Brauer groups of schemes associated to symmetric powers of smooth projective curves in arbitrary characteristics

TL;DR

This paper investigates the structure of Brauer groups related to symmetric powers of smooth projective curves, establishing isomorphisms of their l^n-torsion parts across various associated schemes in arbitrary characteristics.

Contribution

It extends the understanding of Brauer groups by showing isomorphisms of their torsion parts for symmetric powers, Picard schemes, and Quot-schemes over different fields, including arbitrary characteristics.

Findings

Isomorphism of l^n-torsion Brauer groups for symmetric powers and related schemes over separably closed fields.

Results hold in arbitrary characteristics when l is invertible in the base field.

Additional results for Prym varieties over complex numbers.

Abstract

In this paper we show that the l^n-torsion part of the cohomological Brauer groups of certain schemes associated to symmetric powers of a projective smooth curve over a separably closed field k are isomorphic, when `l is invertible in k. The schemes considered are the Symmetric powers themselves, then the corresponding Picard schemes and also certain Quot-schemes. We also obtain similar results for Prym varieties associated to certain finite covers of such curves: we prove such results only for curves defined over the field of complex numbers.