The worst destabilizing 1-parameter subgroup for toric rational curves with one unibranch singularity

TL;DR

This paper investigates the worst destabilizing 1-parameter subgroup in the GIT stability analysis of toric rational curves with a unibranch singularity, translating the problem into convex geometry and providing explicit combinatorial descriptions.

Contribution

It provides an explicit combinatorial description of the worst 1-PS for certain unstable points in GIT, applicable to high embedding dimensions.

Findings

Worst 1-PS characterized by a combinatorial description

Problem reduced to convex geometry: closest point on a cone

Results hold for sufficiently large embedding dimensions

Abstract

Kempf proved that when a point is unstable in the sense of Geometric Invariant Theory, there is a ``worst'' destabilizing 1-parameter subgroup $\lambda^{*}$. It is natural to ask: what are the worst 1-PS for the unstable points in the GIT problems used to construct the moduli space of curves $\overline{M}_g$? Here we consider Chow points of toric rational curves with one unibranch singular point. We translate the problem as an explicit problem in convex geometry (finding the closest point on a polyhedral cone to a point outside it). We prove that the worst 1-PS has a combinatorial description that persists once the embedding dimension is sufficiently large, and present some examples.