Convegno
“ALGEBRAIC AND TROPICAL GEOMETRY OF MODULI SPACES OF CURVES”

The study of Jacobians and their compactifications is central to understanding the geometry of algebraic curves, especially in the context of degenerations and moduli. Compactified Jacobians provide essential tools for extending classical theorems and constructions to singular curves, revealing deep connections between algebraic geometry, combinatorics, and tropical geometry. This mini-course will explore these rich interactions. We will begin with the theory of compactified Jacobians for nodal curves. We will then introduce tropical Jacobians, showing how they arise combinatorially from metric graphs. A central focus will be the Abel map for nodal curves and its resolution, highlighting the interplay between algebraic and tropical geometry. Finally, we will discuss the Torelli theorem for nodal curves, addressing the question of reconstructing a curve from its compactified Jacobian. The course aims to provide a bridge between the classical geometric theory of Jacobians and the modern perspectives offered by tropical geometry and degeneration techniques.

Tropical geometry studies a piecewise linear combinatorial shadow of degenerations and compactifications of algebraic varieties. A typical phenomenon is that many of the usual algebro-geometric objects have a tropical analogue that is intimately tied to its classical counterpart. An example is the theory of divisors and line bundles on algebraic curves, whose tropical counterparts have been crucial in numerous surprising applications to classical Brill—Noether theory and the birational geometry of moduli spaces. One classical object that has resisted the effort of tropical geometers so far is the geometry of vector bundles beyond rank one. In this talk, I will outline an elementary approach to tropical vector bundles that builds on earlier work of Allermann. Although limited in scope, this theory leads to a satisfying tropical story for semistable vector bundles on elliptic curves and, more generally, semihomogeneous vector bundles on abelian varieties. The engines in the background that make these cases accessible to our methods are Atiyah's classification of vector bundles on elliptic curves, Fourier-Mukai transforms on abelian varieties, and the interactions with non-Archimedean uniformization. This talk is based on joint work with Andreas Gross and Dmitry Zakharov (and, in parts, Arne Kuhrs) as well as with Andreas Gross, Inder Kaur, and Annette Werner.

Severi varieties parametrize integral curves of fixed geometric genus in a given linear system on a surface. In this talk, I will present some results on the irreducibility of these varieties in the case of toric surfaces, and their application to the irreducibility of other moduli spaces of curves. This is done using tropical methods and I will indicate some of these aspects. The new results are from ongoing joint work with Xiang He and Ilya Tyomkin.

I will explain several new ideas leading to a better understanding of the cohomology of moduli spaces of (stable) curves of all genera. Joint with Hannah Larson, Sam Payne, and Thomas Willwacher.