Cohomology of Moduli Space of Multiscale Differentials in Genus 0

Abstract

We prove that the rational cohomology ring of moduli space of multiscale differentials in genus 0 is generated by the boundary divisors. The main idea is the technique of the Chow-K\"unneth generation Property and the observation that the intersection of a collection of boundary divisors in the moduli space is irreducible. We observe that the relations between the boundary strata in cohomology are generated by the pullback of the WDVV relations and the relations between the torus-invariant subvarieties in the fiber over $\overline{M}_{0,n}$. We also characterize the cases in which the moduli space is a smooth variety, and in these cases, we prove that the integral cohomology ring is generated by the boundary divisors.