1 Introduction↩︎

Cubic fourfolds are a central object in algebraic geometry, studied with respect to rationality questions and for their connections to hyperkähler manifolds. In a very general cubic fourfold, any algebraic surface is homologous to a complete intersection, but a countably infinite collection of divisors \(\mathcal{C}_d\) in the moduli space of cubic fourfolds parametrize cubics containing extra algebraic surfaces [1]. Cubics contained in these divisors are often more interesting from the point of view of rationality questions: it is conjectured that the rational cubic fourfolds are contained in the union of certain divisors \(\mathcal{C}_d\). The Fano variety of lines \(F\) on a cubic fourfold \(X\) is a hyperkähler manifold by [2], and when \(X\in \mathcal{C}_d\), typically \(F\) exhibits richer birational geometry, which can be studied via the Global Torelli theorem (due to Huybrechts, Markman and Verbitsky). Here, we focus on the Fano varieties of lines on cubic fourfolds belonging to the family \(\mathcal{C}_{12}\), whose general member contains a cubic scroll.

The divisor \(\mathcal{C}_{12}\) was first studied in [3], where the authors studied the birational models of the corresponding Fano variety of lines, and exhibited a birational transformation of infinite order. More recently, it has been proved that \(\mathcal{C}_{12}\) contains many geometrically interesting families of cubic fourfolds. For example, there is a ten dimensional family of cubics admitting an involution fixing a line pointwise, denoted by \(\mathcal{M}_{\phi_2}\) in the notation of [4]. For \(X\in \mathcal{M}_{\phi_2},\) the middle algebraic cohomology of \(X\) is spanned by classes represented by cubic scrolls, along with the square of the hyperplane class. Any pair of cubic scrolls spanning different hyperplane sections of \(X\) is either syzygetic or non-syzygetic, meaning they intersect in three or one points, respectively (2). Further, such a cubic fourfold is conjecturally irrational [4], and one could hope this is reflected in the birational geometry of the associated hyperkähler manifolds.

Motivated by this, we study the birational geometry of the Fano variety \(F\) of lines on a very general cubic fourfold \(X\) containing either a syzygetic or a non-syzygetic pair of cubic scrolls. In both cases, \(F\) has Picard rank three, and we describe the movable and ample cones of \(F\) by identifying the wall divisors following techniques from [5]. We first consider a cubic fourfold \(X\) with a syzygetic pair of cubic scrolls — in this case, the movable cone is bounded by infinitely many walls corresponding to prime exceptional divisors (see 2.4). We prove the following result:

In the non-syzygetic case, the movable cone coincides with the positive cone, but surprisingly the birational geometry of \(F(X)\) is more complicated. This is partially explained by the fact that cubics containing a non-syzygetic pair of cubic scrolls automatically contain a third family of cubic scrolls (see 3).

As a consequence, we obtain the first examples of the following phenomenon:

The relationship between these cubics is explored more fully in two later papers. In [6], they are shown to be birationally equivalent Fourier–Mukai partners, and the authors of [7] use Gale duality to relate the equations of the two cubics.

These results are of interest for three main reasons. First, they provide examples of hyperkähler fourfolds of Picard rank three whose birational geometry is explicitly understood; to our knowledge, this has only been done in the significantly easier case of Picard rank two. Second, our techniques also allow us to deduce information about the birational automorphism group of \(F\), even though understanding the structure of \(\mathrm{Bir}(F)\) is in general more difficult than enumerating the birational models. Specifically, we obtain the following structural result for \(\mathrm{Bir}(F)\):

Finally, the birational equivalence between \(F\) and double EPW sextics continues a beautiful story of associations between cubics in \(\mathcal{C}_{12}\) and Gushel–Mukai fourfolds. In [8], the authors show that a general \(X\in \mathcal{C}_{12}\) is birational to a Hodge-theoretically associated to a GM fourfold \(Z\) containing a plane; in [9], \(X\) and \(Z\) are moreover shown to be Fourier–Mukai partners. The fourfold \(Z\) in turn determines a double EPW sextic, a hyperkähler manifold introduced by O’Grady ([10]–[12]). In [13], the authors show these manifolds can be constructed from considering conics on the general Gushel–Mukai fourfold. This result was extended to all smooth ordinary Gushel–Mukai fourfolds in [14]. Combining both the birational map between \(X\) and \(Z\) and this geometric description of the double EPW sextic in terms of conics on \(Z\), we construct birational maps between \(F\) and two non-isomorphic double EPW sextics explicitly. Along with correcting the census of birational models of \(F\) given in [3], this yields a clearer geometric interpretation of each model and of the birational automorphism group, persisting even when \(X\) specializes to both the syzygetic and non-syzygetic cases.

Outline↩︎

In Section 2, we recall the relevant definitions and results about cubic threefolds and fourfolds containing cubic scrolls, the Fano variety of lines on such a cubic fourfold, and the birational geometry of hyperkähler manifolds of K3\(^{[2]}\)-type. We also introduce Gushel–Mukai fourfolds and double EPW sextics. In Section 3, we study the birational geometry of the Fano variety of lines \(F\) on a very general cubic fourfold \(X\) containing a cubic scroll, proving that \(F\) is birational to two non-isomorphic double EPW sextics. In particular, Theorem 7 corrects the census of birational models of \(F\) originally provided in [3]. In Sections 4 and 5, we prove Theorems 1 and 2 (Theorems 8 and 10), respectively, as well as Theorem 3 (Theorems 9 and 11), by carrying out an analysis of the chambers of the movable cone of \(F\) when \(X\) contains a syzygetic and non-syzygetic, respectively, pair of cubic scrolls. Explicit examples in Appendix 6 illustrate generic behavior of such cubic fourfolds.

Acknowledgments↩︎

We thank Nicolas Addington, Brendan Hassett, Alex Perry, Jack Petok, and Yuri Tschinkel for valuable conversations. In particular, Nicolas Addington suggested the possible connection to double EPW sextics explored in Section 3.3, and Brendan Hassett suggested techniques that streamlined our arguments enumerating the birational models of \(F\). The computations in Appendix 6 were done in Magma [15]. Finally, we thank the anonymous referees and Alexander Kuznetsov for their comments and remarks that helped the exposition.

This material is based upon work supported by the NSF Grant DMS-1929284 while the authors were in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the Hyperkähler Manifolds and Special Cubic Fourfolds Collaborate@ICERM Program. The authors were also partially supported by the Hausdorff Research Institute for Mathematics, while in residence for the Junior Trimester Program on Algebraic geometry: derived categories, Hodge theory, and Chow groups, funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC-2047/1 – 390685813. S.F. was supported in part by an AMS-Simons travel grant and NSF grant DMS-2401601. L.M. was supported in part by NSF grant DMS-2503390.

2 Preliminaries↩︎

We consider complex cubic threefolds and fourfolds that contain rational normal cubic scrolls, hereafter referred to as cubic scrolls.

