The Gorenstein property and Pixton's conjecture for compact type moduli

We show that the tautological ring of ${\mathcal{M}}_{g,n}^{\mathrm{c}\mathrm{t}}$ is not Gorenstein for $g\ge 2$ and $2g+n\ge 12$. We prove new cases of Pixton’s conjecture that the $3$-spin relations are a complete set of relations for the tautological ring, including ${\mathcal{M}}_6^{\mathrm{c}\mathrm{t}}$, ${\mathcal{M}}_{5,2}^{\mathrm{c}\mathrm{t}}$, and ${\mathcal{M}}_7^{\mathrm{c}\mathrm{t}}$. These are the first known cases where Pixton’s conjecture is true, but the tautological ring is not Gorenstein. These results are also a key ingredient in recent work on non-tautological cycles on the moduli space of principally polarized abelian varieties.