The rational Chow ring of the moduli space M g of curves of genus g is known for g ≤ 6. Here, we determine the rational Chow rings of M 7 , M 8 , and M 9 by showing they are tautological. One key ingredient is intersection theory on Hurwitz spaces of degree 4 and 5 covers of P 1 , as developed by the authors in [1]. The main focus of this paper is a detailed geometric analysis of special tetragonal and pentagonal covers whose associated vector bundles on P 1 are so unbalanced that they fail to lie in the large open subset considered in [1]. In genus 9, we use work of Mukai [23] to present the locus of hexagonal curves as a global quotient stack, and, using equivariant intersection theory, we show its Chow ring is generated by restrictions of tautological classes.
In this paper, we ask: for which (g, n) is the rational Chow or cohomology ring of 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 [34]) and genus 1 by Belorousski (the rings are tautological if and only if n ≤ 10 [4]). For g ≥ 2, work of van Zelm [53] shows the Chow and cohomology rings are not tautological once 2g + n ≥ 24, leaving finitely many open cases. Here, we prove that the Chow and cohomology rings of M g,n are isomorphic and generated by tautological classes for g = 2 and n ≤ 9 and for 3 ≤ g ≤ 7 and 2g + n ≤ 14. For such (g, n), this implies that the tautological ring is Gorenstein and M g,n has polynomial point count.
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