## Equisingular deformations of normal surface singularities. I.(English)Zbl 0358.14007

The author tackles the problem of defining equisingular infinitesimal deformations of normal singular points of algebraic surfaces $$S$$. The technique is to study deformations of a resolution $$f: X\to S$$ of the singularity $$x \in S$$ which preserve the essential numerical (topological) data of the exceptional curve $$E=f^{-1}(x)$$, and which blow down to deformations of $$S$$. Much of this interesting paper consists of discussing the weaknesses as well as the strengths of this approach, for instance in comparison with the theory based on simultaneous blowing up along normally flat sections, which is generally more special than the authorts notion but coincides for certain well-studied classes of singularities.
Reviewer: L. Brenton

### MSC:

 14E15 Global theory and resolution of singularities (algebro-geometric aspects) 14J15 Moduli, classification: analytic theory; relations with modular forms 14B10 Infinitesimal methods in algebraic geometry 14D15 Formal methods and deformations in algebraic geometry 32G05 Deformations of complex structures
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