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

TL;DR

This paper computes the rational Chow ring and Picard group of the universal Picard stack over the hyperelliptic locus, revealing its generators, relations, and Brauer class properties, thus extending prior work in algebraic geometry.

Contribution

It determines the Chow ring and Picard group of the universal Picard stack over hyperelliptic curves, including relations and Brauer class distinctions based on parity.

Findings

Chow ring generated by tautological classes

All relations among these classes are explicitly determined

The Picard group is computed, revealing Brauer class behavior

Abstract

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