Abstract
In these notes, we describe some of the main features of an explicit proof of canonical desingularization (of algebraic varieties or analytic spaces X) in characteristic zero. Full details will appear in [7]. The proof is a variation on our proof of local desingularization (“uniformization”) [4], [5], and justifies the philosophy that “a sufficiently good local choice [of centre of blowing-up] should globalize automatically” [5, p. 901]. The final version is surprisingly elementary; these notes, for example, include an essentially self-contained presentation of the hypersurface case. The general case involves a “reduction to the hypersurface case” result from [5].
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
S.S. Abhyankar, “Weighted expansions for canonical desingularization,” Lecture Notes in Math. No. 910, Springer, Berlin-Heidelberg-New York, 1982.
S.S. Abhyankar, Good points of a hypersurface, Adv. in Math. 68 (1988), 87–256.
J.M. Aroca, H. Hironaka and J.L. Vicente, Desingularization theorems, Mem. Math. Inst. Jorge Juan No. 30, Consejo Superior de Investigaciones Científicas, Madrid, 1977.
E. Bierstone and P.D. Milman, Semianalytic and subanalytic sets, Publ. Math. I.H.E.S. 67 (1988), 5–42.
E. Bierstone and P.D. Milman, Uniformization of analytic spaces, J. Amer. Math. Soc. 2 (1989), 801–836.
E. Bierstone and P.D. Milman, Arc-analytic functions, Invent. Math. 101(1990), 411–424.
E. Bierstone and P.D. Milman, Canonical desingularization in characteristic zero: a simple constructive proof, (to appear).
H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero: I, II, Ann. of Math. (2) 79 (1964), 109–326.
H. Hironaka, Idealistic exponents of singularity, in “Algebraic Geometry,” J.J. Sylvester Sympos., John Hopkins Univ., Baltimore, Md.,1976, John Hopkins Univ. Press, Baltimore, Md., 1977, pp. 52–125.
T.T. Moh, Canonical resolution of hypersurface singularities of characteristic zero, preprint, Purdue University, 1990.
M. Spivakovsky, A solution to Hironaka’s polyhedra game, “Arithmetic and geometry, Vol. II,” Prog. Math. No. 36, Birkhäuser, Boston, Mass., 1983, pp. 419–432.
O. Villamayor, Constructiveness of Hironaka’s resolution, Ann. Scient. Ecole Norm. Sup. (4e série) 22 (1989), 1–32.
B. Youssin, Newton polyhedra without coordinates, Mem. Amer. Math. Soc. 433 (1990), 1–74.
B. Youssin, Newton polyhedra of ideals, Mem. Amer. Math. Soc. 433 (1990), 75–99.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1991 Springer Science+Business Media New York
About this chapter
Cite this chapter
Bierstone, E., Milman, P.D. (1991). A simple constructive proof of Canonical Resolution of Singularities. In: Mora, T., Traverso, C. (eds) Effective Methods in Algebraic Geometry. Progress in Mathematics, vol 94. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0441-1_2
Download citation
DOI: https://doi.org/10.1007/978-1-4612-0441-1_2
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-6761-4
Online ISBN: 978-1-4612-0441-1
eBook Packages: Springer Book Archive