Vanishing of Brauer groups of moduli stacks of stable curves

TL;DR

This paper proves the vanishing of the cohomological Brauer groups for moduli stacks of stable curves of genus at least 2 over various fields, extending to certain marked cases and discussing related finiteness results.

Contribution

It establishes new vanishing results for the cohomological Brauer groups of moduli stacks of stable curves, including marked cases and over different base fields.

Findings

Cohomological Brauer groups vanish for genus g ≥ 2 over integers and algebraic closures.

Vanishing extends to certain marked moduli stacks with specific genus and marking conditions.

Finiteness results are discussed for Brauer groups of proper smooth Deligne-Mumford stacks.

Abstract

We show that the cohomological Brauer groups of the moduli stacks of stable genus $g$ curves over the integers and an algebraic closure of the rational numbers vanish for any $g\geq 2$. For the $n$ marked version, we show the same vanishing result in the range $(g,n)=(1,n)$ with $1\leq n \leq 6$ and all $(g,n)$ with $g\geq 4.$ We also discuss several finiteness results on cohomological Brauer groups of proper and smooth Deligne-Mumford stacks over the integers.