Worst Singularities of Plane Curves of Given Degree

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