Low degree Hurwitz stacks in the Grothendieck ring

Abstract

For $2\le d\le 5$, we show that the class of the Hurwitz space of smooth degree $d$, genus $g$ covers of ${\mathbb{P}}^1$ stabilizes in the Grothendieck ring of stacks as $g\to \infty$, and we give a formula for the limit. We also verify this stabilization when one imposes ramification conditions on the covers, and obtain a particularly simple answer for this limit when one restricts to simply branched covers.

Theorem 3.2 was previously missing the hypothesis that the covers were arithmetically Gorenstein, as pointed out to us by Enrico Schlesinger. We have now added in this hypothesis and issued an erratum for the published version in Compositio