In 2.1 we recall generalities on nodal cubic threefolds \(Y\subset \mathbb{P}^4\) that contain cubic scrolls. In particular, we recall the description of the components of the Fano variety of lines \(F(Y)\) for such a threefold due to [3]. In 2.2, we focus on smooth cubic fourfolds \(X\subset \mathbb{P}^5\) containing cubic scrolls, defining the notion of syzygyetic and non-syzygetic pairs of scrolls (2). In 2.3, we describe the Abel–Jacobi map relating the cohomology of \(X\) to its Fano variety of lines. In 2.4, we describe the structure of the movable and ample cone of a hyperkähler manifold of K3\(^{[2]}\)-type (5). Finally, in 2.5 we recall the relevant results on Gushel–Mukai fourfolds and associated double EPW sextics, emphasizing a geometric point of view.

2.1 Cubic threefolds containing cubic scrolls↩︎

Let \(Y\) be a cubic threefold containing a cubic scroll \(T\), as studied in detail in [3]. A generic such cubic threefold has six nodes in general position. Conversely, any cubic threefold with six nodes in general position contains a cubic scroll. Indeed, such a cubic threefold is determinental, and the cubic scrolls in \(Y\) correspond to the degeneracy loci of 2 by 3 and 3 by 2 minors of this determinental matrix. We summarize some of their geometric properties in this section.

By [3], \(Y\) contains another cubic scroll \(T^\vee\) residual to \(T\) in a quadric, and two nets of cubic scrolls homologous to \(T\) and \(T^\vee\) respectively. Each node of \(Y\) is contained in all of the cubic scrolls in \(Y\), and the complete linear system \(|\mathcal{I}_{T/Y}(2)|\) induces a rational map \(q\colon Y\dashrightarrow\mathbb{P}^2\), resolved by passing to a small resolution \(Y^+\) of \(Y\) as in the diagram below.

As noted in [16], the map \(p\) is a \(\mathbb{P}^1\)-bundle, and for a general line \(\ell\subset \mathbb{P}^2\), \(f_*p^*(\ell)\) is a cubic scroll homologous to \(T^\vee\).

Similarly, the linear system \(|\mathcal{I}_{T^\vee/Y}(2)|\) induces a map \(q^\vee\colon Y\dashrightarrow\mathbb{P}^2\) resolved by a small resolution \(Y^-\) which differs from \(Y^+\) by six Atiyah flops [16].

The Fano variety of lines on \(Y\) has three components:

More specifically, the normalization \(S\to S'\) resolves fifteen apparent double points on \(S'\), corresponding to the fifteen lines passing through two nodes of \(Y\) [3].

2.2 Special cubic fourfolds in \(\mathcal{C}_{12}\)↩︎

For a smooth cubic fourfold \(X\), we define the lattice of algebraic cycles (using the fact that cubic fourfolds satisfy the integral Hodge conjecture [18]): \[A(X)=H^4(X,\mathbb{Z})\cap H^{2,2}(X,\mathbb{C}).\] Letting \(\eta_X\) be the square of the hyperplane class, we also define \[H^4(X,\mathbb{Z})_{prim}=\langle\eta_X\rangle^\perp\subset H^4(X,\mathbb{Z})\] and \[A(X)_{prim}=H^4(X,\mathbb{Z})_{prim}\cap A(X).\] Each of the above is a lattice under the intersection pairing. For a very general cubic fourfold, \(A(X)_{prim}=0\), but countably many divisors \(\mathcal{C}_d\) in the moduli space of cubic fourfolds parametrize those with \(A(X)_{prim}\neq0\). Whenever \(\eta_X\in K\subset A(X)\) where \(K\) is a primitive sublattice of rank two, and the discriminant of the intersection form on \(K\) is \(d\), we say \(X\in\mathcal{C}_d\), and \(K\) is called a labeling of \(X\).

We focus in particular on smooth cubic fourfolds \(X\subset \mathbb{P}^5\) that contain a cubic scroll \(T\), or equivalently contain a hyperplane section \(Y=X\cap H\) with six nodes in general position [3]. A cubic scroll \(T\subset X\) determines a labelling \(K_{12}\hookrightarrow A(X)\) (see [1]), so \(X\in \mathcal{C}_{12}.\) Indeed, the intersection pairing on \(\langle\eta_X,T\rangle\subset A(X)\) is given by:

As a consequence of Section 2.1, we immediately see the following:

In much of what follows, we take \(X\) to contain two cubic scrolls \(T_1\) and \(T_2\) spanning different hyperplanes \(H_1\) and \(H_2\). For a very general such \(X\), the classes \(\eta_X\), \([T_1]\), and \([T_2]\) span \(A(X)\), and the intersection form is given by

where \(\tau\in\{1,2,3,4,5\}\) [4]. Note that if \([T_1]\cdot [T_2]=\tau,\) then \([T_1^\vee]\cdot [T_2]=6-\tau\) since \([T_1]+[T_1^\vee]=2\eta_X\); hence we can take \(\tau\in\{1,2,3\}\).

Thus, one can interpret the intersection of \(T_1\) and \(T_2\) in terms of the intersection of two twisted cubics on \(\Sigma=X\cap H_1\cap H_2\), (see [4]). This observation motivates the following definition:

Indeed, the twisted cubics \(T_1\cap H_2\) and \(T_2\cap H_1\) on the cubic surface \(\Sigma=X\cap H_1\cap H_2\) form a syzygetic duad in the first case, as defined in [21], and an azygetic duad in the second. We study these twisted cubics in Lemma 9.

In the non-syzygetic case, there are exactly two other algebraic cycles with the same numerics as a cubic scroll, namely: \[\begin{align} \label{eqn:T3def} \begin{aligned} [T_3]&=3\eta_X-[T_1]-[T_2],\\ [T_3^\vee]&=[T_1]+[T_2]-\eta_X. \end{aligned} \end{align}\tag{2}\]

We will also make use of the following fact:

2.3 The Fano variety of lines on a cubic fourfold↩︎

Let \(X\) be a smooth cubic fourfold and \(F\) its Fano variety of lines, a hyperkähler fourfold of K3\(^{[2]}\)-type [2].

The Abel–Jacobi map \[\alpha\colon H^4(X,\mathbb{Z})\rightarrow H^2(F,\mathbb{Z})\] restricts to an isomorphism \(H^4(X,\mathbb{Z})_{prim}\rightarrow H^2(F,\mathbb{Z})_{prim}\) compatible with the Hodge filtrations and (up to sign) with the quadratic forms on each lattice; more specifically, we have \[q(\alpha(x),\alpha(y))=-x\cdot y\] where \(q\) is the Beauville–Bogomolov–Fujiki form (BBF form) of \(F\). We write \(v^2\mathrel{\vcenter{:}}= q(v,v)\) for \(v \in H^2(F,\mathbb{Z})\). Letting \(g\) be the Plücker polarization on \(F\), we have that \(\alpha(\eta_X)=g\) and, moreover, \(g^2=6\) [2].

