The worst destabilizing 1-parameter subgroup for toric rational curves with one unibranch singularity
Joshua Jackson, David Swinarski
TL;DR
This work analyzes the worst destabilizing 1-parameter subgroup for Chow points of toric rational curves with a single unibranch singularity by translating Geometric Invariant Theory stability into an explicit convex optimization problem. The authors show that the worst 1-PS corresponds to the nearest point on a convex cone W to a strategically defined vector a, with the Hilbert-Mumford function reducible to a simple inner product via a·w, leveraging the secondary (Chow) polytope description. A key result is the persistence phenomenon: for large embedding dimension $N$, the initial segment of the worst-PS weights becomes independent of $N$, while the tail follows a precise linear formula $w_i^* = m i + b$ with $m$ and $b$ given by explicit expressions. The paper combines convex analysis (KKT conditions), combinatorial geometry (Chow/secondary polytopes), and numerical semigroup data to derive explicit descriptions and examples, including higher-order cusps, and outlines future directions towards moduli spaces of unstable curves and extensions to other singularities.
Abstract
Kempf proved that when a point is unstable in the sense of Geometric Invariant Theory, there is a ``worst'' destabilizing 1-parameter subgroup $λ^{*}$. 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.