Anna Barbieri (Università di Verona) - Moduli spaces of stability conditions and of quadratic differentials

The space of Bridgeland stability conditions is a complex manifold attached to a triangulated category D, parametrizing some t-structures of the category. In some cases, it is isomorphic to a moduli space of meromorphic quadratic differentials on a Riemann surface. I will review this correspondence, which is due to Bridgeland-Smith in the simple zeroes case and was extended to higher order zeroes in a joint work with M.Moeller, Y.Qiu, and J.So.

Dario Faro (Università di Pavia) - Gaussian maps and curves on Enriques surfaces

Let C be a complex projective algebraic curve and L and M be two line bundles on C.  One can associate L and M with some natural maps between spaces of global sections of certain sheaves on C. These are called Gaussian-Wahl maps. These maps have been classically studied in connection with extendability questions of curves on surfaces. In this talk I will focus on the case of Enriques surfaces, presenting some natural questions that arise in this situation.

Giulia Gugiatti (ICTP) - Hypergeometric elliptic surfaces

Hypergeometric differential equations have a rich mathematical tradition dating back to the 17th century. A feature that makes them especially interesting in algebraic geometry is that they are motivic: in simple terms, their solution spaces arise from the variation of cohomology of a family of algebraic varieties. Candidate families appear in works by Katz, Gelfand, Kapranov, Zelevinski, and others. However, in general, such realisations are not of the expected dimension. In this talk, I will classify all hypergeometric local systems of weight one and rank two defined over the rationals and show that they can be realised as the variation of the first cohomology of elliptic surfaces. The talk is based on work in progress with F. Rodriguez Villegas and M. Mereb. 

Andrés Rojas González (Humboldt Universität zu Berlin) - Hurwitz-Brill-Noether theory via curves on K3 surfaces 

The geometry of Brill-Noether loci for general curves is described by a collection of theorems dating back to the 70s and 80s. One of these results, the Gieseker-Petri theorem, found a remarkable proof by Lazarsfeld by specializing to curves on suitable K3 surfaces, obtaining in this way concrete examples of Brill-Noether general curves.

On the other hand, Brill-Noether theory for curves of a fixed gonality k has not been understood until recent times, when analogues of the classic theorems have been obtained by several authors.

In this talk I will explain how, by using Bridgeland stability on K3 surfaces with an elliptic pencil, one can find concrete examples of k-gonal curves which behave generically from this “Hurwitz-Brill-Noether” perspective, thus establishing a parallel to Lazarsfeld’s approach. This is a joint work with G. Farkas and S. Feyzbakhsh.

Benedetta Piroddi (Università degli Studi di Milano) - Transcendental lattices of HK manifolds

We introduce the notion of a Hyper-Kaehler manifold X induced by a Hodge structure of K3-type. We explore this notion for the known deformation types of Hyper-Kaehler manifolds studying those that are induced by a K3 or abelian surface, giving lattice-theoretic criteria to decide whether or not they are birational to a moduli space of sheaves over said surface. This is a joint work with Ángel David Ríos Ortiz.

Angelina Zheng (Università di Roma Tre) -  Stable cohomology of moduli spaces of hyperelliptic curves on Hirzebruch surfaces

In this talk we present the stable cohomology of the moduli space of genus g hyperelliptic curves, embedded in a Hirzebruch surface. This is done using two different methods. The first one is Gorinov-Vassiliev’s method, which computes the cohomology of complements of discriminants. Similar information on the cohomology of a moduli space can be obtained through point counting over finite fields.

In the trigonal case, the moduli spaces of genus g trigonal curves embedded in Hirzebruch surfaces give a stratification of the whole moduli space of genus g trigonal curves, without the datum of the embedding. This is no longer true for hyperlelliptic curves, but our moduli space can still be related to the one parametrizing genus g hyperelliptic curves with marked points.

This is a joint work with Jonas Bergström.