Topics on real and complex singularities. An introduction. (English) Zbl 0628.14001

Dimca, Alexandru

Advanced Lectures in Mathematics. Braunschweig/Wiesbaden: Friedr. Vieweg & Sohn. XVII, 242 p.; DM 54.00 (1987).

Singlarity theory developed during the 1960’s from work of Whitney, of thom (catastrophe theory), of Arnol’d (mostly singularities of functions) and of Mather, on stability theory. Most books on the subject up till now have been strongly influenced by one or more of these areas. The author’s is perhaps the first book written in the spirit of the consolidation and expansion that the subject has undergone since 1970.
It contains the expected introductory material on germs and jets, and on equivalence relations via group actions. Material (nowadays routine) is included touching on semialgebraic sets and transversals (slices) to orbits, and there is a chapter of elementary classifications in projective geometry to be used for examples later on. This already sets this book apart as giving an improved all-round introduction to the subject for a student. The first half of the book concludes with a chapter on finite determinacy. The author here omits discussion of the case of right-left equivalence, so that the necessary algebra becomes much simpler: on the other hand, he brings in Artin’s approximation theorem, proves equivalence of contact finite determinacy (in the complex case) to the preimage \(f^{-1}(0)\) having an isolated (complete intersection) singularity, and also shows that this holds “in general”.
The second half of the book treats several independent topics: again while these are fairly basic and accessible, it differs markedly from the selection made by previous authors. There are chapters on weighted homogeneous singularities, includng a statement of Saito’s criterion; classifications of simple singularities of functions (standard) and of 0- dimensional complete intersections (less so); on curve and surface singularities, including a proof that plane curve singularities can be resolved by blowing-up, but not the classification by Puiseux pairs; also with some description of resolution of surface singularities, but treating only rational double points in any detail; and a final chapter on hyperplane sections and projective duals of projective duals of projective hypersurfaces.
The book internationally designed to lead the reader into the theory: a number of topics are introduced and not taken too far, and many substantial results are quoted, with references given for proofs. Although the real case is often mentioned, the main emphasis is on the complex analytic case, which allows many references to other types of mathematics.
In general, the book reads well. The reviewer noted few mistakes (e.g. the claim (p. 52) that a cuspidal cubic specialises to a triangle). In all, this is certainly a book I would recommend to a student learning the subject.

Reviewer: C.T.C.Wall

MSC:

14B05	Singularities in algebraic geometry
14-02	Research exposition (monographs, survey articles) pertaining to algebraic geometry
58C25	Differentiable maps on manifolds
58K99	Theory of singularities and catastrophe theory
32Sxx	Complex singularities
32-02	Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
14-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry
32-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to several complex variables and analytic spaces
14B12	Local deformation theory, Artin approximation, etc.
58-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to global analysis

Keywords:

Singlarity theory; semialgebraic sets; Artin’s approximation theorem; finite determinacy; weighted homogeneous singularities; plane curve singularities; resolution of surface singularities