The Chow rings of the moduli spaces of curves of genus 7, 8, and 9

The rational Chow ring of the moduli space M g \mathcal {M}_g of curves of genus g g is known for g ≤ 6 g \leq 6 . Here, we determine the rational Chow rings of M 7 , M 8 \mathcal {M}_7, \mathcal {M}_8 , and M 9 \mathcal {M}_9 by showing they are tautological. The key ingredient is intersection theory on Hurwitz spaces of degree 4 4 and 5 5 covers of P 1 \mathbb {P}^1 via their associated vector bundles. The main focus of this paper is a detailed geometric analysis of special tetragonal and pentagonal covers whose associated vector bundles on P 1 \mathbb {P}^1 are highly unbalanced, expanding upon previous work of the authors in the more balanced case. In genus 9 9 , we use work of Mukai 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.