Worst singularities of plane curves of given degree

TL;DR

This paper determines the smallest log canonical thresholds of reduced plane curves of degree d, characterizes the curves attaining these thresholds, and explores their implications for Tian's alpha-invariant and GIT stability.

Contribution

It explicitly computes minimal log canonical thresholds for plane curves, describes the curves achieving these thresholds, and links these thresholds to stability properties under group actions.

Findings

Identifies the smallest log canonical thresholds for degree d curves.

Characterizes curves that attain these minimal thresholds.

Establishes a connection between thresholds and GIT stability.

Abstract

We prove that $\frac{2}{d}$, $\frac{2d-3}{(d-1)^2}$, $\frac{2d-1}{d(d-1)}$, $\frac{2d-5}{d^2-3d+1}$ and $\frac{2d-3}{d(d-2)}$ are the smallest log canonical thresholds of reduced plane curves of degree $d\geqslant 3$, and we describe reduced plane curves of degree $d$ whose log canonical thresholds are these numbers. As an application, we prove that $\frac{2}{d}$, $\frac{2d-3}{(d-1)^2}$, $\frac{2d-1}{d(d-1)}$, $\frac{2d-5}{d^2-3d+1}$ and $\frac{2d-3}{d(d-2)}$ are the smallest values of the $\alpha$-invariant of Tian of smooth surfaces in $\mathbb{P}^3$ of degree $d\geqslant 3$. We also prove that every reduced plane curve of degree $d\geqslant 4$ whose log canonical threshold is smaller than $\frac{5}{2d}$ is GIT-unstable for the action of the group $\mathrm{PGL}_3(\mathbb{C})$, and we describe GIT-semistable reduced plane curves with log canonical thresholds $\frac{5}{2d}$.