Let f = 0 be a plane algebraic curve of degree d > 1 with an isolated singular point at 0 ∈ ℂ2. We show that the Milnor number μ0(f) is less than or equal to (d−1)2 − [d/2], unless f = 0 is a set of d concurrent lines passing through 0, and characterize the curves f = 0 for which μ0(f) = (d−1)2 − [d/2].
@article{bwmeta1.element.doi-10_2478_s11533-013-0378-6,
author = {Arkadiusz P\l oski},
title = {A bound for the Milnor number of plane curve singularities},
journal = {Open Mathematics},
volume = {12},
year = {2014},
pages = {688-693},
zbl = {1310.14010},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0378-6}
}
Arkadiusz Płoski. A bound for the Milnor number of plane curve singularities. Open Mathematics, Tome 12 (2014) pp. 688-693. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0378-6/
[1] Cassou-Noguès P., Płoski A., Invariants of plane curve singularities and Newton diagrams, Uni. Iagel. Acta Math., 2011, 49, 9–34 | Zbl 1275.32025
[2] Fulton W., Algebraic Curves, Adv. Book Classics, Addison-Wesley, Redwood City, 1989
[3] Garcia Barroso E.R., Płoski A., An approach to plane algebroid branches, preprint available at http://arxiv.org/abs/1208.0913 | Zbl 1308.32032
[4] Greuel G.-M., Lossen C., Shustin E., Plane curves of minimal degree with prescribed singularities, Invent. Math., 1998, 133(3), 539–580 http://dx.doi.org/10.1007/s002220050254 | Zbl 0924.14013
[5] Gusein-Zade S.M., Nekhoroshev N.N., Singularities of type A k on plane curves of a chosen degree, Funct. Anal. Appl., 2000, 34(3), 214–215 http://dx.doi.org/10.1007/BF02482412 | Zbl 0980.32008
[6] Gwozdziewicz J., Płoski A., Formulae for the singularities at infinity of plane algebraic curves, Univ. Iagel. Acta Math., 2001, 39, 109–133 | Zbl 1015.32026
[7] Huh J., Milnor numbers of projective hypersurfaces with isolated singularities, preprint available at http://arxiv.org/abs/1210.2690 | Zbl 1308.14009
[8] Teissier B., Resolution Simultanée I, II, In: Séminaire sur les Singularités des Surfaces, Lecture Notes in Math., 777, Springer, Berlin, 1980, 71–146 http://dx.doi.org/10.1007/BFb0085880
[9] Wall C.T.C., Singular Points of Plane Curves, London Math. Soc. Stud. Texts, 63, Cambridge University Press, Cambridge, 2004 693 | Zbl 1057.14001