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

I'll start by defining the Chow ring, which is an important invariant of a scheme (or stack). Next, I will define the Picard variety and Picard stack of a curve, and then introduce their universal versions $J^d_g$ and $\mathscr{J}^d_g$ over the moduli space of curves $M_g$. Recently, progress has been made studying the Chow ring of $M_g$ in low genus by stratifying the moduli space by gonality (the minimal degree of a map to $\mathbb{P}^1$). The smallest piece in this stratification is the hyperelliptic locus. Motivated by this, I'll present several results about the restriction of $\mathscr{J}^d_g$ to the hyperelliptic locus, denoted $\mathscr{J}^d_{2,g}$. These include a presentation of the rational Chow ring of $\mathscr{J}^d_{2,g}$. I also determine the integral Picard group of $\mathscr{J}^d_{2,g}$, completing (and extending to the $PGL_2$-equivariant case) prior work of Erman and Wood.

(Notice unusual time of seminar; the seminar lunch will be beforehand, at 11:30 am.)