If \(A(X)=\langle \eta_X,T\rangle\) where \(T\) is a cubic scroll, then setting \(\lambda=\alpha(T-\eta_X)\) and applying the compatibilities of the Abel–Jacobi map as described above, we see that the BBF form on \(\mathrm{NS}(F)\) is given by

and the discriminant group of \(\mathrm{NS}(F)\) is \[D_{\mathrm{NS}(F)}\cong\mathbb{Z}/6\times\mathbb{Z}/4,\] with factors generated by \([\frac{g}{6}]\) and \([\frac{\lambda}{4}]\in D_{\mathrm{NS}(F)}\).

Next, suppose \(A(X)=\langle \eta_X,T_1,T_2\rangle\) where \(T_1\) and \(T_2\) are cubic scrolls spanning different hyperplanes. Let \(\lambda_i=\alpha(T_i-\eta_X)\). If \(T_1\) and \(T_2\) are a syzygetic pair, then the BBF form on \(\mathrm{NS}(F)\) is

and the discriminant group \[D_{\mathrm{NS}(F)}\cong\mathbb{Z}/6\times(\mathbb{Z}/4)^2\] has factors generated by \([\frac{g}{6}]\), \([\frac{\lambda_1}{4}]\), and \([\frac{\lambda_2}{4}]\). On the other hand, if \(T_1\) and \(T_2\) are a non-syzygetic pair labeled so that \([T_1]\cdot[T_2]=1\), the BBF form on \(\mathrm{NS}(F)\) is

and the discriminant group \[D_{\mathrm{NS}(F)}\cong\mathbb{Z}/6\times\mathbb{Z}/2\times\mathbb{Z}/6\] has factors generated by \([\frac{g}{6}]\), \([\frac{\lambda_1}{2}]\), and \([\frac{\lambda_1}{3}+\frac{\lambda_2}{6}]\).

2.4 Birational geometry of hyperkähler fourfolds↩︎

In order to study the birational geometry of \(F\), we use the Global Torelli theorem for hyperkähler manifolds of K3\(^{[2]}\)-type due to Verbitsky [23]. It was reformulated by Huybrechts [24] and Markman [25]; in particular, we use Markman’s Hodge-theoretic version. We denote by \(\mathrm{Mon}^2_{Hdg}(F)\) the subgroup of monodromy operators in \(\mathrm{Mon}^2(F)\) that preserve the Hodge structure. Recall that for a manifold \(F\) of K3\(^{[2]}\)-type, \(\mathrm{Mon}^2(F)\) is equal to the subgroup of elements of \(\mathrm{O}^+(H^2(F,\mathbb{Z}))\) that act by \(\pm \mathrm{Id}\) on the discriminant group [25].

The structure of the nef and movable cones of hyperkähler manifolds are well known: for K3\(^{[m]}\)-type, they are described in [25] and [26] using the extended Mukai-lattice. Here, we focus on the simpler case of manifolds of K3\(^{[2]}\)-type.

Let \(F\) be a projective hyperkähler manifold of K3\(^{[2]}\)-type. Let \(\overline{\mathrm{Pos}(F)}\) be the component of the cone \(\{x\in \mathrm{NS}(F)\otimes \mathbb{R}\mid x^2 \geq 0\}\) that contains an ample class. We denote by \(\mathrm{Mov}(F)\subset \overline{\mathrm{Pos}(F)}\) the (closed) convex cone generated by classes of line bundles on \(F\) whose base locus has codimension at least \(2\). By [27], \[\mathrm{Mov}(F) = \overline{\bigcup_{f\colon F\dashrightarrow F'} f^*\mathrm{Nef}(F')},\] where \(f\colon F\dashrightarrow F'\) is a birational map with \(F'\) a hyperkähler manifold.

We define the following set of divisors: \[\begin{align} \mathcal{W}_{\mathrm{pex}}&\mathrel{\vcenter{:}}= \{\rho \in \mathrm{NS}(F) \mid \rho^2=-2 \}\\ \mathcal{W}_{\mathrm{flop}}&\mathrel{\vcenter{:}}= \{\rho\in \mathrm{NS}(F) \mid \rho^2=-10, \textrm{div}(\rho)=2\}. \end{align}\] A divisor \(\rho\in \mathcal{W}_{\mathrm{pex}}\cup \mathcal{W}_{\mathrm{flop}}\) is called a wall divisor [5], and \(\rho\in \mathcal{W}_{\mathrm{pex}}\) is moreover referred to as a stably prime exceptional divisor. The following structure theorem for \(\mathrm{Mov}(F)\) and \(\mathrm{Amp}(F)\) in the case of hyperkähler manifolds of K3\(^{[2]}\)-type combines the results of Markman [25] and Bayer, Hassett and Tschinkel [26].

Note that each connected component of \[\textrm{Int}(\mathrm{Mov}(F))\setminus \bigcup_{\rho\in \mathcal{W}_{\mathrm{flop}}} \rho^\perp\] corresponds to \(f^*(\mathrm{Amp}(F'))\) for a birational map \(f\colon F\dashrightarrow F'\) with \(F'\) a hyperkähler manifold.

To obtain this chamber decomposition of \(\overline{\mathrm{Pos}(F)}\) and of \(\mathrm{Mov}(F),\) Markman uses a more explicit description of the group \(\mathrm{Mon}^2_{Hdg}(F)\) as follows. Let \(W_{Exc}\subset \mathrm{Mon}^2_{Hdg}(F)\) be the subgroup generated by reflections \(R_\rho\) for all \(\rho\in \mathcal{W}_{\mathrm{pex}}\), and let \(\mathrm{Mon}_{Bir}^2\subset \mathrm{Mon}^2_{Hdg}(F)\) be the subgroup generated by monodromy operators induced from birational transformations of \(F\). Let \(\pi\colon \mathrm{Mon}_{Hdg}^2(F)\rightarrow \mathrm{O}(\mathrm{NS}(F))\) be the restriction homomorphism. We record some useful facts:

2.5 GM fourfolds and double EPW sextics↩︎

Later, we will exhibit birational models of the Fano variety \(F\) of lines on a general cubic fourfold \(X\in\mathcal{C}_{12}\) as double Eisenbud-Popescu-Walter (EPW) sextics constructed from smooth Gushel–Mukai (GM) fourfolds. In this section we recall the relevant background and terminology.

Let \(V_5\) be a complex vector space of dimension \(5\).

Such a fourfold is a Fano variety with Picard number \(1\), index \(2\) and degree \(10\) [34]. In [13], the authors relate a GM fourfold \(Z\) to an EPW sextic \(W\) in the following way. Let \(I\mathrel{\vcenter{:}}= |\mathcal{O}_Z(2)|\) be the linear system of quadrics in \(\mathbb{P}^8\) containing \(Z\), and let \(\mathrm{Disc}(Z) \subset I\) be the irreducible component of the discriminant hypersurface that parametrizes singular quadrics that are not restrictions of the Plücker quadrics.

