F. Assume that X contains a smooth cubic scroll T with

H^{4}(X, ℤ) ∩ H^{2,2}(X, ℂ) = ℤh^{2} + ℤT

or equivalently

H^{2}(F, ℤ) ∩ H^{1,1}(F, ℂ) = ℤg + ℤτ.

The nef cone of F equals the cone dual to Cone(ρ_{1}, ρ_{1}^{∨}), i.e., Cone(α_{1}, α_{1}^{∨}) where α_{1} = 7g − 3τ and α_{1}^{∨} = g + 3τ.

Proof. The main theorem of [23] asserts that a divisor class f on F is nef if (f, ρ) ≥ 0 for each divisor class ρ on F satisfying (g, ρ) > 0 and either (ρ, ρ) ≥ −2 or (ρ, ρ) − 10 and (ρ, H^{2}(F, ℤ)) = 2ℤ. Thus the nef cone contains Cone(α_{1}, α_{1}^{∨}).

It remains to show that α_{1} and α_{1}^{∨} are at the boundary of the nef cone. We show they induce nontrivial contractions of F. It suffices to prove this for a generic cubic fourfold containing a cubic scroll, so we may assume Y has exactly six ordinary double points in linear general position. It follows that there exist Lagrangian planes

P, P^{∨} ⊂ Y ⊂ F

such that lines λ_{1} ⊂ P and λ_{1}^{∨} ⊂ P^{∨} both have degree three. The classes of these lines are dual to ρ_{1} and ρ_{1}^{∨} respectively. The divisors α_{1} and α_{1}^{∨} induce small contractions

γ_{10} : F = F_{0} → F̄_{01}, γ_{01}^{∨} : F_{0} → F̄_{10}^{∨}

of P and P^{∨} respectively. □

We may flop the plane P (resp. P^{∨}) in F to obtain new holomorphic symplectic fourfold F_{1} (resp. F_{1}^{∨}). The birational maps between F, F_{1}, and F_{1}^{∨} induce identifications of their Picard groups. The first step in analyzing their nef cones is to enumerate the orbit of α_{1} under Γ:

class  intersection with g
⋮  ⋮
α_{2}^{∨} = R_{2}R_{1}(α_{1}) = 9τ − g  (α_{2}^{∨}, g) = 48
α_{1}^{∨} = R_{1}(α_{1}) = g + 3τ  (α_{1}^{∨}, g) = 24
α_{1} = 7g − 3τ (α_{1}, g) = 24
α_{2} = R_{1}R_{2}(α_{1}^{∨}) = 17g − 9τ (α_{2}, g) = 48
⋮  ⋮

For j ≥ 3 we define recursively

α_{j} = R_{1}R_{2}(α_{j−2}) and α_{j}^{∨} = R_{2}R_{1}(α_{j−2}^{∨}).

Observe that (α_{i}^{∨}, ρ_{i}^{∨}) = (α_{i}, ρ_{i}) = 0 for each i ≥ 1.

Proposition 30. The nef cone of F_{1} (resp. F_{1}^{∨}) equals Cone(α_{1}, α_{2}) resp. Cone(α_{2}^{∨}, α_{1}^{∨}).

Figure 1 illustrates the relative positions of the nef cones of F and its flop F_{1}^{∨}.

Proof. Since F_{1} is the flop of F along P, it is clear that α_{1} (which induces the contraction of P) is one generator of the nef cone. The factorization given in Theorem 24 shows that ι induces a regular involution on F_{1}. Proposition 27 implies that ι fixes the divisor

τ^{∨} = α(T^{∨}) = α(2h^{2} − T) = 2g − τ,

where the middle equality uses Proposition 23. Consequently, ι acts on J_{12} via the reflection R_{3} through the line orthogonal to 2g − τ

R_{3}(g) = 11g − 6τ, R_{3}(τ) = 20g − 11τ.

Figure 1. The nef cones of F and F_{1}^{∨}

The second generator of the nef cone of F_{1} is thus

R_{3}(α_{1}) = 17g − 9τ = α_{2}.
□

Theorem 24 gives isomorphisms between every second model, so Propositions 29 and 30 suffice to describe the nef cone of every model of F (see Figure 2). We summarize our whole discussion:

Theorem 31. Suppose that X is a smooth cubic fourfold containing a smooth cubic scroll T with

H^{4}(X, ℤ) ∩ H^{2,2}(X, ℂ) = ℤh^{2} + ℤT

and let F = F_{0} denote the variety of lines on X. Then we have an infinite sequence of Mukai flops

⋯ F_{2}^{∨} −−→ F_{1}^{∨} −−→ F_{0} −−→ F_{1} −−→ F_{2} ⋯

with isomorphisms between every other flop in this sequence

⋯ F_{2}^{∨} →^{∼} F_{0} →^{∼} F_{2} ⋯ and ⋯ F_{1}^{∨} →^{∼} F_{1} ⋯

The positive cone of F can be expressed as the union of the nef cones of the models {⋯, F_{1}^{∨}, F_{0}, F_{1}, ⋯ }:

⋯, Cone(α_{2}^{∨}, α_{1}^{∨}), Cone(α_{1}^{∨}, α_{1}), Cone(α_{1}, α_{2}), ⋯

The isomorphisms induce an action of ℤ ≃ ⟨R_{1}R_{2}⟩ ⊂ Γ on the Picard group of F.

Figure 2 is a schematic illustration of the partition of the positive cone into ample cones for isomorphism classes of minimal models. This verifies the conjectures of [22] for the birational models of F. It

Figure 2. Partition of the positive cone into ample cones for various minimal models

also illustrates the Finiteness of Models Conjecture of Kawamata and Morrison–here we have two birational models up to isomorphism.

Remark 32 (Application to rational points). Let F be the variety of lines of a cubic fourfold X containing a cubic scroll T, both defined over a field k. Assume that the hyperplane section containing T has precisely six ordinary double points in linear general position and X does not contain a plane. Then k-rational points on F are Zariski dense. Indeed, the infinite collection of Lagrangian planes defined over k is Zariski dense.

If the Picard group of F has rank two then • F does not admit regular automorphisms, and

• F is not birational to an abelian fibration. Potential density of rational points on varieties of lines on generic

cubic fourfolds over number fields has recently been established in [2].

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