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 $g\geq 2$. 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.