Taking an appropriate double cover of \(W\mathrel{\vcenter{:}}= \mathrm{Disc}(Z)\), one obtains a double EPW sextic associated to \(Z\) coinciding with the one constructed and studied by O’Grady in [10]–[12]. We instead review the construction of the double EPW sextic dual to this double cover. Let \(W^\vee\subset I^\vee\) be the hypersurface dual to \(\mathrm{Disc}(Z)\); when \(Z\) contains no decomposable vectors, \(W^\vee\) is again an EPW sextic [36] (see also [35]). We make use of the following geometric interpretation of \(W^\vee\), explained in [13] (see the construction of the map \(\alpha\) preceding Proposition 4.9); see [14] (specifically Sections 4 and 7.6) for a generalization.

3 Cubic fourfolds containing one cubic scroll↩︎

Let \(X\) be a smooth cubic fourfold containing a cubic scroll \(T\), and let \(F\) be its Fano variety of lines. We assume \(X\) is very general, by which we mean \(\mathrm{rank}(A(X))=2\), \(\mathrm{Aut}(X)=0\), and \(\mathrm{End_{Hdg}}(T(F))=\{\pm1\}\); the third assumption can be made in light of [37]. In [3], the authors compute the ample and movable cone of \(F\), and they exhibit a birational automorphism of \(F\) of infinite order. They also describe hyperkähler fourfolds birational to \(F\) as follows. The scroll \(T\) spans a hyperplane \(H\), and by Proposition 1, the Fano variety of lines on \(Y\mathrel{\vcenter{:}}= X \cap H\) is a subvariety of \(F\) decomposing as \(F(Y)=P\cup S'\cup P^\vee\), where \(P\) and \(P^\vee\) are Lagrangian planes. Let \(F_1\) and \(F_1^\vee\) be the Mukai flops of \(F\) along \(P\) and \(P^\vee\), respectively.

We expand on their study using the more recent techniques outlined in 2.4 for studying the birational geometry of hyperkähler manifolds. More specifically, we prove the following:

In 3.1, we study isometries of \(\mathrm{NS}(F)\), and in particular those induced by birational automorphisms of \(F\). In 3.2, we provide a correction to Hassett and Tschinkel’s count on the number of non-isomorphic birational hyperkähler models, showing that \(F\) has exactly three birational models up to isomorphism, represented by \(F\), \(F_1\), and \(F_1^\vee\); we also complete their description of \(\mathrm{Bir}(F)\). Finally, in 3.3, we prove that \(F_1\) (and similarly \(F_1^\vee\)) is in fact isomorphic to a double EPW sextic associated to the pair \((X,T^\vee)\) (similarly, the pair \((X, T)\)).

3.1 The Lattice \(\mathrm{NS}(F)\)↩︎

Recall from Section 2.3 that \(\mathrm{NS}(F)\) is isomorphic to the lattice \(J_{12}\). The discriminant group is \(\mathbb{Z}/6\times\mathbb{Z}/4\), with factors generated by \([\frac{g}{6}]\) and \([\frac{\lambda}{4}]\). The BBF form represents \(-10\) but not \(-2\), so \(F\) contains no prime exceptional divisors, and \(\mathrm{Mov}(F)=\overline{\mathrm{Pos}(F)}\).

As noted in [3], the isometry group of \(\mathrm{NS}(F)\) is the product \(\{\pm1\}\times\Gamma\), where \(\Gamma\) is the infinite dihedral group \[\Gamma=\langle R_1,R_2\;|\; R_1^2=R_2^2=1\rangle;\] explicitly, the generators are \[R_1=\begin{pmatrix*}[r] 1 & 0 \\ 0 & -1 \end{pmatrix*}\;\; \mathrm{and} \;\; R_2=\begin{pmatrix*}[r] 5 & -4 \\ 6 & -5 \end{pmatrix*}.\] We determine which isometries are induced by birational automorphisms of \(F\):

3.2 Census of birational models of \(F\).↩︎

The movable cone of \(F\) contains infinitely many chambers, corresponding to the nef cones of birational models of \(F\), as outlined in [3] and 2.4. To enumerate the chambers, we follow [3] by enumerating the wall divisors, i.e. \(v\in\mathrm{NS}(F)\) with \(v^2=-10\). Let \(\rho_1=g-2\lambda\), \(\rho_2=3g-4\lambda\), \(\rho_i^\vee=R_1(\rho_i)\), and for all integers \(n\), \[\rho_{2n+i}=(R_1R_2)^n\rho_i\] where we interpret \(\rho_{i}=\rho_{-i}^\vee\) for \(i<0\). Using standard propagation techniques, one sees that all the classes \(v\) with \(v^2=-10\) and \(q(v,g)\ge0\) are of the form \(\rho_i\) or \(\rho_i^\vee\) for some \(i\).

Let \(\alpha_i\) be the class spanning \(\rho_i^\perp\) and pairing positively with \(g\); similarly, let \(\alpha_i^\vee\) span \((\rho_i^\vee)^\perp\) and pair positively with \(g\). Then the walls of \(\mathrm{Mov}(F)\) are spanned by the \(\alpha_i\) and \(\alpha_i^\vee\). In [3], it is shown that \(\mathrm{Nef}(F)=\mathrm{Cone}(\alpha_1,\alpha_1^\vee)\), and by [3], we have \(\mathrm{Nef}(F_1)=\mathrm{Cone}(\alpha_1,\alpha_2)\) and \(\mathrm{Nef}(F_1^\vee)=\mathrm{Cone}(\alpha_1^\vee,\alpha_2^\vee)\). Moreover, for all \(n\), \(\mathrm{Cone}(\alpha_n,\alpha_{n+1})\) and \(\mathrm{Cone}(\alpha_n^\vee,\alpha_{n+1}^\vee)\) are chambers of \(\mathrm{Mov}(F)\) representing nef cones of birational models \(F_n\) and \(F_n^\vee\).

Essentially by construction, \[R_1R_2\cdot\mathrm{Nef}(F_n)=\mathrm{Nef}(F_{n+2}),\] so the authors of [3] conclude in Theorem 7.4 that \(F_n\simeq F_{n+2}\) for all \(n\) (again interpreting \(F_n=F_{-n}^\vee\) when \(n<0\)). In particular, this would mean that \(F\) has at most two birational models up to isomorphism, represented by \(F\) and \(F_1\). However, by Lemma 6, we see that \(R_1R_2\) is not induced by a birational automorphism of \(F\) so by Theorem 4 need not send the nef cone of one model to the nef cone of some isomorphic model. Instead, we provide the following correction to [3]:

3.3 Connection with Gushel–Mukai fourfolds↩︎

Here, we show that \(F_1\) and \(F_1^\vee\) are isomorphic to double EPW sextics. This relates each birational model of \(F\) to a familiar family of hyperkähler fourfolds and explains the birational involutions \(\iota\) and \(\iota^\vee\) on \(F\)—they are induced by the covering involutions associated to the double EPW sextics.

