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.

arXiv comment48 pages, final version, to appear in Journal of Algebraic Geometry

References in corpus (8)

• On the tautological ring of $\M _g$
• admcycles -- a Sage package for calculations in the tautological ring of the moduli space of stable curves
• Equivariant geometry and the cohomology of the moduli space of curves
• Nontautological Bielliptic Cycles
• The integral Chow ring of the stack of smooth non-hyperelliptic curves of genus three
• The Picard rank conjecture for the Hurwitz spaces of degree up to five
• The Q-Picard group of the moduli space of curves in positive characteristic
• On the Chow ring of the Hurwitz space of degree three covers of ${\mathbf{P}}^1$

Cited by in corpus (1)

• Extensions of tautological rings and motivic structures in the cohomology of ${\overline{\mathcal{M}}}_{g,n}$