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Chow rings of low-degree Hurwitz spaces
Samir Canning, Hannah Larson
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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 ℋ_(k, g) parametrizing smooth degree k, genus g covers of ℙ¹. Let k = 3, 4, 5. We prove that the rational Chow rings of ℋ_(k,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 ℋˢ_(k,g) ⊂ℋ_(k,g) are zero in codimension up to roughly g/k. In subsequent work, results developed in this paper are used to prove that the Chow rings of ℳ₇, ℳ₈, and ℳ₉ are tautological.
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