Since \(X\) is a smooth cubic fourfold containing a smooth cubic scroll \(T\), we can apply the following construction, due to [8] and [9]. The complete linear system of quadrics containing the scroll \(T\) induces a rational map \(q\colon X\dashrightarrow\mathbb{P}^8\) which is birational onto its image, a GM fourfold \(Z_T\subset \mathrm{Gr}(2,V_5)\) containing a plane \(\Pi\). Projection from \(\Pi\) induces a rational inverse \(f\colon {Z_T}\dashrightarrow X\). We obtain the following diagram, where \(Y^+\to Y\) is the small resolution from 1 , and \(E\) is the exceptional divisor.

Note that the isomorphism \(\mathrm{Bl}_TX\simeq \mathrm{Bl}_\Pi Z_T\) identifies \(Y^+\) with \(E\), so the image of \(Y^+\) in \(Z_T\) is \(\Pi\).

Let \(W\) be the EPW sextic associated to \(Z_T\) and \(W^\vee\) its dual, as defined in 2.5. While \(Z_T\) depends on the choice of scroll \(T\) in its homology class, \(W\) does not [8]. Let \(\widetilde{W}^\vee\) be the double cover of \(W^\vee\), also introduced in 2.5, equipped with its covering involution \(\tau\).

Let us briefly mention how the rational map \(\beta\) we constructed relates to recent work in [14], generalizing a construction from [13]. In [14], the authors prove that the Hilbert scheme of conics on \(Z_T\) has multiple irreducible components, one for each plane in \(Z_T\) and another denoted \(\overline{G_1^0(Z_T)}\). They moreover show that the component \(\overline{G_1^0(Z_T)}\) is birational to a fivefold \(G_1^+(Z_T)\) admitting a morphism \(f^+\colon G_1^+(Z_T)\to \widetilde{W}^\vee\) whose general fiber is \(\mathbb{P}^1\). Since \(q\) sends a general line on \(X\) to a conic on \(Z_T\) not lying on any plane of \(Z_T\), it induces a rational map \(F\dashrightarrow G_1^+(Z_T)\). The arguments above imply that the image of this rational map is a rational section of \(f^+\).

Starting instead with \(\mathrm{Bl}_{T^\vee}X\), one obtains an isomorphism between \(F_1\) and a double EPW sextic whose covering involution induces \(\iota\). By Lemma 6 and Remark 1, the covering involutions on the two double EPW sextics generate \(\mathrm{Bir}(F)\). This observation, together with Propositions 1 and Proposition 1, completes the proof of Theorem 7.

4 A syzygetic pair of cubic scrolls↩︎

Let \(X\subset \mathbb{P}^5\) be a smooth cubic fourfold containing a syzygetic pair of cubic scrolls \(T_1\) and \(T_2\). We also assume \(X\) is very general, meaning \(\mathrm{rank}(A(X))=3\), \(\mathrm{Aut}(X)=0\), and \(\mathrm{End_{Hdg}}(T(F))=\{\pm1\}\). In this section, we completely describe the birational geometry of the Fano variety \(F\) of lines on \(X\). Specifically, we prove the following:

Furthermore, we determine the birational automorphism group of \(F\):

The outline is as follows: in Section 4.1, we study the arrangement of Lagrangian planes in \(F\) and show that any two planes in \(F\) parametrizing lines contained in the hyperplanes spanned by \(T_1\) and \(T_2\) must intersect. We then identify which isometries of \(\mathrm{NS}(F)\) are induced by birational automorphisms of \(F\) in Section 4.2, including identifying how the involutions from Remark 1 act. After enumerating the (infinitely many) walls of the movable cone of \(F\) in Section 4.3, we enumerate the birational models and identify the birational automorphism group in Section 4.4.

4.1 Lagrangian planes in \(F\)↩︎

Let \(H_i\) be the hyperplane spanned by \(T_i\), and \(Y_i=X\cap H_i\). Recall from Section 2.1 that \[F(Y_i)=P_i\cup S_i'\cup P_i^\vee\] where \(P_i\) and \(P_i^\vee\) are Lagrangian planes. The intersection \(F(Y_1)\cap F(Y_2)\) parametrizes lines on the cubic surface \(\Sigma=X\cap H_1\cap H_2\). For a general cubic fourfold \(X\) satisfying the hypotheses above, \(\Sigma\) is smooth by Lemma 2. The following lemma implies the intersection \(F(Y_1)\cap F(Y_2)\) is transverse in general.

We also compare intersections on \(X\) to intersections on \(\Sigma\):

Recall from Remark 1 that a line \(\ell\subset Y_i\) is bisecant to \(A_i\) or \(A_i^\vee\) if and only if \([\ell]\in P_i\) or \(P_i^\vee\), respectively. Conversely, \([\ell]\in S_i'\) if and only if \(\ell\) meets both \(A_i\) and \(A_i^\vee\). So, one can tell which components of \(F(Y_1)\) and \(F(Y_2)\) a line \(\ell\subset\Sigma\) lies in by looking at the incidence relations between \(\ell\) and the curves \(\gamma_i\) and \(\gamma_i^\vee\).

In particular, since \(P_1\cap P_1^\vee\) and \(P_2\cap P_2^\vee\) are nonempty (see for example [3]), any two of the four Lagrangian planes in \(F\) studied here intersect. This clarifies what we prove later: after flopping any one of the planes in \(F\), one can no longer flop any of the other three.

4.2 The lattice \(\mathrm{NS}(F)\)↩︎

Letting \(\lambda_i=\alpha(T_i-\eta_X)\), recall from Section 2.3 that \(\mathrm{NS}(F)\) with the BBF form is isomorphic to the lattice \(J_{syz}\) and has discriminant group \(D_{\mathrm{NS}(F)}\cong\mathbb{Z}/6\times(\mathbb{Z}/4)^2\) with factors generated by \([\frac{g}{6}]\), \([\frac{\lambda_1}{4}]\), and \([\frac{\lambda_2}{4}]\). The BBF form represents both \(-2\) and \(-10\).

The isometry group of \(\mathrm{NS}(F)\) with basis \(\{g,\lambda_1,\lambda_2\}\) is generated by \(\pm1\) and the four reflections below. We calculated the isometry group \(\mathrm{O}(\mathrm{NS}(F))\) via the algorithm outlined in [40] and implemented using the Magma package AutHyp.m. \[R_1=\begin{pmatrix*}[r] 5 & 4 & 0 \\ -6 & -5 & 0 \\ 0 & 0 & -1 \end{pmatrix*}, R_2=\begin{pmatrix*}[r] 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{pmatrix*},\]

\[R_3=\begin{pmatrix*}[r] 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix*}, R_4=\begin{pmatrix*}[r] 7 & -4 & 4 \\ 6 & -3 & 4 \\ -6 & 4 & -3 \end{pmatrix*}.\]

The following lemma helps distinguish which isometries of the lattice \(\mathrm{NS}(F)\) are induced by birational automorphisms of \(F\).

