Chow Rings of Hurwitz Spaces with Marked Ramification
Emily Clader, Zhengning Hu, Hannah Larson, Amy Q. Li, Rose Lopez
TL;DR
The paper addresses the Chow rings of Hurwitz spaces and their admissible-cover compactifications, focusing on the first nontrivial degree where the Chow ring is unknown, k=3. It develops a CKgP-based, vector-bundle framework to analyze codimension-1 boundary strata as images of gluing maps from marked-ramification Hurwitz spaces, proving those factors have trivial Chow rings. By excision and explicit excess-intersection calculations, it shows the codimension-1 boundary strata of \\overline{\\mathscr{H}}_{3,g} have trivial Chow rings, implying A^2(\\overline{\\mathscr{H}}_{3,g}) is generated by codimension-2 boundary strata. The work employs principal parts bundles, evaluation maps, and a detailed analysis of boundary stratifications to derive a complete description of the Chow ring in genus g, establishing a foundation for higher-codimension investigations and potential extensions to larger degrees. The results have significance for intersection theory on Hurwitz spaces and for understanding how boundary geometry controls Chow groups in moduli problems.
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 $\mathscr{H}_{k',g'}(μ)$ with a single marked fiber of prescribed (not necessarily simple) ramification profile $μ$, and to prove that the spaces $\mathscr{H}_{k',g'}(μ)$ with $k'=2,3$ have trivial Chow ring.