Chow Rings of Hurwitz Spaces with Marked Ramification

TL;DR

This paper explores the Chow rings of Hurwitz spaces with marked ramification, establishing foundational results for the case when the degree is three, and revealing the structure of boundary strata.

Contribution

It introduces a method to analyze the Chow rings of Hurwitz spaces, especially for degree three, by relating boundary strata to products of simpler Hurwitz spaces with trivial Chow rings.

Findings

Chow ring of $ar{ ext{Hurwitz space}}_{3,g}$'s codimension-2 part is generated by boundary classes.

Boundary strata can be realized via gluing maps from products of Hurwitz spaces with trivial Chow rings.

Spaces $ ext{Hurwitz space}_{k',g'}( ext{profile})$ with $k'=2,3$ have trivial Chow rings.

Abstract

The Hurwitz space $\overline{\mathscr{H}}_{k,g}$ is a compactification of the space of smooth genus-$g$ curves with a simply-branched degree-$k$ map to $\mathbb{P}^1$. In this paper, we initiate a study of the Chow rings of these spaces, proving in particular that when $k=3$ (which is the first case in which the Chow ring is not already known), the codimension-2 Chow group is generated by the fundamental classes of codimension-2 boundary strata. The key tool is to realize the codimension-1 boundary strata of $\overline{\mathscr{H}}_{3,g}$ as the images of gluing maps whose domains are products of Hurwitz spaces $\mathscr{H}_{k',g'}(\mu)$ with a single marked fiber of prescribed (not necessarily simple) ramification profile $\mu$, and to prove that the spaces $\mathscr{H}_{k',g'}(\mu)$ with $k'=2,3$ have trivial Chow ring.