Torsors under the universal Jacobian over ${\mathcal{M}}_g$

Abstract

We consider the universal family of smooth genus-$g$ curves over $\mathbb{C}$. We show that for $g\ge 4$, every torsor under the relative Jacobian is isomorphic to a connected component of the relative Picard scheme. As a byproduct, we show that $\mathrm{Br}({\mathcal{M}}_{3,1})=\mathbb{Z}/2\mathbb{Z}$ over $\mathbb{C}$ and ${\mathrm{H}}_2({\Gamma }_{3,1},\mathbb{Z})=\mathbb{Z}/2\mathbb{Z}$, pinning down the torsion subgroup in \cite[Theorem 1.2]{zbMATH01991000}.

arXiv comment11 pages