Compact Moduli Spaces of Marked Cubic Plane Curves

TL;DR

This paper explores the GIT-based compactifications of the moduli space of marked cubic plane curves, analyzing wall-crossings through curve singularities and point configurations.

Contribution

It provides a complete description of GIT walls and links moduli space wall-crossing to singularities and point positions on cubic curves.

Findings

Complete GIT wall description for marked cubic plane curves

Connection between wall-crossing and curve singularities

Analysis of point configurations affecting moduli space stability

Abstract

We study compactifications of the moduli space of a plane cubic curve marked by \(n\) labeled points up to projective equivalence via Geometric Invariant Theory (GIT). Specifically, we provide a complete description of the GIT walls and show that the moduli-theoretic wall-crossing can be understood through analysis of the singularities of the plane curves and the position of the points.