Let \(F_i\) and \(F_i^\vee\) denote the flops of \(F\) along \(P_i\) and \(P_i^\vee\), respectively. As mentioned in Remark 1, each of these models admits a regular involution: \(\iota_i\) and \(\iota_i^\vee\), respectively. These involutions can also be regarded as birational involutions on \(F\).

4.3 Wall and chamber structure of \(\mathrm{Mov}(F)\)↩︎

In order to study the birational geometry of \(F\) and, in particular, to enumerate the birational hyperkähler models of \(F\) up to isomorphism, we give a description of the wall and chamber decomposition of the movable cone of \(F\).

As before, we fix the orthogonal basis \(\{g,\lambda_1,\lambda_2\}\) on \(\mathrm{NS}(F)\), and we denote a class \[v=ag+b\lambda_1+c\lambda_2\in \mathrm{NS}(F)\otimes \mathbb{R}\] by the vector \(v=(a,b,c)\).

We observe a nice geometric consequence:

Next, we describe the chambers of \(\mathrm{Mov}(F)\) neighboring \(\mathrm{Nef}(F)\).

Figure 1 illustrates the movable cone with labels on the chambers we have described so far. Figure 2 gives more detail further out from the central chamber.

We finish the section by demonstrating how to propagate all of the (infinitely many) walls of the movable cone. This is not necessary for enumerating the birational models of \(F\) up to isomorphism, but we will obtain a description of the birational automorphism group of \(F\) as a byproduct.

Let \(\Gamma\subset\mathrm{O}(\mathrm{NS}(F))\) be the subgroup generated by \(\iota_1^*,\) \((\iota_1^\vee)^*\), \(\iota_2^*\), and \((\iota_2^\vee)^*\). Since these lattice automorphisms are induced by birational automorphisms of \(F\), they preserve \(\mathrm{Mov}(F)\) and act on the sets \(\mathcal{W}_{\mathrm{flop}}\) and \(\mathcal{W}_{\mathrm{pex}}\) of wall divisors (see Section 2.4 for the definitions). In particular, they also act on the sets \[\; \Delta_{\mathrm{flop}}\mathrel{\vcenter{:}}= \{\rho\in\mathcal{W}_{\mathrm{flop}}\;|\;\rho^\perp\cap\overline{\mathrm{Mov}(F)}\neq\varnothing\}\] and \[\; \Delta_{\mathrm{pex}}\mathrel{\vcenter{:}}= \{\rho\in\mathcal{W}_{\mathrm{pex}}\;|\;\rho^\perp\cap\overline{\mathrm{Mov}(F)}\neq\varnothing\},\] which define walls between chambers in \(\overline{\mathrm{Mov}(F)}\) and walls bounding \(\overline{\mathrm{Mov}(F)}\), respectively.

We will return to the group \(\Gamma\) at the end of this section, proving that it is isomorphic to \(\mathrm{Bir}(F)\) in 1.

4.4 Birational geometry of \(F\)↩︎

With our understanding of the geometry of \(\mathrm{Mov}(F)\), we conclude by enumerating the birational hyperkähler models of \(F\) up to isomorphism, completing the proof of Theorem 8. We also describe the birational automorphism group of \(F\) completely.

Together with the content of Section 3.3, identifying the flops of \(F\) with pairs of dual double EPW sextics, Propositions 1 and 1 complete the proof of Theorem 8.

As mentioned previously, Proposition 1 explains geometrically why, after flopping any one of the four planes in \(F\), the other three planes cannot also be flopped. We see this in Figure 1, where the nef cones of \(F_i, F_i^\vee\) have two flopping walls and two prime exceptional walls.

We conclude by characterizing the birational automorphism group of \(F\).

Along with Corollary 3, Proposition 1 proves Theorem 9.

5 A non-syzygetic pair of cubic scrolls↩︎

Let \(X\subset \mathbb{P}^5\) be a smooth cubic fourfold containing a non-syzygetic pair of cubic scrolls. We also assume \(X\) is very general, meaning \(\mathrm{rank}(A(X))=3\), \(\mathrm{Aut}(X)=0\), and \(\mathrm{End_{Hdg}}(T(F))=\{\pm1\}\). As with the syzygetic case, we describe the birational geometry of the Fano variety \(F\) of lines on \(X\), proving the following main result:

As before, we also obtain generators for \(\mathrm{Bir}(F)\):

In contrast to Theorem 9, this surjection has nontrivial kernel; for further discussion, see Remark 1.

The outline of this section is similar to that of Section 4: in Section 5.1, we study the arrangement of Lagrangian planes in \(F\), showing that some pairs of planes intersect and others do not. We outline in Section 5.2 properties of the lattice \(\mathrm{NS}(F)\) and describe the actions on this lattice by the various involutions on the flops of \(F\). This allows us to describe the structure of the movable cone of \(F\) in Section 5.3, which further enables a census of the birational hyperkähler models of \(F\) and a description of \(\mathrm{Bir}(F)\) in Section 5.4.

5.1 Lagrangian planes in \(F\)↩︎

For a general cubic \(X\) containing a non-syzygetic pair of cubic scrolls \(T_1\) and \(T_2\), we saw in Lemma 3 that there is a third cubic scroll \(T_3\subset X\) and \([T_i]\cdot[T_j]=1\) for all \(i,j\). Let \(H_i\) be the hyperplane spanned by \(T_i\), and \(Y_i=X\cap H_i\). As in Section 2.1, we have \[F(Y_i)= P_i\cup S_i'\cup P_i^\vee\subset F,\] where \(P_i\) and \(P_i^\vee\) are Lagrangian planes. For \(i\neq j\), let \(\Sigma_{ij}=X\cap H_i\cap H_j\); for general \(X\), all three cubic surfaces \(\Sigma_{ij}\) are smooth, by Lemma 2. The lines on \(\Sigma_{ij}\) are parametrized by \(F(Y_i)\cap F(Y_j)\).

In particular, there are six pairs of disjoint planes among the \(P_i\) and \(P_i^\vee\), namely, \(P_i\cup P_j^\vee\) for \(i\neq j\). This means that in addition to the six flops of \(F\) along a plane \(P_i\) or \(P_i^\vee\), there are six models which can be obtained by flopping a pair of disjoint planes. Perhaps surprisingly, we will show in 23 that all six models of the last type are isomorphic to one another.

5.2 The lattice \(\mathrm{NS}(F)\)↩︎

Again, let \(\lambda_i=\alpha(T_i-\eta_X)\). Recall from Section 2.3 that \(\mathrm{NS}(F)\) with the BBF form is isomorphic to the lattice \(J_{nonsyz}\) and has discriminant group \(D_{\mathrm{NS}(F)}\cong\mathbb{Z}/6\times\mathbb{Z}/2\times\mathbb{Z}/6\) with factors generated by \([\frac{g}{6}]\), \([\frac{\lambda_1}{2}]\), and \([\frac{\lambda_1}{3}+\frac{\lambda_2}{6}]\). It is easy to check that this quadratic form represents \(-10\) but not \(-2\).

