Configurations of points on degenerate varieties and properness of moduli spaces

Abstract

Consider a smooth variety $X$ and a smooth divisor $D\subset X$. Kim and Sato (arXiv:0806.3819) define a natural compactification of $(X\setminus D)^n$, denoted $X_D^{[n]}$, which is a moduli space of stable configurations of $n$ points lying on expansions of $(X,D)$ in the sense of Jun Li (arXiv:math/0009097, arXiv:math/0110113). The purpose of this note is to generalize Kim and Sato's construction to the case where $X$ is an algebraic stack; and to construct an analogous projective moduli space $W_π^{[n]}$ for a degeneration $π:W \to B$. We construct $X^n_D$ and $W_π^{[n]}$ and prove their properness using a universal construction introduced in our paper arXiv:1110.2976 with Cadman and Wise. We then use these spaces for a concrete application, as explained in the next paragraph. In arXiv:1103.5132, a degeneration formula for Gromov--Witten invariants of schemes and stacks is developed, generalizing the approach of Jun Li. This in particular requires proving properness of Li's stack of pre-deformable stable maps in the case where the target $(X,D)$ or $W\to B$ is a Deligne--Mumford stack. One could simply adapt Li's proof, or follow the age-old tradition of imposing such endeavor as an exercise on "the interested reader". Instead, we prefer to provide a different proof here, which uses the properness of $X_D^{[n]}$ and $W_π^{[n]}$. Similar ideas are used by Kim, Kresch and Oh (arXiv:1105.6143) to prove the properness of their space of ramified maps. This note is identical to the text available on our web pages since March 2013. It is posted now as it has become an essential ingredient in others' work.