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Uniformization of analytic spaces. (English) Zbl 0685.32007
The authors give an explicit elementary proof of a local version of resolution of singularities in characteristic zero. “Local” means that the centres of blowing up are chosen locally, so that a finite number of finite sequences of local blowings-up may be required to cover a neighbourhood of a given point.
Those investigators who want to start the theory of (resolution of) singularities, and who want to find the next materials, shall look over this paper. I think that this paper feast their eyes upon.
Reviewer: S.Ohyanagi

MSC:
32S45 Modifications; resolution of singularities (complex-analytic aspects)
32S05 Local complex singularities
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14B05 Singularities in algebraic geometry
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[1] Shreeram S. Abhyankar, Weighted expansions for canonical desingularization, Lecture Notes in Mathematics, vol. 910, Springer-Verlag, Berlin-New York, 1982. With a foreword by U. Orbanz. · Zbl 0479.14009
[2] 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
[3] José M. Aroca, Heisuke Hironaka, and José L. Vicente, Desingularization theorems, Memorias de Matemática del Instituto ”Jorge Juan” [Mathematical Memoirs of the Jorge Juan Institute], vol. 30, Consejo Superior de Investigaciones Científicas, Madrid, 1977. · Zbl 0366.32009
[4] Bruce Michael Bennett, On the characteristic functions of a local ring, Ann. of Math. (2) 91 (1970), 25 – 87. · Zbl 0198.06101
[5] E. Bierstone and P. D. Milman, Relations among analytic functions. I, Ann. Inst. Fourier (Grenoble) 37 (1987), no. 1, 187 – 239 (English, with French summary). E. Bierstone and P. D. Milman, Relations among analytic functions. II, Ann. Inst. Fourier (Grenoble) 37 (1987), no. 2, 49 – 77 (English, with French summary). · Zbl 0611.32002
[6] Edward Bierstone and Pierre D. Milman, Ideals of holomorphic functions with \?^{\infty } boundary values on a pseudoconvex domain, Trans. Amer. Math. Soc. 304 (1987), no. 1, 323 – 342. · Zbl 0631.32015
[7] Edward Bierstone and Pierre D. Milman, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 5 – 42. · Zbl 0674.32002
[8] Heisuke Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109 – 203; ibid. (2) 79 (1964), 205 – 326. · Zbl 0122.38603
[9] Heisuke Hironaka, Certain numerical characters of singularities, J. Math. Kyoto Univ. 10 (1970), 151 – 187. · Zbl 0214.20003
[10] Heisuke Hironaka, Introduction to the theory of infinitely near singular points, Consejo Superior de Investigaciones Científicas, Madrid, 1974. Memorias de Matematica del Instituto ”Jorge Juan”, No. 28. · Zbl 0366.32007
[11] Heisuke Hironaka, Idealistic exponents of singularity, Algebraic geometry (J. J. Sylvester Sympos., Johns Hopkins Univ., Baltimore, Md., 1976) Johns Hopkins Univ. Press, Baltimore, Md., 1977, pp. 52 – 125. · Zbl 0496.14011
[12] Mark Spivakovsky, A solution to Hironaka’s polyhedra game, Arithmetic and geometry, Vol. II, Progr. Math., vol. 36, Birkhäuser Boston, Boston, MA, 1983, pp. 419 – 432. · Zbl 0531.14009
[13] Jean-Claude Tougeron, Idéaux de fonctions différentiables, Springer-Verlag, Berlin-New York, 1972. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 71. · Zbl 0188.45102
[14] Boris Youssin, Newton polyhedra without coordinates, Mem. Amer. Math. Soc. 87 (1990), no. 433, i – vi, 1 – 74. · Zbl 0709.14028
[15] Boris Youssin, Newton polyhedra of ideals, Mem. Amer. Math. Soc. 87 (1990), no. 433, i – vi, 75 – 99. · Zbl 0709.14029
[16] Oscar Zariski, Local uniformization on algebraic varieties, Ann. of Math. (2) 41 (1940), 852 – 896. · Zbl 0025.21601
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