The Brauer groups of moduli of genus three curves, abelian threefolds and plane curves

TL;DR

This paper computes the $ ext{ell}$-primary torsion of the Brauer groups of various moduli stacks of genus three curves, plane curves, and abelian threefolds by analyzing their low degree cohomological invariants.

Contribution

It provides explicit calculations of the Brauer groups for these moduli stacks, extending understanding of their cohomological properties in various characteristics.

Findings

Computed $ ext{ell}$-primary torsion of Brauer groups for genus three curves

Determined Brauer groups of moduli stacks of plane curves of degree $d$

Analyzed cohomological invariants to achieve these results

Abstract

We compute the $\ell$-primary torsion of the Brauer group of the moduli stack of smooth curves of genus three over any field of characteristic different from two and the Brauer group of the moduli stacks of smooth plane curves of degree $d$ over any algebraically closed field of characteristic different from two, three and coprime to $d$. We achieve this result by computing the low degree cohomological invariants of these stacks. As a corollary we are additionally able to compute the $\ell$-primary torsion of the Brauer group of the moduli stack of principally polarized abelian varieties of dimension three over any field of characteristic different from two.