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

We compute the $\mathrm{\ell }$-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 $\mathrm{\ell }$-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.