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

The rational Chow ring of the moduli space ${\mathcal{M}}_g$ of curves of genus $g$ is known for $g\le 6$. Here, we determine the rational Chow rings of ${\mathcal{M}}_7,{\mathcal{M}}_8,$ and ${\mathcal{M}}_9$ by showing they are tautological. One key ingredient is intersection theory on Hurwitz spaces of degree $4$ and $5$ covers of ${\mathbb{P}}^1$, 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 ${\mathbb{P}}^1$ 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.