Geometry of complex surface singularities. (English) Zbl 1026.32056

Brasselet, Jean-Paul (ed.) et al., Singularities-Sapporo 1998. Proceedings of the international symposium on singularities in geometry and topology, Sapporo, Japan, July 6-10, 1998. Tokyo: Kinokuniya Company Ltd. Adv. Stud. Pure Math. 29, 163-180 (2000).
A geometric technique to study a singular point \(x\) of a complex analytic space \(X\) (embedded in affine space) is to consider the limits of tangent hyperplanes, i.e. hyperplanes \(H_u\) containing the tangent space \(T_{X,u}\) to \(X\) at a variable regular point \(u\in X\) near \(x\). Work in this direction was done by Lê, Teissier, Gonzalez-Sprinberg, Lejeune-Jalabert, Snoussi, etc. The case best understood is that of surfaces. This article is a survey of results about limits of tangent hyperplanes of a normal surface singularity.
The first part of the paper reviews results in the case where the surface is contained in \(\mathbb{C}^3\). Milnor numbers are introduced and criteria to conclude that a plane is not a limit of tangents, based on these numbers, are discussed. Other results, relating these limits to certain generatrices of the tangent cone to \(X\) at \(x\) are also explained.
Then the author moves to the case of a normal surface, not necessarily embeddable in three-space. He explains how many of the previous results can be extended to this situation. Much of this was done by Jawad Snoussi (dissertation, University of Marseille).
Finally, the theory of resolution of normal surface singularities is reviewed. Some connections of the previous notions to this theory are explained and, at the end, the author suggests the use of these techniques to attack certain unsolved problems. For instance, that of getting effective bounds for the number of normalized blowing ups (with zero-dimensional centers) necessary to resolve a normal surface singularity, or the similar one when one uses normalized Nash modifications.
This paper is written with great clarity and precision and it should be an eczellent introduction to this subject.
For the entire collection see [Zbl 0963.00026].


32S25 Complex surface and hypersurface singularities
32S45 Modifications; resolution of singularities (complex-analytic aspects)
32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
32S10 Invariants of analytic local rings
14J17 Singularities of surfaces or higher-dimensional varieties