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 (1992)) and genus 1 by Belorousski (the rings are tautological if and only if n \leq 10 (1998)). For g \geq 2 , work of van Zelm (2018) shows the Chow and cohomology rings are not tautological once 2g + n \geq 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 \leq 9 and for 3 \leq g \leq 7 and 2g + n \leq 14 . For such (g, n) , this implies that the tautological ring is Gorenstein and \bar{\mathcal{M}}_{g,n} has polynomial point count.