Universal compactified Jacobians: cohomological invariance and boundary combinatorics

TL;DR

This paper proves the cohomological invariance of universal compactified Jacobians across different degrees and stability conditions using combinatorial methods, and explores boundary geometry and isomorphism conditions.

Contribution

It provides a combinatorial proof of cohomological invariance and analyzes boundary combinatorics and isomorphism conditions of compactified Jacobians.

Findings

Cohomology of compactified Jacobians is independent of degree and stability condition.

A combinatorial argument sums contributions of individual strata to prove invariance.

Results include characterizations of when Jacobians are isomorphic and their classes in the Grothendieck ring.

Abstract

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 $K_0(\text{Var}_{\mathbb{C}})$.