Intersection theory on Hurwitz spaces of low-degree covers

dc.title: Intersection theory on Hurwitz spaces of low-degree covers dc.description.abstract: We show that the first cohomology group of the Hurwitz space of fully-marked admissible covers H^1(H_{d,g}(μ)) vanishes for covers of degree d = 3 and deduce the same result for the classical Hurwitz space of simply-branched covers H^s_{3,g}. In degrees 4 and 5, we compute examples where H^1(H_{d,g}(μ)) is nonzero. We describe the stratification of the boundary of H_{d,g}(μ) by lower-dimensional H_{d',g'}(μ), and set up an inductive framework which may be used for future arguments involving the odd cohomology of H_{d,g}(μ).

Furthermore, we summarize recent work with Clader et. al. showing that the Chow group A^2(H^s_{3,g}) is generated by classes of codimension-2 boundary strata. Together, these results make progress on determining the extent to which the coho- mology and Chow rings of Hurwitz spaces are tautological.