Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2016; 53(2): 461-473

Printed March 1, 2016

https://doi.org/10.4134/JKMS.2016.53.2.461

Copyright © The Korean Mathematical Society.

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

Jaesun Shin

KAIST

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.

Keywords: Milnor sum, polar degree, GIT

MSC numbers: Primary 14H50

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