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

TL;DR

This paper establishes an upper bound for the Milnor sum of projective plane curves based on degree and GIT stability, characterizing equality cases as P{2}ski curves, thus linking singularity invariants with geometric invariant theory.

Contribution

It extends P{2}ski's bound on Milnor numbers to the sum of Milnor numbers using GIT, providing a new criterion for classifying P{2}ski curves.

Findings

Milnor sum is bounded by (d-1)^2 - floor(d/2).

Equality characterizes P{2}ski curves.

GIT provides bounds for singularity invariants.

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.