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

We show that mod $2$ cohomological invariants of the moduli stack $\mathscr{M}_{3,n}$ 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 $\mathrm{Sp}_6(2)$. We achieve this by showing that the torsor of full level two structures $\mathscr{M}_{3,n}(2) \to \mathscr{M}_{3,n}$ 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 $\mathscr{M}_{3,n}$ is non-zero in degree three. Our main result holds also for the stack $\mathscr{A}_3$ of principally polarized abelian threefolds.