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Equimultiplicity and blowing up. An algebraic study. With an appendix by B. Moonen. (English) Zbl 0649.13011
Berlin etc.: Springer-Verlag. xvii, 629 p. DM 164.00 (1988).
In the desingularization theory (developed by Zariski and Hironaka) one is led to blow up regular centers D contained in the singular locus of a given algebraic variety X. In such a case all points of D have the same $$(1)\quad multiplicity,$$ $$(2)\quad Hilbert\quad polynomial,$$ and $$(3)\quad Hilbert$$ function. The corresponding three conditions are the same if X is a hypersurface, but in general they are different. One can ask for generalizations of the numerical conditions corresponding to (1), (2) and (3). And indeed, it is the main aim of the book under review to develop a general theory concerning these questions in the frame of commutative algebra. In this sense this book may be considered as a special course in commutative algebra. It has nine chapters and ends with a long appendix written by B. Moonen.
The first three chapters develop the basic facts and techniques concerning multiplicities, Hilbert functions, reduction of ideals, generalities about graded rings and blowing ups, characterizations of quasi-unmixed ideals, etc. - In chapter four various notions of equimultiplicity are presented (all of them coinciding in the hypersurface case), while in the next chapter one illustrates how these notions can be used in order to study the behaviour of the Cohen-Macaulay property undergoing a blowing up. - In chapter six one continues the idea of the previous chapter by using these conditions of equimultiplicity in the study of the numerical behaviour of singularities of blowing up singular centers. - In the last three chapters one discusses the local cohomology and the duality of graded rings (somehow in a parallel way with the case of local rings). For example, one studies local rings (A,m) with finite local cohomology $$H^ i_ m(A)$$ for $$i<\dim (A)$$, with applications to the affine cones over a projective variety; one also studies the Cohen-Macaulay properties of the Rees rings.
The appendix by B. Moonen has three parts: the first one treats the fundamentals of the local complex-analytic geometry, the second one deals with the geometric description of the multiplicity of a complex space- germ as the local mapping degree of a generic projection, and the third one with the theory of compact Stein neighbourhoods and the properties of normal flatness in the analytic case. The book is selfcontained and written with all details needed for a beginner.
Reviewer: L.Bădescu

##### MSC:
 13H15 Multiplicity theory and related topics 14E15 Global theory and resolution of singularities (algebro-geometric aspects) 14-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry 13-02 Research exposition (monographs, survey articles) pertaining to commutative algebra 14B15 Local cohomology and algebraic geometry 32Bxx Local analytic geometry 13-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra 13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)