The tautological ring of ${\overline{\mathcal{M}}}_{g,n}$ is rarely Gorenstein

Abstract

We prove that the tautological rings ${\mathsf{R}}^{\ast }({\overline{\mathcal{M}}}_{g,n})$ and ${\mathsf{RH}}^{\ast }({\overline{\mathcal{M}}}_{g,n})$ are not Gorenstein when $g\ge 2$ and $2g+n\ge 24$, extending results of Petersen and Tommasi in genus $2$. The proof uses the intersection of tautological classes with non-tautological bielliptic cycles. We conjecture the converse: the tautological rings should be Gorenstein when $g=0,1$ or $g\ge 2$ and $2g+n<24$. The conjecture is known for $g=0,1$ by work of Keel and Petersen, and we prove several new cases of this conjecture for ${\mathsf{RH}}^{\ast }({\overline{\mathcal{M}}}_{g,n})$ when $g\ge 2$.

arXiv comment13 pages, comments welcome!

References in corpus (9)

• admcycles -- a Sage package for calculations in the tautological ring of the moduli space of stable curves
• The structure of the tautological ring in genus one
• The Gorenstein conjecture fails for the tautological ring of ${\bar{\mathcal{M}}}_{2,n}$
• Relations on ${\overline{M}}_{g,n}$ via equivariant Gromov-Witten theory of ${\mathbb{P}}^1$
• Pixton's double ramification cycle relations
• The tautological ring of the space of pointed genus two curves of compact type
• Nontautological Bielliptic Cycles
• Frobenius Manifolds near the Discriminant and Relations in the Tautological Ring
• The eleventh cohomology group of ${\bar{\mathcal{M}}}_{g,n}$