The Gorenstein conjecture fails for the tautological ring of $\mathcal{\bar M}_{2,n}$

TL;DR

This paper demonstrates that the tautological ring of the moduli space of genus 2 curves with N marked points is not Gorenstein, providing evidence that the minimal N with non-tautological classes is 20.

Contribution

It shows that the tautological ring of ar M_{2,N} is not Gorenstein and identifies the degree where non-tautological classes appear, supporting N=20 as the minimal such N.

Findings

Non-tautological classes exist on ar M_{2,20}

Non-tautological cohomology appears only outside the middle degree

The tautological ring of ar M_{2,N} is not Gorenstein

Abstract

Let $N$ be the smallest integer such that there is a non-tautological cohomology class of even degree on $\mathcal{\bar M}_{2,N}$. We remark that there is such a non-tautological class on $\mathcal{\bar M}_{2,20}$, by work of Graber and Pandharipande. We show that $\mathcal{\bar M}_{2,N}$ has non-tautological cohomology only in one degree, which is not the middle degree. In particular, it follows that the tautological ring of $\mathcal{\bar M}_{2,N}$ is not Gorenstein. We present some evidence suggesting that N=20 holds.