Brauer groups of tame stacky curves and their $\mu_r$-gerbes

TL;DR

This paper investigates the structure of Brauer groups associated with $bc_r$-gerbes over stacky curves, providing exact sequences, conditions for splitting, and explicit formulas especially for smooth base curves.

Contribution

It offers new characterizations and formulas for Brauer groups of $bc_r$-gerbes over stacky curves, including singular cases and smooth base curves.

Findings

Provides an exact sequence for Brauer groups of $bc_r$-gerbes over stacky curves.

Characterizes when the Brauer group sequence is short exact and when it splits.

Derives a precise formula for the Brauer group when the base curve is smooth.

Abstract

We fit the Brauer group of a $\mu_r$-gerbe over a (possibly arbitrarily singular) stacky curve into an exact sequence and give characterizations for when it is short exact and conditions for when it splits. We also give a precise formula for the Brauer group of a $\mu_r$-gerbe in the case that the base curve is smooth.