Wonderful compactifications of the moduli space of points in affine and projective space

TL;DR

This paper introduces new smooth compactifications of moduli spaces of weighted points in affine and projective spaces, generalizing existing constructions and establishing their properties as GIT quotients with normal crossings boundaries.

Contribution

It constructs and analyzes a class of wonderful compactifications for moduli spaces of weighted points, extending their applicability and connecting to Hassett's moduli spaces in dimension one.

Findings

GIT quotients of these compactifications are also wonderful under certain conditions

The compactifications have normal crossings boundary

In dimension one, they coincide with Hassett's moduli spaces

Abstract

We introduce and study smooth compactifications of the moduli space of n labeled points with weights in projective space, which have normal crossings boundary and are defined as GIT quotients of the weighted Fulton-MacPherson compactification. We show that the GIT quotient of a wonderful compactification is also a wonderful compactification under certain hypotheses. We also study a weighted version of the configuration spaces parametrizing n points in affine space up to translation and homothety. In dimension one, the above compactifications are isomorphic to Hassett's moduli space of rational weighted stable curves.