Chow Rings of Hurwitz Spaces with Marked Ramification

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.