Worst Singularities of Plane Curves of Given Degree

We prove that 2d,2d−3(d−1)2,2d−1d(d−1),2d−5d2−3d+1 and 2d−3d(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 2d,2d−3(d−1)2,2d−1d(d−1),2d−5d2−3d+1 and 2d−3d(d−2) are the smallest values of the α-invariant of Tian of smooth surfaces in P3 of degree d⩾3. We also prove that every reduced plane curve of degree d⩾4 whose log canonical threshold is smaller than 52d is GIT-unstable for the action of the group PGL3(C), and we describe GIT-semistable reduced plane curves with log canonical thresholds 52d.