A bound for the Milnor sum of projective plane curves in terms of GIT

Abstract

Let $C$ be a projective plane curve of degree $d$ whose singularities are all isolated. Suppose $C$ is not concurrent lines. P{\l}oski proved that the Milnor number of an isolated singlar point of $C$ is less than or equal to $(d-1)^{2}-\lfloor \frac{d}{2} \rfloor$. In this paper, we prove that the Milnor sum of $C$ is also less than or equal to $(d-1)^{2}-\lfloor \frac{d}{2} \rfloor$ and the equality holds if and only if $C$ is a P{\l}oski curve. Furthermore, we find a bound for the Milnor sum of projective plane curves in terms of GIT.