Worst Singularities of Plane Curves of Given Degree.

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 $d⩾3$ , 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 $P^3$ of degree $d⩾3$ . We also prove that every reduced plane curve of degree $d⩾4$ whose log canonical threshold is smaller than $\frac{5}{2d}$ is GIT-unstable for the action of the group ${\mathrm{PGL}}_3\left(C\right)$ , and we describe GIT-semistable reduced plane curves with log canonical thresholds  $\frac{5}{2d}$ .