The isometry group of \(\mathrm{NS}(F)\) with basis \(\{g,\lambda_1,\lambda_2\}\) is generated by \(\pm1\) and the four transformations below. We calculated \(\mathrm{Aut}(\mathrm{NS}(F))\) via [40] and the Magma package AutHyp.m. \[R_1=\begin{pmatrix*}[r] 5 & 4 & -2 \\ -6 & -5 & 2 \\ 0 & 0 & -1 \end{pmatrix*}, R_2=\begin{pmatrix*}[r] 3 & 2 & -2\\ -2 & -1 & 2 \\ 2 & 2 & -1 \end{pmatrix*},\] \[R_3=\begin{pmatrix*}[r] 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix*}, R_4=\begin{pmatrix*}[r] 1 & 0 & 0 \\ 0 & 1 & -1 \\ 0 & 1 & 0 \end{pmatrix*}.\] Note that \(R_1\), \(R_2\), and \(R_3\) are reflections whereas \(R_4\) has order \(6\). The transformations \(R_3\) and \(R_4\) generate the dihedral group \(D_{12}\).

Let \(F_i\) and \(F_i^\vee\) denote the flops of \(F\) along \(P_i\) and \(P_i^\vee\), respectively, for \(i=1,2,3\). Recall from Remark 1 that each of these models admits a regular involution \(\iota_i\) or \(\iota_i^\vee\). These involutions can also be regarded as birational involutions on \(F\).

5.3 Wall and chamber structure of \(\mathrm{Mov}(F)\)↩︎

As in the syzygetic case, we enumerate birational models of \(F\) by studying the geometry of the movable cone. Unlike before, there are no prime exceptional divisors, so \(\mathrm{Mov}(F)=\overline{\mathrm{Pos}(F)}\).

Using the involutions on the six flops of \(F\), we can also describe the nef cones of the flops:

For \(1\le i,j\le3\), let \(F_{ij}\) be the flop of \(F\) along \(P_i\) and \(P_j^\vee\).

We illustrate the movable cone as described so far in Figure 3. In the next section, we will need to know the chambers bordering \(\mathrm{Nef}(F_{12})\), enumerated by the following lemma and illustrated analogously for \(\mathrm{Nef}(F_{32})\) in Figure 4.

5.4 Birational geometry of \(F\)↩︎

We now classify birational hyperkähler models of \(F\) up to isomorphism, proving Theorem 10.

The content of Section 3.3 identifies each flop of \(F\) with a double EPW sextic. To complete the proof of Theorem 10, we identify \(F_{12}\) with the Fano variety of lines on another smooth cubic fourfold:

We end by providing some information about the structure of \(\mathrm{Bir}(F)\). Let \[\Gamma=\langle\iota_i^*,(\iota_i^\vee)^*\;|\;i=1,2,3\rangle\subset\mathrm{O}(\mathrm{NS}(F)).\] Unlike in the syzygetic case, we are not able to make use of the action of \(\Gamma\) on \(\Delta_{\mathrm{flop}}\) in order to deduce generators and relations for \(\mathrm{Bir}(F)\); largely, this is because in the syzygetic case, each wall of \(\mathrm{Mov}(F)\) was adjacent to finitely many chambers whereas in the non-syzygetic case, each wall borders infinitely many chambers. On the other hand, the fact that the movable cone coincides with the positive cone somewhat streamlines the argument in the following lemma, analogous to Lemma 14.

This completes the proof of Theorem 11.

6 Examples↩︎

Here, we provide explicit examples of cubic fourfolds containing pairs of cubic scrolls in order to justify earlier assertions about generic behavior (cf. Lemma 2). The computational claims in the proof below can be verified with Magma code provided on the arXiv as an ancillary file.

We work over the field \(\mathbb{F}_{29}\), but the choices made in producing our examples amount to picking a point in a tower of projective bundles, as can be seen in the Magma code. Hence our examples lift to characteristic zero.

6.1 An explicit cubic fourfold with a syzygetic pair of cubic scrolls↩︎

Let \(X\) be the smooth cubic fourfold with defining equation \[\begin{align} f= 17x_0x_1x_2 &+ 19x_1^2x_2 + 9x_0x_2^2 + 10x_1x_2^2 + 18x_2^3 + 12x_0^2x_3 + 10x_0x_1x_3 + 8x_0x_2x_3 \\ &+ 4x_1x_2x_3 + 27x_2^2x_3 + 2x_0x_3^2 + 3x_2x_3^2 + 20x_0^2x_4 + 11x_0x_1x_4 \\ &+ 23x_1^2x_4 + 11x_0x_2x_4 + 24x_1x_2x_4 + 14x_2^2x_4 + 7x_0x_3x_4 + 26x_1x_3x_4 \\ &+ 19x_2x_3x_4 + 15x_0x_4^2 + 10x_1x_4^2 + 7x_0^2x_5 + 16x_0x_1x_5 + 18x_1^2x_5\\ &+ 22x_0x_3x_5 + 8x_1x_3x_5 + 23x_3^2x_5 + 18x_0x_4x_5 + 5x_1x_4x_5 + 7x_3x_4x_5\\ &+ 22x_4^2x_5 + 21x_1x_5^2 + 5x_3x_5^2 + 28x_4x_5^2 + 2x_5^3. \end{align}\] The hyperplanes \(H_1\) and \(H_2\) defined by \(x_5=0\) and \(x_2=0\), respectively, intersect \(X\) in six-nodal cubic threefolds \(Y_1\) and \(Y_2\). The cubic scroll \(T_1\subset H_1\), defined by the vanishing of the minors of the matrix \[M_1=\left(\begin{matrix} x_0& x_1& x_2\\ x_2 &x_3 &x_4 \end{matrix}\right),\] is contained in \(Y_1\). Similarly, the cubic scroll \(T_2\subset H_2\), defined by the vanishing of the minors of the matrix \[M_2=\left(\begin{matrix} l_0 & l_1 & l_2\\ l_2 & l_3 & l_4 \end{matrix}\right),\] where \[\begin{align} l_0&\mathrel{\vcenter{:}}= 17x_0 + 12x_1 + 12x_2 + 17x_3 + 7x_4 + 6x_5,\\ l_1&\mathrel{\vcenter{:}}= 17x_0 + 4x_1 + 17x_2 + 25x_3 + 18x_4 + 13x_5,\\ l_2&\mathrel{\vcenter{:}}= x_5,\\ l_3&\mathrel{\vcenter{:}}= 10x_0 + 13x_1 + 12x_2 + 15x_3 + 14x_4 + 17x_5, \\ l_4&\mathrel{\vcenter{:}}= 16x_0 + 13x_1 + 9x_2 + 10x_3 + 19x_4 + 7x_5, \\ \end{align}\]

is contained in \(Y_2\). The pair of cubic scrolls is syzygetic; \(T_1\) and \(T_2\) intersect transversely in three points, namely \((1:1:0:0:0:0)\), \((0:0:0:1:1:0)\), and \((0:1:0:1:0:0)\). The cubic surface \(\Sigma=Y_1\cap Y_2\) is smooth, as desired.

