Universal compactified Jacobians: cohomological invariance and boundary combinatorics

Pagani and Tommasi have introduced a class of smoothable fine compactified Jacobians $\overline{\mathcal{J}}_{g,n}^d(\sigma)\rightarrow \overline{\mathcal{M}}_{g,n}$ over the moduli space of stable curves, depending nontrivially on the degree $d$ and the choice of a stability condition $\sigma$. A theorem of Migliorini-Shende-Viviani implies that the cohomology of $\overline{\mathcal{J}}_{g,n}^d(\sigma)$ is independent of $d$ and $\sigma$, a statement which is quite unexpected from the point of view of the boundary geometry of these spaces. We reprove this independence statement using a direct combinatorial argument, summing up contributions of individual strata. The Appendix includes a result by J. Feusi characterizing when $\mathcal{J}_{g,n}^d$ and $\mathcal{J}_{g,n}^{d'}$ are $S_n$-equivariantly isomorphic over $\mathcal{M}_{g,n}$, and a result by Q. Yin showing that $[\mathcal{J}^d_g]$ and $[\mathcal{J}^{d'}_g]$ are not always equal in $K_0(\text{Var}_{\mathbb{C}})$.