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Regular version of the site

Working paper

Worst singularities of plane curves of given degree

I 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 $d\geqslant 3$. I describe reduced plane curves of degree $d$ whose log canonical thresholds are these numbers. I 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 I describe GIT-semistable reduced plane curves with log canonical thresholds $\frac{5}{2d}$. I 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 $d\geqslant 3$.