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

2024/04/19 by Larson, Hannah
#14C15 #14H40 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2404.12607

Abstract

Let \mathscrJ^d_g → \mathscrM_g be the universal Picard stack parametrizing degree d line bundles on genus g curves, and let \mathscrJ^d_2,g be its restriction to locus of hyperelliptic curves \mathscrH_2,g ⊂ \mathscrM_g. We determine the rational Chow ring of \mathscrJ^d_2,g for all d and g. In particular, we prove it is generated by restrictions of tautological classes on \mathscrJ^d_g and we determine all relations among the restrictions of such classes. We also compute the integral Picard group of \mathscrJ^d_2,g, completing (and extending to the PGL₂-equivariant case) prior work of Erman and Wood. As a corollary, we prove that \mathscrJ^d_2,g is either a trivial \mathbbG_m-gerbe over its rigidification, or has Brauer class of order 2, depending on the parity of d - g.

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