The Brauer group of $BG$ and gerbe structures of moduli spaces

TL;DR

This paper investigates the Brauer group of certain algebraic stacks related to curves with automorphisms, determining Brauer classes of gerbes and computing the Brauer group of classifying stacks for algebraic groups.

Contribution

It provides a complete determination of the Brauer classes of $oldsymbol{ extmu}_N$-gerbes of curves with automorphisms and calculates the Brauer group of classifying stacks for smooth connected semisimple groups.

Findings

Brauer classes of specific gerbes are fully characterized.

The Brauer group of $BG$ for semisimple groups is explicitly computed.

Results connect automorphism structures of curves to cohomological invariants.

Abstract

We study the $\mu _N$-gerbe of curves of genus $g$ with an order $N$ automorphism, and explore what corresponding $H^2$-cohomology classes the components of this stack can have. In particular, we look at curves whose quotients by the order $N$ automorphism are genus 0, and completely determine the Brauer classes of these gerbes. The key technical input is the calculation of the Brauer group of $BG$, for $G$ a smooth connected semisimple linear algebraic group.