The tautological ring of $\overline{\mathcal{M}}_{g,n}$ is rarely Gorenstein
Samir Canning
TL;DR
The paper proves that the tautological rings $igl\mathsf{R}^*(\bar{\mathcal{M}}_{g,n})$ and $igl\mathsf{RH}^*(\bar{\mathcal{M}}_{g,n})$ fail to be Gorenstein when $g\ge 2$ and $2g+n\ge 24$, by constructing a non-tautological bielliptic cycle in genus $2$ and pushing it through self-gluing to reach the critical locus $2g+n=24$, where it annihilates the tautological–complementary pairings. The proof uses admissible $G$-covers ($G=\mathbb{Z}/2\mathbb{Z}$), a detailed SvZ fibre-product framework for pullbacks of bielliptic loci, and the excess-intersection formula to isolate a single non-tautological contribution after pulling back to $ar{\mathcal{M}}_{2,20}$. A fully worked genus-$2$ cohomology input (via Petersen–Tommasi) feeds the construction, allowing a clear propagation to higher genus without requiring explicit global cohomology knowledge. The paper then proves partial positive results, showing Gorenstein property for many pairs $(g,n)$ with $g\ge 2$ and $n$ small, by an induction that leverages boundary stratifications and the vanishing of odd cohomology on the boundary in certain ranges. Altogether, the work delineates the finite, highly structured set of cases where the tautological rings may be Gorenstein and frames a precise conjecture for the remaining instances.
Abstract
We prove that the tautological rings $\mathsf{R}^*(\overline{\mathcal{M}}_{g,n})$ and $\mathsf{RH}^*(\overline{\mathcal{M}}_{g,n})$ are not Gorenstein when $g\geq 2$ and $2g+n\geq 24$, 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 $g\geq 2$ 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 $\mathsf{RH}^*(\overline{\mathcal{M}}_{g,n})$ when $g\geq 2$.