Chow rings of low-degree Hurwitz spaces

While there is much work and many conjectures surrounding the intersection theory of the moduli space of curves, relatively little is known about the intersection theory of the Hurwitz space Hk,g parametrizing smooth degree k, genus g covers of P. Let k = 3, 4, 5. We prove that the rational Chow rings of Hk,g stabilize in a suitable sense as g tends to infinity. In the case k = 3, we completely determine the Chow rings for all g. We also prove that the rational Chow groups of the simply branched Hurwitz space H k,g ⊂ Hk,g are zero in codimension up to roughly g/k. In [8], results developed in this paper are used to prove that the Chow rings of M7,M8, and M9 are tautological.

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