Cohomology Rings of Moduli of Point Configurations on the Projective Line

TL;DR

This paper characterizes the Chow rings of moduli spaces of ordered point configurations on the projective line and demonstrates the existence of two small desingularizations with different cohomology rings.

Contribution

It provides a detailed description of Chow rings for these moduli spaces and reveals non-isomorphic cohomology rings for different desingularizations.

Findings

Chow rings of moduli spaces are explicitly described.

Existence of two small desingularizations with distinct cohomology rings.

Illustrates complexity of the topology of moduli spaces.

Abstract

We describe the Chow rings of moduli spaces of ordered configurations of points on the projective line for arbitrary (sufficiently generic) stabilities. As an application, we exhibit such a moduli space admitting two small desingularizations with non-isomorphic cohomology rings.