On a Theorem of Greuel and Steenbrink

  • Duco van StratenEmail author


A famous theorem of Greuel and Steenbrink states that the first Betti number of the Milnor fibre of a smoothing of a normal surface singularity vanishes. In this paper we prove a general theorem on the first Betti number of a smoothing that implies an analogous result for weakly normal singularities.


Singularities Topology of smoothings Weakly normal spaces 

2010Mathematics Subject Classification:

14B07 32S25 32S30 



The basis of the above text is part of my PhD thesis [9], but the results were never properly published. For this version only minor cosmetic changes have been made. I thank D. Siersma for asking me about the result and the idea of writing it up as a contribution to the volume on occasion of Gert-Martins 70th birthday.


  1. 1.
    Clemens, H.: Degeneration of Kähler manifolds. Duke Math. J. 44 (2), 215–290 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Esnault, H.: Fibre de Milnor d’ un cone sur une courbe plane singulière. Invent. Math. 68 (3), 477–496 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Greuel, G.-M., Steenbrink, J.: On the topology of smoothable singularities. In: Singularities, Part 1 (Arcata, CA, 1981). Proceedings of Symposia in Pure Mathematics, vol. 40, pp. 535–545. American Mathematical Society, Providence, RI (1983)Google Scholar
  4. 4.
    Milnor, J.: Singular Points of Complex Hypersurfaces. Annals of Mathematics Studies, vol. 61. Princeton University Press, Princeton, NJ, University of Tokyo Press, Tokyo (1968)Google Scholar
  5. 5.
    Mumford, D.: The topology of normal singularities of an algebraic surface and a criterion for simplicity. Inst. Hautes Études Sci. Publ. Math. 9, 5–22 (1961)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Siersma, D.: Singularities with critical locus a 1-dimensional complete intersection and transversal type A1. Topol. Appl. 27 (1), 51–73 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Steenbrink, J.: Mixed Hodge structures associated with isolated singularities. In: Singularities, Part 2 (Arcata, CA, 1981). Proceedings of Symposia in Pure Mathematics., vol. 40, pp. 513–536. American Mathematical Society, Providence, RI (1983)Google Scholar
  8. 8.
    van Straten, D.: On the Betti numbers of the Milnor fibre of a certain class of hypersurface singularities. In: Singularities, Representation of Algebras, and Vector Bundles (Lambrecht, 1985). Lecture Notes in Mathematics vol. 1273, pp. 203–220. Springer, Berlin (1987)Google Scholar
  9. 9.
    van Straten, D.: Weakly normal surface singularities and their improvements. Thesis, Leiden (1987)Google Scholar
  10. 10.
    Wahl, J.: Smoothings of normal surface singularities. Topology 20 (3), 219–246 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Zariski, O.: On the problem of existence of algebraic functions of two variables possessing a given branch curve. Am. J. Math. 51 (2), 305–328 (1929)MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institut für MathematikJohannes Gutenberg UniversitätMainzGermany

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