Deformations of Singularities

Jan Stevens (auth.)3540005609, 9783540005605

These notes deal with deformation theory of complex analytic singularities and related objects.

The first part treats general theory. The central notion is that of versal deformation
in several variants. The theory is developed both in an abstract way and in a concrete way suitable for computations.

The second part deals with more specific problems, specially on curves and surfaces. Smoothings of singularities are the main concern.

Examples are spread throughout the text.

Table of contents :
Introduction….Pages 1-4
1. Deformations of singularities….Pages 5-14
2. Standard bases….Pages 15-22
3. Infinitesimal deformations….Pages 23-31
4. Example: the fat point of multiplicity four….Pages 33-38
5. Deformations of algebras….Pages 39-44
6. Formal deformation theory….Pages 45-53
7. Deformations of compact manifolds….Pages 55-61
8. How to solve the deformation equation….Pages 63-66
9. Convergence for isolated singularities….Pages 67-70
10. Quotient singularities….Pages 71-77
11. The projection method….Pages 79-92
12. Formats….Pages 93-104
13. Smoothing components of curves….Pages 105-111
14. Kollár’s conjectures….Pages 113-124
15. Cones over curves….Pages 125-136
16. The versal deformation of hyperelliptic cones….Pages 137-146
References….Pages 147-153
Index….Pages 155-157