Tautological classes on low-degree Hurwitz spaces

Let Hk,g be the Hurwitz stack parametrizing degree k, genus g covers of P. We define the tautological ring of Hk,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 ≤ 5. The set-up developed here is also used in our subsequent work [3], wherein we prove new results about the structure of the Chow ring for k ≤ 5.

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