Torsors of the Jacobians of the universal Fermat curves

TL;DR

This paper investigates torsors of Jacobians of universal Fermat curves, revealing their structure as components of the Picard scheme and demonstrating the abundance of non-isomorphic torsors for generic cases.

Contribution

It establishes that all torsors of these Jacobians are connected components of the Picard scheme and shows the existence of uncountably many non-isomorphic torsors for generic curves.

Findings

Torsors are connected components of the Picard scheme.

Uncountably many non-isomorphic torsors exist for generic Fermat curves.

Results contribute to the Franchetta type problem for torsors.

Abstract

Let $m\geq3$ be an integer. We show that every torsor of the Jacobian of the universal family of degree-$m$ Fermat curve is necessarily a connected component of the Picard scheme. We show that the Jacobian of the generic degree-$m$ Fermat curve has uncountably many non-isomorphic torsors. We give some results towards the Franchetta type problem for torsors of the Jacobian of the universal family of genus-$g$ curves over $\mathcal{M}_g$.