6.2 An explicit cubic fourfold with a non-syzygetic pair of cubic scrolls↩︎

Let \(X\) be the smooth cubic fourfold with defining equation \[\begin{align} f = 20x_0x_1x_2 &+ 4x_1^2x_2 + 15x_0x_2^2 + 17x_1x_2^2 + 6x_2^3 + 9x_0^2x_3 + 25x_0x_1x_3 + 27x_0x_2x_3 \\ &+ 15x_1x_2x_3 + 19x_2^2x_3 + 5x_0x_3^2 + 19x_2x_3^2 + 14x_0^2x_4 + 14x_0x_1x_4 \\ &+ 9x_1^2x_4 + 23x_0x_2x_4 + 25x_1x_2x_4 + 21x_2^2x_4 + 14x_0x_3x_4 + 10x_1x_3x_4 \\ &+ 18x_2x_3x_4 + 8x_0x_4^2 + 11x_1x_4^2 + 10x_0^2x_5 + 4x_0x_1x_5 + 24x_1^2x_5 \\ &+ 22x_0x_3x_5 + 16x_1x_3x_5 + 17x_3^2x_5 + 5x_0x_4x_5 + 18x_1x_4x_5 + 25x_3x_4x_5 \\ &+ 27x_4^2x_5 + 2x_0x_5^2 + 28x_1x_5^2 + 21x_3x_5^2 + 28x_4x_5^2 + 13x_5^3. \end{align}\] We consider the following hyperplanes: \[\begin{align} H_1&\mathrel{\vcenter{:}}= \{x_5=0\},\\ H_2&\mathrel{\vcenter{:}}= \{x_2=0\}, \\ H_3&\mathrel{\vcenter{:}}= \{x_0 + 24x_1 + x_2 + x_3 + 20x_4 + 9x_5=0\}. \end{align}\] One sees that \(H_1, H_2\) and \(H_3\) intersect \(X\) in six-nodal cubic threefolds \(Y_1\), \(Y_2\), and \(Y_3\) respectively. The cubic scroll \(T_1\subset H_1\) defined by the vanishing of the minors of the matrix \[M_1=\left(\begin{matrix} x_0& x_1& x_2\\ x_2 &x_3 &x_4 \end{matrix}\right),\] is contained in \(Y_1\). The cubic scroll \(T_2 \subset H_2\), defined by the vanishing of the minors of the matrix \[M_2=\left(\begin{matrix} l_0 & l_1 & l_2\\ l_2 & l_3 & l_4 \end{matrix}\right),\] where \[\begin{align} l_0&\mathrel{\vcenter{:}}= 26x_0 + 2x_1 + 8x_2 + 4x_3 + 5x_4 + 24x_5,\\ l_1&\mathrel{\vcenter{:}}= 13x_0 + 11x_1 + 22x_2 + 18x_3 + 6x_4 + 15x_5,\\ l_2&\mathrel{\vcenter{:}}= 12x_0 + 19x_1 + 15x_2 + 16x_3 + 17x_4 + 14x_5,\\ l_3&\mathrel{\vcenter{:}}= 18x_0 + 3x_1 + 18x_2 + 26x_3 + 18x_4 + 10x_5, \\ l_4&\mathrel{\vcenter{:}}= 28x_0 + 14x_1 + 5x_2 + 21x_3 + x_4 + 3x_5. \\ \end{align}\] is contained in \(Y_2\). The cubic scroll \(T_3\subset H_3\), cut out by the quadrics \[\begin{align} Q_{31}&\mathrel{\vcenter{:}}= x_1^2 + 17x_1x_3 + 27x_2x_3 + 9x_3^2 + 27x_1x_4 + 23x_2x_4 + 11x_3x_4 + 14x_4^2 + 24x_1x_5 \\ & + 13x_2x_5 + 10x_3x_5 + 2x_4x_5 + 8x_5^2,\\ Q_{32}&\mathrel{\vcenter{:}}= x_1x_2 + 25x_1x_3 + 20x_2x_3 + 5x_3^2 + 5x_1x_4 + 6x_2x_4 + 23x_3x_4 + 5x_4^2 + 20x_1x_5 \\ & + 2x_2x_5 + 24x_3x_5 + 3x_4x_5 + 8x_5^2,\\ Q_{33}&\mathrel{\vcenter{:}}= x_2^2 + 28x_1x_3 + 5x_2x_3 + 25x_3^2 + 20x_1x_4 + 14x_2x_4 + 15x_3x_4 + x_4^2 + 27x_1x_5 \\ &+ 15x_2x_5 + 10x_3x_5 + 2x_4x_5 + x_5^2, \end{align}\] is contained in \(Y_3\). The pairs \((T_1,T_2)\), \((T_1,T_3)\), and \((T_2,T_3)\) all form non-syzygetic pairs. Indeed, any two intersect transversely in one point; explicitly, \[\begin{align} T_1\cap T_2&=\{(0:1:0:1:0:0)\},\\ T_1\cap T_3&=\{(22 : 19 : 15 : 9 : 1 : 0)\},\\ T_2\cap T_3&=\{(15 : 9 : 0 : 15 : 9 : 1)\}. \end{align}\]

To verify that \([T_3]=3\eta_X-[T_1]-[T_2]\), it suffices to check \([T_3]\in\langle\eta_X,[T_1],[T_2]\rangle\). If not, then each of the non-syzygetic pairs gives rise to another \(\mathbb{F}_{29}\)-rational hyperplane slicing \(X\) in a cubic threefold singular along at least a length \(6\) zero-dimensional subscheme. In that case, \(X\) has at least six such hyperplane sections. Direct computation verifies that \(X\) has only four six-nodal hyperplane sections, cut out by \(H_1\), \(H_2\), \(H_3\), and \[x_0 + 16x_1 + 8x_2 + 11x_3 + 26x_4 + 13x_5=0.\] So, \(T_3\) represents the desired class in cohomology.

Now, let \(\Sigma_{ij}=Y_i\cap Y_j\). All three cubic surfaces \(\Sigma_{ij}\) are smooth, so the same is true for a general cubic fourfold containing a non-syzygetic pair of cubic scrolls.

References↩︎

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1. In Section 2 of [13], the authors include a generality assumption on the quadric \(Q\) used to define the GM fourfold \(Z_T\) and the double EPW sextic \(\widetilde{W}^\vee\) to ensure that \(Z_T\) is smooth and of the correct dimension; here, that generality assumption is satisfied.↩︎