Universal compactified Jacobians: cohomological invariance and boundary combinatorics

Abstract

Pagani and Tommasi have introduced a class of smoothable fine compactified Jacobians ${\overline{\mathcal{J}}}_{g,n}^d(\sigma )\to {\overline{\mathcal{M}}}_{g,n}$ 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 ${\mathcal{J}}_{g,n}^{d^{\prime }}$ are $S_n$-equivariantly isomorphic over ${\mathcal{M}}_{g,n}$, and a result by Q. Yin showing that $[{\mathcal{J}}_g^d]$ and $[{\mathcal{J}}_g^{d^{\prime }}]$ are not always equal in $K_0({\text{Var}}_{\mathbb{C}})$.

22 pages, comments welcome