Worst Singularities of Plane Curves of Given Degree

We prove that $$ and $$ are the smallest log canonical thresholds of reduced plane curves of degree $$ , and we describe reduced plane curves of degree d whose log canonical thresholds are these numbers. As an application, we prove that $$ and $$ are the smallest values of the $$ -invariant of Tian of smooth surfaces in $$ of degree $$ . We also prove that every reduced plane curve of degree $$ whose log canonical threshold is smaller than $$ is GIT-unstable for the action of the group $$ , and we describe GIT-semistable reduced plane curves with log canonical thresholds  $$ .