Jacobians, Pryms and degenerations

The $\Gamma_{00}$-conjecture predicts that an indecomposable principally polarized abelian variety $(A,\Theta)$ is the Jacobian of a smooth curve if and only if the base locus $\Gamma_{00}$ of certain linear subsystem of $|2\Theta|$ has positive dimension. This conjecture is known to be true under an additional genericity hypothesis. In contrast, in this talk I will discuss a recent result which asserts that such a base locus is positive-dimensional as soon as there are low-degree hyperelliptic curves contained in $A.$ Along the way we discuss new results about Seshadri constants and curves on abelian varieties. 

Given an étale double cover of smooth curves one can associate a principally polarized abelian variety called the Prym variety. The map from the moduli space of double covers R_g to the moduli space of pp abelian varieties A_{g-1} is called Prym-Torelli, in analogy with the Jacobian case. This map does not extend to a morphism from the boundary of R_g to any of the standard toroidal compactifications of A_{g-1}. In this talk I will present their tropical counterpart and prove that, with a slight modification of the definition, the tropical Prym-Torelli map is continuous.

We classify all the modular compactifications of the universal Jacobian over the moduli space of pointed stable curves by torsion-free rank 1 sheaves. We do so by introducing V-functions and showing an isomorphism of posets. In particular, we show that in the absence of marked points the only compactified universal Jacobians are the ones constructed by Caporaso. This is joint work with N. Pagani and F. Viviani.

A classical result of Deninger–Murre implies that the class of the zero section of a family of principally polarized abelian varieties of dimension g is a multiple of the g-th power of the universal theta divisor. In this talk I will present joint work with Younghan Bae, Aitor Iribar-López, and Sam Molcho extending this formula to families with torus rank 1 degenerations. Working over the partial compactification of A_g parametrizing torus rank 1 degenerations, we compute the class of the zero section on a toroidal compactification of the universal family in terms of the theta divisor together with explicit boundary corrections. In particular, we show that this class lies in the tautological ring. The main tools are extensions of the Fourier–Mukai transform and the multiplication by N maps to toroidal compactifications of torus rank 1 degenerations.

Last year, O. de Gaay Fortman, P. Engel and S. Schreieder disproved the integral Hodge conjecture for abelian varieties. They showed that any very general fiber of a matroidal family of abelian varieties with its associated matroid not cographic fails the integral Hodge conjecture when the dimension is greater than or equal 4. We will show that the matroids associated to matroidal families of Prym varieties are cosigned-graphic, which are matroids that can be viewed as a generalization of cographic matroids. In particular, using the criterion provided by the aforementioned authors, we show that any very general Prym variety of dimension greater than or equal 6 fails the integral Hodge conjecture.

Let X be a genus 2 curve and \pi: X \to E a non-constant map to an elliptic curve. The Jacobian of X is then isogenous to E \times E' for a complementary elliptic curve E′ that comes with a map \pi': X \to E'. Given X, E, and \pi explicitly, how can one compute E′ and \pi'?

 In this talk, I will present an algorithm that answers this question, which is both conceptually simple and computationally efficient. This is joint work with Lombardo, Naccarato, and Zannier.

Divisors on a ‘general’ algebraic curve are well understood: the Brill–Noether theorem tells us that they have the expected rank. Considering degenerations of algebraic curves we get tropical curves, however, the Brill–Noether theory of tropical curves is not entirely understood. In this talk we will focus on tropical curves with a divisorial behaviour similar to the one of general algebraic curves. These are called Brill–Noether general and we will introduce a new family of such curves, known as chains of theta-graphs and loops.

The Prym–Torelli problem asks whether a cover can be recovered from its associated polarised Prym variety. For double covers this question has a long history, beginning with the classical theory of Prym varieties and leading to a detailed understanding of the corresponding Prym maps. However, much less is known for cyclic covers of higher degree. In this talk, I will consider étale cyclic covers of hyperelliptic curves and study to what extent the cover is determined by its Prym variety. I will describe several reconstruction results and explain how they translate into injectivity and generic injectivity of the corresponding Prym maps. The talk is based on joint work with P. Borówka, J. C. Naranjo and A. Ortega.