Book Review
The AMS does not provide abstracts of book reviews. You may download the entire review from the links below.
MathSciNet review: 2731655
Full text of review: PDF This review is available free of charge.
Book Information:
Author: Steven Dale Cutkosky
Title: Resolution of singularities
Additional book information: Graduate Studies in Mathematics, 63, American Mathematical Society, Providence, RI, 2004, viii+186 pp., ISBN 978-0-8218-3555-5, US$41.00
Author: János Kollár
Title: Lectures on resolution of singularities
Additional book information: Annals of Mathematics Studies, 166, Princeton University Press, Princeton, New Jersey, 2007, vi+208 pp., ISBN 978-0-691-12923-5, US$42.00
- Shreeram Shankar Abhyankar, Resolution of singularities of embedded algebraic surfaces, Pure and Applied Mathematics, Vol. 24, Academic Press, New York-London, 1966. MR 0217069
- D. Abramovich and A. J. de Jong, Smoothness, semistability, and toroidal geometry, J. Algebraic Geom. 6 (1997), no. 4, 789–801. MR 1487237
- D. Abramovich and K. Karu, Weak semistable reduction in characteristic 0, Invent. Math. 139 (2000), no. 2, 241–273. MR 1738451, DOI https://doi.org/10.1007/s002229900024
- 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, 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
- Fedor A. Bogomolov and Tony G. Pantev, Weak Hironaka theorem, Math. Res. Lett. 3 (1996), no. 3, 299–307. MR 1397679, DOI https://doi.org/10.4310/MRL.1996.v3.n3.a1
- 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
- Vincent Cossart, Uwe Jannsen, and Shuji Saito. Canonical embedded and non-embedded resolution of singularities for excellent two-dimensional schemes, 2009. preprint arXiv:0905.2191.
- Vincent Cossart and Olivier Piltant, Resolution of singularities of threefolds in positive characteristic. I. Reduction to local uniformization on Artin-Schreier and purely inseparable coverings, J. Algebra 320 (2008), no. 3, 1051–1082. MR 2427629, DOI https://doi.org/10.1016/j.jalgebra.2008.03.032
- Vincent Cossart and Olivier Piltant, Resolution of singularities of threefolds in positive characteristic. II, J. Algebra 321 (2009), no. 7, 1836–1976. MR 2494751, DOI https://doi.org/10.1016/j.jalgebra.2008.11.030
- Steven Dale Cutkosky, Resolution of singularities, Graduate Studies in Mathematics, vol. 63, American Mathematical Society, Providence, RI, 2004. MR 2058431
- Steven Dale Cutkosky, The structure of morphisms, Algebra, arithmetic and geometry with applications (West Lafayette, IN, 2000) Springer, Berlin, 2004, pp. 273–281. MR 2037094
- Steven Dale Cutkosky, Resolution of singularities for 3-folds in positive characteristic, Amer. J. Math. 131 (2009), no. 1, 59–127. MR 2488485, DOI https://doi.org/10.1353/ajm.0.0036
- 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
- Jean Giraud, Sur la théorie du contact maximal, Math. Z. 137 (1974), 285–310 (French). MR 460712, DOI https://doi.org/10.1007/BF01214371
- A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Inst. Hautes Études Sci. Publ. Math. 24 (1965), 231 (French). MR 199181
- A. Grothendieck, Travaux de Heisouké Hironaka sur la résolution des singularités, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 7–9 (French). MR 0414283
- Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
- 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
- Herwig Hauser, Three power series techniques, Proc. London Math. Soc. (3) 89 (2004), no. 1, 1–24. MR 2063657, DOI https://doi.org/10.1112/S0024611503014588
- A. J. de Jong, Smoothness, semi-stability and alterations, Inst. Hautes Études Sci. Publ. Math. 83 (1996), 51–93. MR 1423020
- János Kollár, Lectures on resolution of singularities, Annals of Mathematics Studies, vol. 166, Princeton University Press, Princeton, NJ, 2007. MR 2289519
- János Kollár and Shigefumi Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original. MR 1658959
- 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
- 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
- Mircea Mustaţă. Lectures on resolution of singularities, 2010. IAS/Park City Mathematics Series, to appear.
- Kapil H. Paranjape, The Bogomolov-Pantev resolution, an expository account, New trends in algebraic geometry (Warwick, 1996) London Math. Soc. Lecture Note Ser., vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 347–358. MR 1714830, DOI https://doi.org/10.1017/CBO9780511721540.014
- Michael Temkin, Desingularization of quasi-excellent schemes in characteristic zero, Adv. Math. 219 (2008), no. 2, 488–522. MR 2435647, DOI https://doi.org/10.1016/j.aim.2008.05.006
- 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
- 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
Review Information:
Reviewer: Dan Abramovich
Affiliation: Providence, Rhode Island, 02912
Email: abrmovic@math.brown.edu
Journal: Bull. Amer. Math. Soc. 48 (2011), 115-122
DOI: https://doi.org/10.1090/S0273-0979-10-01301-7
Published electronically: April 9, 2010
Review copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.