UC San Diego

We study the Chow rings of the Hurwitz spaces parametrizing degree 3, 4, and 5 covers of the projective line, the Chow rings of the moduli spaces of curves of genus 7, 8, and 9, and the Chow rings of moduli spaces of elliptic surfaces. We prove a stabilization result for the Chow rings of the Hurwitz spaces, and completely determine the Chow ring for degree 3 covers. We use these results to compute the Chow rings of the moduli spaces of curves of genus 7, 8, and 9. Then, we compute the Chow rings of moduli spaces of elliptic surfaces. We show that they satisfy a stability property, and that they satisfy vanishing and dimension properties predicted by Oprea--Pandharipande.