A complete classification of modular compactifications of the universal Jacobian
This is the third paper in a series, following [FPVa] and [FPVb]. We classify all modular compactifications of the universal Jacobian over $\overline{\mathcal{M}}_{g,n}$, both as stacks and as their relative good moduli spaces. Our main result gives a combinatorial parametrization of compactified universal Jacobian stacks by $V$-functions on a stability domain $\mathbb{D}_{g,n}$ of half-vine types (two-components topological types with a chosen side); under this correspondence, fine compactifications are exactly the general $V$-functions. We single out the classical compactified universal Jacobians, namely those induced by numerical polarizations (relative $\mathbb{R}$-line bundles on the universal curve $\overline{\mathcal{C}}_{g,n}/\overline{\mathcal{M}}_{g,n}$), recovering the constructions of Kass-Pagani and Melo in the fine case, and we prove that their good moduli spaces are locally projective over $\overline{\mathcal{M}}_{g,n}$. We determine when two compactified universal Jacobians are isomorphic over $\overline{\mathcal{M}}_{g,n}$ and describe a resolution of the universal family via a compactified Jacobian over $\overline{\mathcal{M}}_{g,n+1}$. Finally, we analyse the poset $\Sigma_{g,n}$ of compactified universal Jacobians, an extension of the poset of regions of the hyperplane arrangement of classical stability conditions $\mathcal{A}_{g,n}$ studied in Kass-Pagani. We prove that for $n=0$ all compactified universal Jacobians are those constructed by Caporaso. We then give an explicit description of the submaximal elements of $\Sigma_{g,n}$ for all $n$, generalizing the stability walls in the classical stability space $\mathcal{A}_{g,n}$ from Kass-Pagani's work.