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. Płoski 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 Płoski curve. Furthermore, we find a bound for the Milnor sum of projective plane curves in terms of GIT.

12 pages, 5 figures

References in corpus (1)

• Milnor numbers of projective hypersurfaces with isolated singularities