Cohomological invariants of ${\mathcal{M}}_{3,n}$ via level structures

We show that mod $2$ cohomological invariants of the moduli stack ${\mathcal{M}}_{3,n}$ of smooth pointed curves of genus three contain a free module with generators in degree $0$, $2$, $3$, $4$ and $6$, formed by the invariants of the symplectic group ${\mathrm{S}\mathrm{p}}_6(2)$. We achieve this by showing that the torsor of full level two structures ${\mathcal{M}}_{3,n}(2)\to {\mathcal{M}}_{3,n}$ is versal. Along the way, we prove that the invariants of the stack of del Pezzo surfaces of degree two contain the invariants of the Weyl group $W({\mathsf{E}}_7)$ and that the mod $2$ cohomology of ${\mathcal{M}}_{3,n}$ is non-zero in degree three. Our main result holds also for the stack ${\mathcal{A}}_3$ of principally polarized abelian threefolds.