Chow rings of low-degree Hurwitz spaces

TL;DR

This paper investigates the intersection theory of Hurwitz spaces for low degrees, proving stabilization of their Chow rings as genus increases and explicitly determining the rings for degree 3 covers.

Contribution

It establishes the stabilization of Chow rings for low-degree Hurwitz spaces and fully determines these rings for degree 3, advancing understanding of their algebraic structure.

Findings

Chow rings of Hurwitz spaces stabilize as genus tends to infinity.

Complete determination of Chow rings for degree 3 Hurwitz spaces.

Chow groups of simply branched Hurwitz spaces are trivial in low codimensions.

Abstract

While there is much work and many conjectures surrounding the intersection theory of the moduli space of curves, relatively little is known about the intersection theory of the Hurwitz space $\mathcal{H}_{k, g}$ parametrizing smooth degree $k$, genus $g$ covers of $\mathbb{P}^1$. Let $k = 3, 4, 5$. We prove that the rational Chow rings of $\mathcal{H}_{k,g}$ stabilize in a suitable sense as $g$ tends to infinity. In the case $k = 3$, we completely determine the Chow rings for all $g$. We also prove that the rational Chow groups of the simply branched Hurwitz space $\mathcal{H}^s_{k,g} \subset \mathcal{H}_{k,g}$ are zero in codimension up to roughly $g/k$. In subsequent work, results developed in this paper are used to prove that the Chow rings of $\mathcal{M}_7, \mathcal{M}_8,$ and $\mathcal{M}_9$ are tautological.