Book Review
The AMS does not provide abstracts of book reviews. You may download the entire review from the links below.
Full text of review: PDF This review is available free of charge.
Book Information:
Authors: K. Kiyek and J. L. Vicente
Title: Resolution of curve and surface singularities in characteristic zero
Additional book information: Kluwer Academic Publishers, Dordrecht/Boston/London, 2004, xxii + 483 pp., ISBN 1-4020-2028-7, US$129.00$, hardcover; ISBN 1-4020-2029-5, e-book
- Shreeram Abhyankar, Local uniformization on algebraic surfaces over ground fields of characteristic $p\ne 0$, Ann. of Math. (2) 63 (1956), 491–526. MR 78017, DOI https://doi.org/10.2307/1970014
- Shreeram S. Abhyankar, Resolution of singularities of arithmetical surfaces, Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963) Harper & Row, New York, 1965, pp. 111–152. MR 0200272
- S. S. Abhyankar, Resolution of singularities of embedded algebraic surfaces, 2nd ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR 1617523
- Shreeram S. Abhyankar, Good points of a hypersurface, Adv. in Math. 68 (1988), no. 2, 87–256. MR 934366, DOI https://doi.org/10.1016/0001-8708%2888%2990015-1
- 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]. MR 0444999
- 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]. MR 0444999
- Edward Bierstone and Pierre D. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. 128 (1997), no. 2, 207–302. MR 1440306, DOI https://doi.org/10.1007/s002220050141
- Edward Bierstone and Pierre D. Milman, Uniformization of analytic spaces, J. Amer. Math. Soc. 2 (1989), no. 4, 801–836. MR 1001853, DOI https://doi.org/10.1090/S0894-0347-1989-1001853-2
- Edward Bierstone and Pierre D. Milman, A simple constructive proof of canonical resolution of singularities, Effective methods in algebraic geometry (Castiglioncello, 1990) Progr. Math., vol. 94, Birkhäuser Boston, Boston, MA, 1991, pp. 11–30. MR 1106412, DOI https://doi.org/10.1007/978-1-4612-0441-1_2
- Bruce Michael Bennett, On the characteristic functions of a local ring, Ann. of Math. (2) 91 (1970), 25–87. MR 252388, DOI https://doi.org/10.2307/1970601 Bodnár, G., Schicho, J.: A computer program for the resolution of singularities. In: Resolution of Singularities (eds. H. Hauser et al.), Progr. in Math. 181, Birkhäuser, 2000, 231-238.
- Ana María Bravo, Santiago Encinas, and Orlando Villamayor U., A simplified proof of desingularization and applications, Rev. Mat. Iberoamericana 21 (2005), no. 2, 349–458. MR 2174912, DOI https://doi.org/10.4171/RMI/425
- A. Bravo and O. Villamayor U., A strengthening of resolution of singularities in characteristic zero, Proc. London Math. Soc. (3) 86 (2003), no. 2, 327–357. MR 1971154, DOI https://doi.org/10.1112/S0024611502013801
- Steven Dale Cutkosky, Resolution of singularities, Graduate Studies in Mathematics, vol. 63, American Mathematical Society, Providence, RI, 2004. MR 2058431
- A. J. de Jong, Smoothness, semi-stability and alterations, Inst. Hautes Études Sci. Publ. Math. 83 (1996), 51–93. MR 1423020
- Santiago Encinas and Herwig Hauser, Strong resolution of singularities in characteristic zero, Comment. Math. Helv. 77 (2002), no. 4, 821–845. MR 1949115, DOI https://doi.org/10.1007/PL00012443
- S. Encinas and O. Villamayor, Good points and constructive resolution of singularities, Acta Math. 181 (1998), no. 1, 109–158. MR 1654779, DOI https://doi.org/10.1007/BF02392749
- Santiago Encinas and Orlando Villamayor, A course on constructive desingularization and equivariance, Resolution of singularities (Obergurgl, 1997) Progr. Math., vol. 181, Birkhäuser, Basel, 2000, pp. 147–227. MR 1748620
- Santiago Encinas and Orlando Villamayor, A new proof of desingularization over fields of characteristic zero, Proceedings of the International Conference on Algebraic Geometry and Singularities (Spanish) (Sevilla, 2001), 2003, pp. 339–353. MR 2023188, DOI https://doi.org/10.4171/RMI/350
- Herwig Hauser, Seventeen obstacles for resolution of singularities, Singularities (Oberwolfach, 1996) Progr. Math., vol. 162, Birkhäuser, Basel, 1998, pp. 289–313. MR 1652479
- Herwig Hauser, The Hironaka theorem on resolution of singularities (or: A proof we always wanted to understand), Bull. Amer. Math. Soc. (N.S.) 40 (2003), no. 3, 323–403. MR 1978567, DOI https://doi.org/10.1090/S0273-0979-03-00982-0
- 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. MR 0199184, DOI https://doi.org/10.2307/1970547
- Vincent Cossart, Jean Giraud, and Ulrich Orbanz, Resolution of surface singularities, Lecture Notes in Mathematics, vol. 1101, Springer-Verlag, Berlin, 1984. With an appendix by H. Hironaka. MR 775681
- 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. MR 0498562 Jung, H.: Darstellung der Funktionen eines algebraischen Körpers zweier unabhängiger Veränderlicher
- Joseph Lipman, Desingularization of two-dimensional schemes, Ann. of Math. (2) 107 (1978), no. 1, 151–207. MR 491722, DOI https://doi.org/10.2307/1971141
- Joseph Lipman, Introduction to resolution of singularities, Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974) Amer. Math. Soc., Providence, R.I., 1975, pp. 187–230. MR 0389901
- Orlando Villamayor, Constructiveness of Hironaka’s resolution, Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 1, 1–32. MR 985852, DOI https://doi.org/10.24033/asens.1573
- O. E. Villamayor U., Patching local uniformizations, Ann. Sci. École Norm. Sup. (4) 25 (1992), no. 6, 629–677. MR 1198092 Villamayor, O.: An introduction to constructive desingularization. ArXiv: math.AG/ 0507537.
- Jarosław Włodarczyk, Simple Hironaka resolution in characteristic zero, J. Amer. Math. Soc. 18 (2005), no. 4, 779–822. MR 2163383, DOI https://doi.org/10.1090/S0894-0347-05-00493-5
- Oscar Zariski, The reduction of the singularities of an algebraic surface, Ann. of Math. (2) 40 (1939), 639–689. MR 159, DOI https://doi.org/10.2307/1968949
- Oscar Zariski, Reduction of the singularities of algebraic three dimensional varieties, Ann. of Math. (2) 45 (1944), 472–542. MR 11006, DOI https://doi.org/10.2307/1969189




Review Information:
Reviewer: Ana Bravo
Affiliation: Universidad Autonóma de Madrid
Email: ana.bravo@uam.es
Reviewer: Herwig Hauser
Affiliation: Institut für Mathematik, Universität Innsbruck
Email: herwig.hauser@uibk.ac.at
Journal: Bull. Amer. Math. Soc. 43 (2006), 241-247
Published electronically: February 15, 2006
Review copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.