Chow rings of low-degree Hurwitz spaces

Abstract

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 ℋk,g{\mathcal{H}_{k,g}} parametrizing smooth degree k, genus g covers of ℙ1{\mathbb{P}^{1}}. Let k=3,4,5{k=3,4,5}. We prove that the rational Chow rings of ℋk,g{\mathcal{H}_{k,g}} stabilize in a suitable sense as g tends to infinity. In the case k=3{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 ℋk,gs⊂ℋk,g{\mathcal{H}^{s}_{k,g}\subset\mathcal{H}_{k,g}} are zero in codimension up to roughly gk{\frac{g}{k}}. In [S. Canning and H. Larson, The Chow rings of the moduli spaces of curves of genus 7,8{7,8} and 9, preprint 2021, arXiv:2104.05820], results developed in this paper are used to prove that the Chow rings of ℳ7,ℳ8,{\mathcal{M}_{7},\mathcal{M}_{8},} and ℳ9{\mathcal{M}_{9}} are tautological.