On a smoothness characterization for good moduli spaces

Abstract

Let $\mathcal{X}$ be a smooth Artin stack with properly stable good moduli space $\ddot{I}€:\mathcal{X}\to X$. The purpose of this paper is to prove that a simple geometric criterion can often characterize when the moduli space $X$ is smooth and the morphism $\ddot{I}€$ is flat.