Research

We study so-called non-syzygetic cubic fourfolds, i.e., smooth cubic fourfolds containing two cubic surface scrolls in distinct hyperplanes with intersection number between the two scrolls equal to $1$. We prove that a very general non-syzygetic cubic fourfold has precisely one nontrivial Fourier-Mukai partner that is also non-syzygetic. We characterise non-syzygetic cubic fourfolds algebraically as those having a special type of equation that is almost linear determinantal, and show that the equation of the Fourier-Mukai partner can be obtained by applying Gale duality. We establish that Gale dual cubics are birational, Fourier-Mukai partners and have birational Fano varieties of lines under suitable genericity assumptions, recovering a result of Brooke-Frei-Marquand. We show that the birationality of the Fano varieties of lines continues to hold in the context of equivariant birational geometry, but birationality of the cubics may not. We exhibit examples of Gale dual cubics with faithful actions of the alternating group on four letters that could provide counterexamples to equivariant versions of a conjecture by Brooke-Frei-Marquand predicting birationality of the cubics if the Fano varieties of lines are birational, and also possibly a related conjecture by Huybrechts predicting birationality of Fourier-Mukai partners.

We study linearizability of actions of finite groups on cubic threefolds with non-isolated singularities.

A very good cubic fourfold is a smooth cubic fourfold that does not contain a plane, a cubic scroll, or a hyperplane section with a corank 3 singularity. We prove that the normalization of the relative compactified Prym variety associated to the universal family of hyperplanes of a very good cubic fourfold is in fact smooth, thereby extending prior results of Laza, Saccà and Voisin. Using a similar argument, we also prove the smoothness of the normalization of the relative compactified Picard of the associated relative Fano variety of relative lines.

We study the equivariant Kuznetsov component Ku_G(X) of a general cubic fourfold X with a symplectic involution. We show that Ku_G(X) is equivalent to the derived category of a K3 surface S, where S is given as a component of the fixed locus of the induced symplectic action on the Fano variety of lines on X

We determine the algebraic and transcendental lattices of a general cubic fourfold with a symplectic automorphism of prime order. We prove that cubic fourfolds admitting a symplectic automorphism of order at least three are rational, and we exihibit two families of rational cubic fourfolds that are not equivariantly rational with respect to their group of automorphisms. As an application, we classify the cohomological action of symplectic birational transformations of manifolds of OG10 type that are induced by prime order sympletic automorphisms of cubic fourfolds.

We give several examples of pairs of non-isomorphic cubic fourfolds whose Fano varieties of lines are birationally equivalent (and in one example isomorphic). Two of our examples, which are special families of conjecturally irrational cubics containing cubic scrolls, provide new evidence for the conjecture that Fourier-Mukai partners are birationally equivalent. We explore how various notions of equivalence for cubic fourfolds are related, and we conjecture that cubic fourfolds with birationally equivalent Fano varieties of lines are themselves birationally equivalent.

We characterize the birational geometry of some hyperkähler fourfolds of Picard rank 3 obtained as the Fano varieties of lines on cubic fourfolds containing pairs of cubic scrolls. In each of the two cases considered, we provide a census of the birational models, relating each model to familiar geometric constructions. We also provide structural results about the birational automorphism groups, giving generators in both cases and a full set of relations in one case. Finally, as a byproduct of our analysis, we obtain non-isomorphic cubic fourfolds whose Fano varieties of lines are birationally equivalent.

We study linearizability of actions of finite groups on cubic threefolds with nonnodal singularities.

Lagrangian fibrations of hyperkähler manifolds are induced by semi-ample line bundles which are isotropic with respect to the Beauville-Bogomolov-Fujiki form. For a non-isotrivial family of hyperkähler manifolds over a complex manifold S of positive dimension, we prove that the set of points in S, for which there is an isotropic class in the Picard lattice of the corresponding hyperkähler manifold represented as a fiber over that point, is analytically dense in S. 

The defect of a cubic threefold X with isolated singularities is a global invariant that measures the failure of Q-factoriality. We compute the defect for such cubics in terms of topological data about the curve of lines through a singular point. We express the mixed Hodge structure on the middle cohomology of X in terms of both the defect and local invariants of the singularities. We then relate the defect to various geometric properties of X: in particular, we show that a cubic threefold is not Q-factorial if and only if it contains either a plane or a cubic scroll. We relate the defect to existence of compactified intermediate Jacobian fibrations with irreducible fibers associated to a cubic fourfold.

We classify finite groups that act faithfully by symplectic birational transformations on an irreducible holomorphic symplectic (IHS) manifold of OG10 type. In particular, if X is an IHS manifold of OG10 type and G a finite subgroup of symplectic birational transformations of X, then the action of G on H2(X, Z) is conjugate to a subgroup of one of 375 groups of isometries. We prove a criterion for when such a group is determined by a group of automorphisms acting on a cubic fourfold, and apply it to our classification. Our proof is computer aided, and our results are available in a Zenodo dataset. 

We give a complete classification of symplectic birational involutions of manifolds of OG10 type. We approach this classification with three techniques - via involutions of the Leech lattice, via involutions of cubic fourfolds and finally lattice enumeration via a modified Kneser's neighbour algorithm. The classification consists of three involutions with an explicit geometric realisation via cubic fourfolds, and three exceptional involutions which cannot be obtained by any known construction.

There are two types of involutions on a cubic threefold: the Eckardt type (which has been studied by the first named and the third named authors) and the non-Eckardt type. Here we study cubic threefolds with a non-Eckardt type involution, whose fixed locus consists of a line and a cubic curve. Specifically, we consider the period map sending a cubic threefold with a non-Eckardt type involution to the invariant part of the intermediate Jacobian. The main result is that the global Torelli Theorem holds for the period map. To prove the theorem, we project the cubic threefold from the pointwise fixed line and exhibit the invariant part of the intermediate Jacobian as a Prym variety of a (pseudo-)double cover of stable curves. The proof relies on a result of Ikeda and Naranjo-Ortega on the injectivity of the related Prym map. We also describe the invariant part of the intermediate Jacobian via the projection from a general invariant line and show that the two descriptions are related by the bigonal construction.

There are three types of involutions on a cubic fourfold; two of anti-symplectic type, and one symplectic. Here we show that cubics with involutions exhibit the full range of behaviour in relation to rationality conjectures. Namely, we show a general cubic fourfold with symplectic involution has no associated K3 surface and is conjecturely irrational. In contrast, we show a cubic fourfold with a particular anti-symplectic involution has an associated K3, and is in fact rational. We show such a cubic is contained in the intersection of all non-empty Hassett divisors; we call such a cubic Hassett maximal. We study the algebraic and transcendental lattices for cubics with an involution both lattice theoretically and geometrically.

Let S be a smooth rational surface whose anticanonical bundle has self intersection⩾3. We show that there exist A-polar cylinders for a polarized pair (S, A) except when S is a smooth cubic surface and A is an anticanonical divisor.