Abstract
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.
To Gert-Martin Greuel on the occasion of his 70th birthday.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Clemens, H.: Degeneration of Kähler manifolds. Duke Math. J. 44 (2), 215–290 (1977)
Esnault, H.: Fibre de Milnor d’ un cone sur une courbe plane singulière. Invent. Math. 68 (3), 477–496 (1982)
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)
Milnor, J.: Singular Points of Complex Hypersurfaces. Annals of Mathematics Studies, vol. 61. Princeton University Press, Princeton, NJ, University of Tokyo Press, Tokyo (1968)
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)
Siersma, D.: Singularities with critical locus a 1-dimensional complete intersection and transversal type A1. Topol. Appl. 27 (1), 51–73 (1987)
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)
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)
van Straten, D.: Weakly normal surface singularities and their improvements. Thesis, Leiden (1987)
Wahl, J.: Smoothings of normal surface singularities. Topology 20 (3), 219–246 (1981)
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)
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
van Straten, D. (2017). On a Theorem of Greuel and Steenbrink. In: Decker, W., Pfister, G., Schulze, M. (eds) Singularities and Computer Algebra. Springer, Cham. https://doi.org/10.1007/978-3-319-28829-1_17
Download citation
DOI: https://doi.org/10.1007/978-3-319-28829-1_17
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-28828-4
Online ISBN: 978-3-319-28829-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)