Resolution of singularities / Steven Dale Cutkosky

Contient des exercices

Bibliogr. p. 179-183. Index

.... The book under review provides as well an introduction as advanced treatment of the resolution problem. Its modern presentation of meanwhile classical ideas interacts with recent research on the topic (cf. e.g. J. Kollar [“Resolution of Singularities - Seattle Lecture", preprint, http://arXiv.org/abs/math/0508332] and results by the author). After a short introduction, Chapter 2 defines basic notions of smoothness, non-singularity, resolution, normalization and local uniformization, followed by chapter 3, containing a discussion of embedded resolution for curve singularities. Chapter 4 starts constructing the blowing up of an ideal and gives the general notion of resolution. The fifth chapter studies resolution of surface singularities and their embedded resolution (again in characteristic 0). Chapter 6 gives a complete proof for resolution of singularities in arbitrary dimension and characteristic 0, based on the work of Encinas and Villamayor. Chapters 7 and 8 cover additional topics: Local uniformization and resolution of surfaces in positive characteristics (in a modern version of Zariski’s original proof) and an introduction to valuation theory in algebraic geometry, together with the problem of local uniformization. An appendix contains technical material on the singular locus and semi-continuity-theorems used in the previous text. This book is pleasant to read and gives with its exercises a well prepared basis for a graduate course. (Zentralblatt)