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

Abstract

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

20 pages

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