Tautological classes on low-degree Hurwitz spaces

TL;DR

This paper introduces the tautological ring of Hurwitz spaces parametrizing low-degree covers and shows most Chow classes are tautological up to a certain codimension, advancing understanding of their algebraic structure.

Contribution

It defines the tautological ring for Hurwitz spaces and demonstrates that nearly all Chow classes are tautological for degrees up to 5, except possibly on factoring covers.

Findings

Most Chow classes are tautological up to codimension g/k for k ≤ 5.

The framework aids in analyzing the Chow ring structure of Hurwitz spaces.

Excludes classes supported on factoring covers from tautological classes.

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, wherein we prove new results about the structure of the Chow ring for $k \leq 5$.