Universal compactified Jacobians: cohomological invariance and boundary combinatorics

Abstract

Abstract Pagani and Tommasi have introduced a class of smoothable fine compactified Jacobians $$\overline{\mathcal {J}}_{g,n}^{\,d}(\sigma )\rightarrow \overline{\mathcal {M}}_{g,n}$$ J ¯ g , n d ( σ ) → M ¯ g , n over the moduli space of stable curves, depending nontrivially on the degree d and the choice of a stability condition $$\sigma $$ σ . A theorem of Migliorini–Shende–Viviani implies that the cohomology of $$\overline{\mathcal {J}}_{g,n}^{\,d}(\sigma )$$ J ¯ g , n d ( σ ) is independent of d and $$\sigma $$ σ , 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 $${\mathcal {J}}_{g,n}^{\,d}$$ J g , n d and $${\mathcal {J}}_{g,n}^{\,d'}$$ J g , n d ′ are $$S_n$$ S n -equivariantly isomorphic over $$\mathcal {M}_{g,n}$$ M g , n , and a result by Q. Yin showing that $$[\mathcal {J}^{\,d}_g]$$ [ J g d ] and $$[\mathcal {J}^{\,d'}_g]$$ [ J g