A bound for the Milnor number of plane curve singularities

Arkadiusz Płoski

Open Mathematics (2014)

  • Volume: 12, Issue: 5, page 688-693
  • ISSN: 2391-5455

Abstract

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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].

How to cite

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Arkadiusz Płoski. "A bound for the Milnor number of plane curve singularities." Open Mathematics 12.5 (2014): 688-693. <http://eudml.org/doc/269739>.

@article{ArkadiuszPłoski2014,
abstract = {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].},
author = {Arkadiusz Płoski},
journal = {Open Mathematics},
keywords = {Milnor number; Plane algebraic curve; plane curve},
language = {eng},
number = {5},
pages = {688-693},
title = {A bound for the Milnor number of plane curve singularities},
url = {http://eudml.org/doc/269739},
volume = {12},
year = {2014},
}

TY - JOUR
AU - Arkadiusz Płoski
TI - A bound for the Milnor number of plane curve singularities
JO - Open Mathematics
PY - 2014
VL - 12
IS - 5
SP - 688
EP - 693
AB - 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].
LA - eng
KW - Milnor number; Plane algebraic curve; plane curve
UR - http://eudml.org/doc/269739
ER -

References

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  1. [1] Cassou-Noguès P., Płoski A., Invariants of plane curve singularities and Newton diagrams, Uni. Iagel. Acta Math., 2011, 49, 9–34 Zbl1275.32025
  2. [2] Fulton W., Algebraic Curves, Adv. Book Classics, Addison-Wesley, Redwood City, 1989 
  3. [3] Garcia Barroso E.R., Płoski A., An approach to plane algebroid branches, preprint available at http://arxiv.org/abs/1208.0913 Zbl1308.32032
  4. [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 Zbl0924.14013
  5. [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 Zbl0980.32008
  6. [6] Gwozdziewicz J., Płoski A., Formulae for the singularities at infinity of plane algebraic curves, Univ. Iagel. Acta Math., 2001, 39, 109–133 Zbl1015.32026
  7. [7] Huh J., Milnor numbers of projective hypersurfaces with isolated singularities, preprint available at http://arxiv.org/abs/1210.2690 Zbl1308.14009
  8. [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. [9] Wall C.T.C., Singular Points of Plane Curves, London Math. Soc. Stud. Texts, 63, Cambridge University Press, Cambridge, 2004 693 Zbl1057.14001

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