On a Theorem of Greuel and Steenbrink
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.
KeywordsSingularities Topology of smoothings Weakly normal spaces
2010Mathematics Subject Classification:14B07 32S25 32S30
The basis of the above text is part of my PhD thesis , 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.
- 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.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
- 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.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.van Straten, D.: Weakly normal surface singularities and their improvements. Thesis, Leiden (1987)Google Scholar