Superisolated Surface Singularities

Details Superisolated Surface Singularities Superisolated Surface Singularities

2005-09-13

arxiv.org/abs/math/0509283v1

Mathematics

Algebraic Geometry

Survey article for the Proceedings of the Conference "Singularities and Computer Algebra" on Occasion of Gert-Martin Greuel's

Scientific paper

In this survey, we review part of the theory of superisolated surface singularities (SIS) and its applications including some new and recent developments. The class of SIS singularities is, in some sense, the simplest class of germs of normal surface singularities. Namely, their tangent cones are reduced curves and the geometry and topology of the SIS singularities can be deduced from them. Thus this class \emph{contains}, in a canonical way, all the complex projective plane curve theory, which gives a series of nice examples and counterexamples. They were introduced by I. Luengo to show the non-smoothness of the $\mu$-constant stratum and have been used to answer negatively some other interesting open questions. We review them and the new results on normal surface singularities whose link are rational homology spheres. We also discuss some positive results which have been proved for SIS singularities.

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