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We study projective models of generalized Kummer fourfolds via O'Grady's theta groups and the classical Coble cubic. More precisely, we establish a duality between two singular models of the generalized Kummer fourfold of a Jacobian abelian…
We compute the subgroup of the monodromy group of a generalized Kummer variety associated to equivalences of derived categories of abelian surfaces. The result was previously announced in arXiv:1201.0031. Mongardi showed that the subgroup…
We prove a sufficient condition for the Jacobian problem in the setting of real, complex and mixed polynomial mappings. This follows from the study of the bifurcation locus of a mapping subject to a new Newton non-degeneracy condition.
We construct an infinite number of Shimura curves contained in the locus of hyperelliptic Jacobians of genus 3. In the opposite direction, we show that in genus 3 the only possible non-complete (in the moduli space of abelian threefolds)…
The paper investigates the locus of non-simple principally polarised abelian $g$-folds. We show that the irreducible components of this locus are $\Is^g_{D}$, defined as the locus of principally polarised $g$-folds having an abelian…
A well known result of Clemens and Griffiths says that a smooth cubic threefold can be recovered from its intermediate Jacobian. In this paper we discuss the possible degenerations of these abelian varieties, and thus give a description of…
We give a solution to the weak Schottky problem for genus five Jacobians with a vanishing theta null, answering a question of Grushevsky and Salvati Manni. More precisely, we show that if a principally polarized abelian variety of dimension…
Motivated by results of Mestre and Voisin, in this note we mainly consider abelian varieties isogenous to hyperelliptic Jacobians In the first part we prove that a very general hyperelliptic Jacobian of genus $g\ge 4$ is not isogenous to a…
Recent developments on the uniformity of the number of rational points on curves and subvarieties in a moving abelian variety rely on the geometric concept of the degeneracy locus. The first-named author investigated the degeneracy locus in…
We prove that the moduli space of Calabi-Yau 3-folds coming from eight planes of $P^3$ in general positions is not modular. In fact we show the stronger statement that the Zariski closure of the monodromy group is actually the whole…
We study the local geometry of the moduli space of intermediate Jacobians of $(2,2)$-threefolds in ${\mathbb P}^2 \times {\mathbb P}^2$. More precisely, we prove that a composition of the second fundamental form of the Siegel metric in…
The intermediate Jacobian map, which associates to a smooth cubic threefold its intermediate Jacobian, does not extend to the GIT compactification of the space of cubic threefolds, not even as a map to the Satake compactification of the…
In this paper we define the notion of a hyperk\"ahler manifold (potentially) of Jacobian type. If we view hyperk\"ahler manifolds as "abelian varieties", then those of Jacobian type should be viewed as "Jacobian varieties". Under a minor…
A new proof of the non-rationality of a generic cubic threefold is given as follows: If a generic cubic threefold were rational then the associated intermediate Jacobian would be a product of Jacobians of curves. We degenerate a generic…
We show that a given rational Shimura curve Y with strictly maximal Higgs field in the moduli space of g-dimensional abelian varieties does not generically intersect the Schottky locus for large g. We achieve this by using a result of…
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,…
We study the locus of intermediate Jacobians of cubic threefolds within the moduli space of complex principally polarized abelian fivefolds, and its generalization to arbitrary genus - the locus of abelian varieties with a singular odd…
This is a slightly revised version of the author's 2010 diploma thesis. It is concerned with the interplay between real multiplication on Jacobian varieties, as the title suggests, and complex geodesics in the moduli space of curves. Large…
It is well known that, fixed an even, unimodular, positive definite quadratic form, one can construct a modular form in each genus; this form is called the theta series associated to the quadratic form. Varying the quadratic form, one…
We prove that the locus of Hilbert schemes of n points on a projective K3 surface is dense in the moduli space of irreducible holomorphic symplectic manifolds of that deformation type. The analogous result for generalized Kummer manifolds…