The stability space of compactified universal Jacobians

In this paper we describe compactified universal Jacobians, i.e., compactifications of the moduli space of line bundles on smooth curves obtained as moduli spaces of rank 1 torsion-free sheaves on stable curves, using an approach due to Oda–Seshadri. We focus on the combinatorics of the stability conditions used to define compactified universal Jacobians. We explicitly describe an affine space, the stability space, with a decomposition into polytopes such that each polytope corresponds to a proper Deligne–Mumford stack that compactifies the moduli space of line bundles. We apply this description to describe the set of isomorphism classes of compactified universal Jacobians (answering a question of Melo) and to resolve the indeterminacy of the Abel–Jacobi sections (addressing a problem raised by Grushevsky–Zakharov).