The geometry and combinatorics of compactified universal Jacobians

We will study the geometry of the compactification of the universal Jacobian, the moduli space that parametrises pairs (C,L) where C is a smooth algebraic curve of given genus and L is a line bundle of some fixed degree on C. A classical modular compactification of the moduli space of curves is given in terms of stable curves. There are several different modular compactifications that admit a map to the moduli space of stable curves. The set of all such compactifications modulo isomorphisms has been recently studied in a series of works by Kass and Pagani.

We will address two natural (and independent) questions.

1) Recent work by Migliorini-Shende-Viviani shows that the cohomology of two compactified Jacobians of the same stable curve are isomorphic as vector spaces. Here we want to ask if the same is true for two compactified universal Jacobians (over the same moduli space). This project will first require to understand the details of the proof of Migliorini-Shende-Viviani's result, which requires developing some understanding of intersection cohomology and of the classical decomposition theorem by Beilinson-Bernstein-Deligne.

2) How many non-isomorphic compactified Jacobians exist for a fixed genus? By the work of Kass-Pagani this question can be reformulated as the combinatorial problem of counting the number of chambers of a certain hyperplane arrangement on a real torus, modulo the action of a certain group. The theory to count the chambers of a hyperplane arrangement in a vector space is classical and due to Zavlasky, and recent results extend that theory to the case of arrangements on a torus.