Ordinarity of configuration spaces and of wonderful compactifications

TL;DR

The paper proves that certain configuration spaces and moduli spaces are ordinary, establishing conditions under which their compactifications preserve this property, with implications for various well-known spaces.

Contribution

It demonstrates that the ordinarity property is preserved in Fulton-MacPherson configuration spaces and wonderful compactifications under specific conditions, extending to many known configuration spaces.

Findings

Fulton-MacPherson configuration spaces are ordinary if the base is ordinary

Moduli spaces of genus zero stable curves are ordinary

Wonderful compactifications are ordinary if and only if the building set is ordinary

Abstract

We prove the following: (1) if $X$ is ordinary, the Fulton-MacPherson configuration space $X[n]$ is ordinary for all $n$; (2) the moduli of stable $n$-pointed curves of genus zero is ordinary. (3) More generally we show that a wonderful compactification $X_\sg$ is ordinary if and only if $(X,\sg)$ is an ordinary building set. This implies the ordinarity of many other well-known configuration spaces (under suitable assumptions).