Ordinarity of configuration spaces and of wonderful compactifications

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 $X_{\mathcal{G}}$ is ordinary if and only if $(X,\mathcal{G})$ is an ordinary building set. This implies the ordinarity of many other well-known configuration spaces (under suitable assumptions).