Bulletin of the American Mathematical Society

Author(s): Herwig Hauser
Journal: Bull. Amer. Math. Soc. 40 (2003), 323-403.
MSC (2000): Primary 14B05, 14E15, 32S05, 32S10, 32S45
Posted: May 6, 2003
Retrieve article in: PDF

Abstract: This paper is a handyman's manual for learning how to resolve the singularities of algebraic varieties defined over a field of characteristic zero by sequences of blowups.

References directly related to the Hironaka Theorem

Citations of the text not included in this list appear in the second list below.

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Aroca, J.-M., Hironaka, H., Vicente, J.-L.: The theory of the maximal contact. Memorias Mat. Inst. Jorge Juan, Madrid 29 (1975). MR 56:3344

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Aroca, J.-M., Hironaka, H., Vicente, J.-L.: Desingularization theorems. Memorias Mat. Inst. Jorge Juan, Madrid 29 (1975). MR 80h:32027

[BEV]

Bravo, A., Encinas, S., Villamayor, O.: A simplified proof of desingularization and applications. Preprint. Download at http://arXiv.org/archive/math, math.AG/0206244.

[BM1]

Bierstone, E., Milman, P.: Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant. Invent. Math. 128 (1997), 207-302. MR 98e:14010

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Bierstone, E., Milman, P.: A simple constructive proof of canonical resolution of singularities. In: Effective Methods in Algebraic Geometry (eds. T. Mora, C. Traverso). Progress in Math. 94, Birkhäuser, Boston, 1991, 11-30. MR 92h:32053

[BM4]

Bierstone, E., Milman, P.: Semianalytic and subanalytic sets. Publ. Math. IHES 67 (1988), 5-42. MR 89k:32011

[BM5]

Bierstone, E., Milman, P.: Resolution of singularities. In: Several complex variables (Berkeley, 1995-1996), 43-78. Math. Sci. Res. Inst. Publ. 37, Cambridge Univ. Press, Cambridge, 1999. MR 2001f:32052

[BM6]

Bierstone, E., Milman, P.: Desingularization algorithms I. The role of exceptional divisors. Preprint. Download at http://arXiv.org/archive/math, math.AG/0207098.

[Bo1]

Bodnár, G.: Algorithmic resolution of singularities. Thesis, Univ. Linz, 2001.

[Bo2]

Bodnár, G.: Computation of blowing up centers. J. Pure Appl. Alg. 179 (2003), 221-233.

[Bo3]

Bodnár, G.: Algorithmic tests for the normal crossings property. Preprint, Univ. Linz.

[BS1]

Bodnár, G., Schicho, J.: Automated Resolution of Singularities for Hypersurfaces. J. Symb. Comp. 30 (2000), 401-428. MR 2001i:14083

[BS 2]

Bodnár, G., Schicho, J.: Two computational techniques for singularity resolution. J. Symbolic Comput. 32 (2001), no. 1-2, 39-54. MR 2002d:14020

[BS 3]

Bodnár, G., Schicho, J.: A computer program for the resolution of singularities. In: Resolution of singularities (ed. H. Hauser at al.), 231-238, Progr. in Math. 181, Birkhäuser, Basel, 2000. MR 2001e:14001

[BV 1]

Bravo, A., Villamayor, O.: Strengthening the theorem of embedded desingularization. Math. Res. Lett. 8 (2001), no. 1-2, 79-89. MR 2002b:14019

[BV 2]

Bravo, A., Villamayor, O.: A strengthening of resolution of singularities in characteristic zero. Proc. London Math. Soc. 86 (2003), 327-357.

[EH]

Encinas, S., Hauser, H.: Strong resolution of singularities in characteristic zero. Comment. Math. Helv. 77 (2002), 821-845.

[EV 1]

Encinas, S., Villamayor, O.: Good points and constructive resolution of singularities. Acta Math. 181 (1998), 109-158. MR 99i:14020

[EV 2]

Encinas, S., Villamayor, O.: A course on constructive desingularization and equivariance. In: Resolution of Singularities (ed.: H. Hauser et al.), Progress in Math. 181, Birkhäuser, Basel, 2000. MR 2001g:14018

[EV 3]

Encinas, S., Villamayor, O.: A proof of desingularization over fields of characteristic zero. Preprint. Download at http://arXiv.org/archive/math, math.AG/0101208.

[Gr 2]

Grothendieck, A.: Traveaux de Heisouké Hironaka sur la résolution des singularités. Actes Congrès International Math., Nice, 1970, Gauthier-Villars, Paris, 1971. MR 54:2386

[Hi 1]

Hironaka, H.: Resolution of singularities of an algebraic variety over a field of characteristic zero. Ann. Math. 79 (1964), 109-326. MR 33:7333

[Hi 6]

Hironaka, H.: Idealistic exponents of singularity. In: Algebraic Geometry, the Johns Hopkins Centennial Lectures. Johns Hopkins University Press, 1977. MR 58:16661

[Hi 8]

Hironaka, H.: Desingularization of complex analytic varieties. Actes Congrès International Math., Nice, 1970, Gauthier-Villars, Paris, 1971, 627-631. MR 54:13127

[Kz]

Kunz, E.: Algebraische Geometrie IV. Vorlesung Univ. Regensburg.

[Lp 3]

Lipman, J.: Introduction to resolution of singularities. Proceedings Symp. Pure Appl. Math. 29 Amer. Math. Soc., 1975, 187-230. MR 52:10730

[M]

Matsuki, K.: Notes on the inductive algorithm of resolution of singularities by S. Encinas and O. Villamayor. Preprint. Download at http://arXiv.org/archive/math, math.AG/0103120.

[Vi 1]

Villamayor, O.: Constructiveness of Hironaka's resolution. Ann. Scient. Ec. Norm. Sup. Paris 22 (1989), 1-32. MR 90b:14014

[Vi 2]

Villamayor, O.: Patching local uniformizations. Ann. Scient. Ec. Norm. Sup. Paris 25 (1992), 629-677.

[Vi 3]

Villamayor, O.: An introduction to the algorithm of resolution. In: Algebraic Geometry and Singularities (eds. A. Campillo, L. Narváez). Proc. Conf. on Singularities La Rábida. Birkhäuser, 1996.

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