Israel Journal of Mathematics

, Volume 115, Issue 1, pp 29–50

Milnor numbers for surface singularities

  • A. Melle-Hernández
Article

Abstract

An additive formula for the Milnor number of an isolated complex hypersurface singularity is shown. We apply this formula for studying surface singularities. Durfee's conjecture is proved for any absolutely isolated surface and a generalization of Yomdin singularities is given.

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References

  1. [1]
    E. Artal-Bartolo,Forme de Jordan de la monodromie des singularitées superisolées de surfaces, Memoirs of the American Mathematical Society, Vol. 109, no. 525, American Mathematical Society, Providence, RI, 1994.Google Scholar
  2. [2]
    A. Dimca,Singularities and Topology of Hypersurfaces, Springer-Verlag, Berlin-Heidelberg-New York, 1992.MATHGoogle Scholar
  3. [3]
    A. Durfee,The signature of smoothings of complex surface singularities, Mathematische Annalen232 (1978), 85–98.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    C. G. Gibson et al.,Topological stability of smooth mappings, Lecture Notes in Mathematics552, Springer-Verlag, Berlin-Heidelberg-New York, 1976.MATHCrossRefGoogle Scholar
  5. [5]
    S. Gusein-Zade, I. Luengo and A. Melle-Hernández,Partial resolutions and the zeta-function of a singularity, Commentarii Mathematici Helvetici72 (1997), 244–256.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    R. Hartshorne,Algebraic Geometry, Graduate Text in Mathematics, Vol. 52, Springer-Verlag, New York, 1977.MATHGoogle Scholar
  7. [7]
    F. Hirzebruch,Topological Methods in Algebraic Geometry, Springer-Verlag, New York, 1966.MATHGoogle Scholar
  8. [8]
    A. G. Kushnirenko,Polyèdres de Newton et nombres de Milnor, Inventiones Mathematicae32 (1976), 1–31.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    H Laufer,On μ for surface singularities, Proceedings of Symposia in Pure Mathematics, Vol. 30, American Mathematical Society, Providence, RI, 1975, pp. 45–49.Google Scholar
  10. [10]
    D. T. Lê,Some remarks on relative monodromy, Proceedings of the Nordic Summer School, Oslo 1976, Sijthaff and Nordhoff, Alpheu aan den Rijn, 1977, pp. 397–403.Google Scholar
  11. [11]
    D. T. Lê,Ensembles analytiques complexes avec lieu singulier de dimension un (d'apres Iomdine), Seminaire sur les Singularités, Publications Matematiqués de l'Université París VII, 1980, pp. 87–95.Google Scholar
  12. [12]
    D. T. Lê, and B. Teissier,Cycles évanescents, sections planes et conditions de Whitney II, Proceedings of Symposia in Pure Mathematics, Vol. 40, American Mathematical Society, Providence, RI, 1983, pp. 65–103.Google Scholar
  13. [13]
    I. Luengo,The μ-constant stratum is not smooth, Inventiones Mathematicae90 (1987), 139–152.MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    I. Luengo and A. Melle-Hernández,A formula for the Milnor number, Comptes Rendus de l'Académie des Sciences, Paris, Série I321 (1995), 1473–1478.MATHGoogle Scholar
  15. [15]
    J. Milnor,Singular points of complex hypersurfaces, Annales of Mathematics Studies, No. 61, Princeton Univ. Press, Princeton, New Jersey, 1968.MATHGoogle Scholar
  16. [16]
    V. P. Palamodov,Multiplicity of a holomorphic mappings, Functional Analysis and its Applications1 (1967), 218–266.MATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    A. Parusiński,A generalization of the Milnor number, Mathematische Annalen281 (1988), 247–254.CrossRefMathSciNetMATHGoogle Scholar
  18. [18]
    A. Parusiński and P. Pragacz,A formula for the Euler characteristic of singular hypersurfaces, Journal of Algebraic Geometry4 (1995), 337–351.MathSciNetMATHGoogle Scholar
  19. [19]
    M. Saito,On Steenbrink's Conjecture, Mathematische Annalen289 (1991), 703–716.MATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    D. Siersma,The monodromy of a series of hypersurface singularities, Commentarii Mathematici Helvetici65 (1990), 181–197.MATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    M. G. Soares and P. J. Giblin,Recognizing singularities of surfaces in ℂℙ, Mathematical Proceedings of the Cambridge Philosophical Society91 (1982), 17–27.MATHMathSciNetCrossRefGoogle Scholar
  22. [22]
    B. Teissier,Cycles évanescents, sections planes et conditions de Whitney, Asterisque7–8 (1973), 363–391.Google Scholar
  23. [23]
    B. Teissier,Variétés polaires I, Inventiones Mathematicae40 (1977), 267–292.MATHCrossRefMathSciNetGoogle Scholar
  24. [24]
    M. Tomari,A p g -formula and elliptic singularities, Publications of the Research Institute for Mathematical Sciences of Kyoto University21 (1985), 297–354.MATHMathSciNetGoogle Scholar
  25. [25]
    S. S. T. Yau,Topological type of isolated hypersurfaces singularities, Contemporary Mathematics, Vol. 101, American Mathematical Society, Providence, RI, 1989, pp. 303–321.Google Scholar
  26. [26]
    Y. N. Yomdin,Complex surfaces with a one-dimensional set of singularities, Siberian Mathematical Journal5 (1975), 748–762.CrossRefGoogle Scholar

Copyright information

© Hebrew University 2000

Authors and Affiliations

  • A. Melle-Hernández
    • 1
  1. 1.Departamento de Geometría y Topología, Facultad de MatemáticasUniversidad Complutense de madridMadridSpain

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