Chow Rings of Hurwitz Spaces with Marked Ramification

Abstract

The Hurwitz space ${\overline{\mathcal{H}}}_{k,g}$ 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 ${\overline{\mathcal{H}}}_{3,g}$ as the images of gluing maps whose domains are products of Hurwitz spaces ${\mathcal{H}}_{k^{\prime },g^{\prime }}(\hat{I}¼)$ with a single marked fiber of prescribed (not necessarily simple) ramification profile $\hat{I}¼$, and to prove that the spaces ${\mathcal{H}}_{k^{\prime },g^{\prime }}(\hat{I}¼)$ with $k^{\prime }=2,3$ have trivial Chow ring.

27 pages, comments welcome!