Chow Rings of Hurwitz Spaces with Marked Ramification

The Hurwitz space H k,g is a compactification of the space of smooth genus-g curves with a simply-branched degree-k map to 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 H 3,g as the images of gluing maps whose domains are products of Hurwitz spaces H k ,g (µ) with a single marked fiber of prescribed (not necessarily simple) ramification profile µ, and to prove that the spaces H k ,g (µ) with k = 2, 3 have trivial Chow ring.