The Chow ring of the universal Picard stack over the hyperelliptic locus

Abstract

Let ${\mathcal{J}}_g^d\to {\mathcal{M}}_g$ be the universal Picard stack parametrizing degree $d$ line bundles on genus $g$ curves, and let ${\mathcal{J}}_{2,g}^d$ be its restriction to locus of hyperelliptic curves ${\mathcal{H}}_{2,g}\subset {\mathcal{M}}_g$. We determine the rational Chow ring of ${\mathcal{J}}_{2,g}^d$ for all $d$ and $g$. In particular, we prove it is generated by restrictions of tautological classes on ${\mathcal{J}}_g^d$ and we determine all relations among the restrictions of such classes. We also compute the integral Picard group of ${\mathcal{J}}_{2,g}^d$, completing (and extending to the ${\mathrm{PGL}}_2$-equivariant case) prior work of Erman and Wood. As a corollary, we prove that ${\mathcal{J}}_{2,g}^d$ is either a trivial ${\mathbb{G}}_m$-gerbe over its rigidification, or has Brauer class of order $2$, depending on the parity of $d-g$.

arXiv comment28 pages