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Resolution of singularities. (English) Zbl 1076.14005

Graduate Studies in Mathematics 63. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3555-6/hbk). viii, 186 p. (2004).
Resolution of singularities is one of the origins of algebraic geometry. There is a long way from Newton’s method to determine branches of a plane curve, Puiseux-series’, the work of M. Noether, Riemann, the geometers of the Italian school, and many others. In the middle of the 20th century, Zariski and Abhyankar prepared the ground to study the general case of arbitrary dimension which has been settled in characteristic 0 by H. Hironaka [Ann. Math. (2) 79, 109–203, 205–326 (1964; Zbl 0122.38603)]. Hironaka’s result (which had not been recognized in its full importance over the first years) is meanwhile one of the most famous, frequently used theorems of algebraic geometry, though apparently only few people have gone through all details of its demanding proof. Apart from the still unsolved problem of resolution in positive characteristics – which motivates a closer look for alternative proofs of the characteristic 0 case – there are other reasons for further studies: The resolution problem admits modifications, one of them due to A. J. de Jong [in: Resolution of singularities. A research textbook in tribute to Oscar Zariski. Prog. Math. 181, 375–380 (2000; Zbl 1022.14005)], which led to a solution of the general problem up to alterations. On the other hand, appearance of computers in the offices of most mathematicians during the last decade of the 20th century has led to increasing interest in algorithmic questions. Refined resolution algorithms have been developed [cf. O. Villamayor, in: Real analytic and algebraic geometry. Proc. int. conf. Trento 1992, 277–291 (1995; Zbl 0930.14039)], and there is a continued interest in better understanding the ideas of Hironaka’s original proof [cf. H. Hauser, Bull. Am. Math. Soc., New Ser. 40, No.3, 323–403 (2003; Zbl 1030.14007)]. Several more recent proofs include results of E. Bierstone and P. D. Milman [Invent. Math. 128, No.2, 207–302 (1997; Zbl 0896.14006)], S. Encinas and O. Villamayor [in: Resolution of singularities. A research textbook in tribute to Oscar Zariski. Prog. Math. 181, 147–227 (2000; Zbl 0969.14007) and Rev. Mat. Iberoam. 19, No.2, 339–353 (2003; Zbl 1073.14021)], T. T. Moh [Commun. Algebra 20, No.11, 3207–3249 (1992; Zbl 0784.14008)], O. Villamayor [Ann. Sci. Éc. Norm. Supér., IV. Sér. 22, No.1, 1–32 (1989; Zbl 0675.14003), Ann. Sci. Éc. Norm. Supér., IV. Sér. 25, No.6, 629–677 (1992; Zbl 0782.14009)], J. Wlodarczyk [J. Am. Math. Soc. 18, No.4, 779–822 (2005; Zbl 1084.14018)]. It should be noted, that there exist first approaches to use computer-algebra systems for performing the resolution process of given singularities (cf. the one by G. Bodnar, J. Schicho [J. Symb. Comput. 30, No.4, 401–428 (2000; Zbl 1011.14005)] using Maple and another one by A. Frühbis-Krüger, G. Pfister [Mitt. Dtsch. Math.-Ver. 13, No.2, 98–105 (2005; Zbl 1084.14036)] for Singular, respectively).
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.

MSC:

14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14B05 Singularities in algebraic geometry
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14J17 Singularities of surfaces or higher-dimensional varieties

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