arXiv subjects
Publications and source records attributed to Lisa Marquand.
We describe a Cremona transformation on $\mathbb{P}^5$ given by the linear system of quintics that are double along the Fano model of an Enriques surface $S$. Using this Cremona transform, we construct a birational equivalence between the very general cubic fourfold $X$ of discriminant 44 and its unique Fourier--Mukai partner.
We construct five irreducible divisors in the moduli space of complex cubic fourfolds parametrising smooth cubic fourfolds that contain highly singular hyperplane sections. We prove that each is not a Noether-Lefschetz (or Hassett) divisor, utilising the computational method developed by Addington-Auel.
We develop an algorithm to count the number of virtual Fourier-Mukai partners for a given cubic fourfold, with initial input the primitive algebraic lattice and transcendental Hodge structure. Under some mild assumptions, we prove that our virtual count is enough to recover the actual count of Fourier-Mukai partners. We apply our algorithm to examples of cubics with a symplectic automorphism. In particular, we prove that a general cubic with a symplectic involution has 1120 non-trivial Fourier-Mukai partners, each admitting an Eckardt involution, and all birational. As a corollary, we prove that admitting a symplectic automorphism is not a Fourier-Mukai invariant for cubic fourfolds.
We develop the notion of Peskine sixfolds with associated K3 surfaces and cubic fourfolds and work out numerical conditions for when these associations occur. In discriminant 24, the first family for which there is an associated cubic fourfold, we identify the cubic explicitly. Moreover, we prove that in this case the Fano variety of lines of the cubic fourfold is isomorphic to the associated Debarre-Voisin hyperk\"ahler fourfold.
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; further, we prove that this partner 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, Sacc\`a 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 $\mathrm{Ku}_G(X)$ of a general cubic fourfold $X$ with a symplectic involution. We show that $\mathrm{Ku}_G(X)$ is equivalent to the derived category $D^b(S)$ 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 determine 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 in $\calC_{12}$, 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 hyperk\"ahler 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 identify all of the birational models, relating each model to familiar geometric constructions, and give explicit birational maps between them. 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 isolated singularities.
Lagrangian fibrations of hyperk\"ahler 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 hyperk\"ahler 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 hyperk\"ahler manifold represented as a fiber over that point, is analytically dense in $S$. We also prove the expected openness and density of the locus of polarised hyperk\"ahler manifolds that admit a nef algebraic isotropic line bundle.
The defect of a cubic threefold $X$ with isolated singularities is a global invariant that measures the failure of $\mathbb{Q}$-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 $\mathbb{Q}$-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.
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.
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 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.