Cohomology of Moduli Space of Multiscale Differentials in Genus 0

TL;DR

This paper proves that the rational cohomology ring of the moduli space of multiscale differentials in genus 0 is generated by boundary divisors, using Chow-K"unneth techniques and analyzing boundary relations.

Contribution

It establishes the generation of the cohomology ring by boundary divisors and characterizes smooth cases, advancing understanding of moduli space topology.

Findings

Cohomology ring generated by boundary divisors in genus 0

Relations between boundary strata derived from WDVV and torus-invariant relations

Integral cohomology ring generated by boundary divisors in smooth cases

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.