The degree d universal Jacobian parametrizes degree d line bundles on smooth curves. There are several approaches on how to extend it to a proper family over the moduli space of stable curves. In this talk, we introduce a simple definition of a fine compactified universal Jacobian. We focus on the case of genus 1 and obtain a combinatorial classification for fine universal compactified Jacobians, which enables us to construct new examples of them. The description we obtain for universal fine compactified Jacobians of genus 1 also yields a formula for their rational cohomology.
This is joint work with Nicola Pagani (Liverpool).