On the Chow and cohomology rings of moduli spaces of stable curves

In this paper, we ask: for which $(g,n)$ is the rational Chow or cohomology ring of ${\bar{\mathcal{M}}}_{g,n}$ generated by tautological classes? This question has been fully answered in genus $0$ by Keel (the Chow and cohomology rings are tautological for all $n$) and genus $1$ by Belorousski (the rings are tautological if and only if $n\le 10$). For $g\ge 2$, work of van Zelm shows the Chow and cohomology rings are not tautological once $2g+n\ge 24$, leaving finitely many open cases. Here, we prove that the Chow and cohomology rings of ${\bar{\mathcal{M}}}_{g,n}$ are isomorphic and generated by tautological classes for $g=2$ and $n\le 9$ and for $3\le g\le 7$ and $2g+n\le 14$. For such $(g,n)$, this implies that the tautological ring is Gorenstein and ${\bar{\mathcal{M}}}_{g,n}$ has polynomial point count.