[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/JKMS.2016.53.2.461

A BOUND FOR THE MILNOR SUM OF PROJECTIVE PLANE CURVES IN TERMS OF GIT

Shin, Jaesun (Department of Mathematical Sciences KAIST)

Publication Information

Journal of the Korean Mathematical Society / v.53, no.2, 2016 , pp. 461-473 More about this Journal

Abstract

Let C be a projective plane curve of degree d whose singularities are all isolated. Suppose C is not concurrent lines. P loski 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 loski curve. Furthermore, we find a bound for the Milnor sum of projective plane curves in terms of GIT.

Keywords

Milnor sum; polar degree; GIT;

Citations & Related Records

Reference

1	I. Cheltsov, Worst singularities of plane curves of given degree, preprint available at http://arxiv.org/abs/1409.6186.
2	I. Dolgachev, Lectures on Invariant Theory, LMS Lecture Note Series 296, Cambridge University Press, Cambridge, 2003.
3	T. Fassarella and N. Medeiros, On the polar degree of projective hypersurfaces, J. Lond. Math. Soc. (2) 86 (2012), no. 1, 259-271. DOI
4	G.-M. Greuel, C. Lossen, and E. Shustin, Introduction to singularities and deformations, Springer Monographs in Mathematics, Springer, Berlin, 2007.
5	J. Huh, Milnor numbers of projective hypersurfaces with isolated singularities, Duke Math. J. 163 (2014), no. 8, 1525-1548. DOI
6	A. P loski, A bound for the Milnor number of plane curve singularities, Cent. Eur. J. Math. 12 (2014), no. 5, 688-693.