Vanishing of Brauer groups of moduli stacks of stable curves

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 $g\ge 2$. For the $n$ marked version, we show the same vanishing result in the range $(g,n)=(1,n)$ with $1\le n\le 6$ and all $(g,n)$ with $g\ge 4.$ We also discuss several finiteness results on cohomological Brauer groups of proper and smooth Deligne-Mumford stacks over the integers.

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