Hauser, Herwig The Hironaka theorem on resolution of singularities (or: A proof we always wanted to understand). (English) Zbl 1030.14007 Bull. Am. Math. Soc., New Ser. 40, No. 3, 323-403 (2003). The resolution of singularities of algebraic varieties defined over a field of characteristic zero by a sequence of blow-ups is a famous result proved by H. Hironaka [Ann. Math. (2) 79, 109-203, 205-326 (1964; Zbl 0122.38603)], which has a lot of applications. Probably, relatively few mathematicians read the 200 page proof. This can change now. The article is arranged in a similar way to a talk in a colloquium (25% should be understood by everyone, 25% is for people who are interested, the next 25% is for the specialists and the rest only the speaker will understand) with one difference: that also the rest is understandable. It starts with an overview explaining the result and giving rough ideas of the proof. The author suggests that very busy people should only read this. The next chapter gives an introduction to the main problems (choice of the centre of the blow-up, equiconstant points, improvement of singularities under blow-up including many examples). This is for the next 25. The next chapter (constructions and proofs) gives the technical details. It contains also several examples and a section on problems in positive characteristic. In an appendix, necessary basic facts from commutative algebra and the theory of blow-ups used in the previous chapters are collected. It is advisable for everybody interested in resolution of singularities to read this article. Reviewer: Gerhard Pfister (Kaiserslautern) Cited in 3 ReviewsCited in 43 Documents MSC: 14E15 Global theory and resolution of singularities (algebro-geometric aspects) 14B05 Singularities in algebraic geometry 32S05 Local complex singularities 32S10 Invariants of analytic local rings 32S45 Modifications; resolution of singularities (complex-analytic aspects) Keywords:Hironaka resolution of singularities; blowing up Citations:Zbl 0122.38603 Software:desing PDFBibTeX XMLCite \textit{H. Hauser}, Bull. Am. Math. Soc., New Ser. 40, No. 3, 323--403 (2003; Zbl 1030.14007) Full Text: DOI References: [1] Jose M. Aroca, Heisuke Hironaka, and José L. Vicente, The theory of the maximal contact, Instituto ”Jorge Juan” de Matemáticas, Consejo Superior de Investigaciones Cientificas, Madrid, 1975. Memorias de Matemática del Instituto ”Jorge Juan”, No. 29. [Mathematical Memoirs of the ”Jorge Juan” Institute, No. 29]. · Zbl 0366.32008 [2] José M. Aroca, Heisuke Hironaka, and José L. 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