On a combinatorial classification of fine compactified universal Jacobians

Abstract: Extending the definition of $V$-stability conditions, given by Viviani in a recent preprint, we introduce the notion of universal stability conditions. Building on results by Pagani and Tommasi, we show that fine compactified universal Jacobians, that is, fine compactified Jacobians over the moduli spaces of stable pointed curves $\overline{\mathcal{M}}_{g,n}$, are combinatorially classified by universal stability conditions. We use these stability conditions to show the following. The inclusion of fine compactified universal Jacobians of type $(g,n)$ whose fibres over geometric points are classical, that is, they are constructed by some numerical polarisations, into the class of all fine compactified universal Jacobians, is strict, in general, for any $g\geq 2$. This answers a question of Pagani and Tommasi.