Tautological classes on low-degree Hurwitz spaces

Abstract

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 $k \leq 5$. The set-up developed here is also used in our subsequent work [6], wherein we prove new results about the structure of the Chow ring for $k \leq 5$.