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

Abstract

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.

48 pages, final version, to appear in Journal of Algebraic Geometry