Tautological classes on low-degree Hurwitz spaces

Abstract

Let ${\mathcal{H}}_{k,g}$ be the Hurwitz stack parametrizing degree $k$, genus $g$ covers of ${\mathbb{P}}^1$. We define the tautological ring of ${\mathcal{H}}_{k,g}$ and we show that all Chow classes, except possibly those supported on the locus of "factoring covers," are tautological up to codimension roughly $g/k$ when $k\le 5$. The set-up developed here is also used in our subsequent work, wherein we prove new results about the structure of the Chow ring for $k\le 5$.

30 pages. This paper is part of our work on low-degree Hurwitz spaces, which has been split off from a previous version because of length