Chow rings, cohomology rings, and point counts of moduli spaces of curves
Abstract
In this expository article, we present on state-of-the art results regarding three closely related invariants of moduli spaces of curves: their Chow rings, cohomology rings, and point counts over finite fields. We study the moduli space Mg,n, parameterizing smooth genus g curves with n marked points, as well as its compactification by stable curves Mg,n. After explaining the relationship between these different invariants, we survey what is know regarding the following related questions: When are the Chow rings tautological? When are the cohomology groups tautological? And when are the point counts over fields of size q given by a polynomial in q?
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