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Статья

Worst Singularities of Plane Curves of Given Degree

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