Variation of stability for moduli spaces of unordered points in the plane

TL;DR

This paper investigates how the stability of moduli spaces of unordered points in the plane varies under different GIT quotients, revealing new insights into their birational geometry and stability behavior.

Contribution

It provides a detailed analysis of GIT stability variations for moduli spaces of points, using Hilbert schemes as Mori dream spaces and Bridgeland stability, including explicit descriptions for five and seven points.

Findings

Identification of GIT walls for specific point configurations

Distinct stability behaviors for seven points compared to Chow quotients

Complete description of moduli space stability for five points

Abstract

We study compactifications of the moduli space of unordered points in the plane via variation of GIT quotients of their corresponding Hilbert scheme. Our VGIT considers linearizations outside the ample cone and within the movable cone. For that purpose, we use the description of the Hilbert scheme as a Mori dream space, and the moduli interpretation of its birational models via Bridgeland stability. We determine the GIT walls associated with curvilinear zero-dimensional schemes, collinear points, and schemes supported on a smooth conic. For seven points, we study a compactification associated with an extremal ray of the movable cone, where stability behaves very differently from the Chow quotient. Lastly, a complete description for five points is given.