Research

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 by projecting from a singularity. We determine 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 classify finite groups that act faithfully by symplectic birational transformations on an irreducible holomorphic symplectic (IHS) manifold of OG10 type. In particular, up to deformation there are 367 birational conjugacy classes of pairs (X,G), consisting of an IHS manifold X and saturated, finite subgroup G of symplectic birational transformations of X. We determine the action of G on the second cohomology for each case. 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.

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.

In this paper we give a 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. In particular, 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 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.