Local monomialization of transcendental extensions

Dale CUTKOSKY, Steven

doi:10.5802/aif.2132

Local monomialization of transcendental extensions
[Monomialisations locales des extensions transcendantes]

Dale CUTKOSKY, Steven ¹

¹ University of Missouri, department of mathematics, Columbia, MO 65211 (USA)

Annales de l'Institut Fourier, Tome 55 (2005) no. 5, pp. 1517-1586.

Résumé
Abstract

Soient $R \subset S$ deux anneaux locaux réguliers, essentiellement de type fini sur un corps $k$ de caractéristique zéro. Si $V$ est un anneau de valuation du corps des fractions $K$ de $S$ dominant $S$ , nous montrons qu’il existe des suites de transformés monoidaux (éclatements d’idéaux premiers réguliers) $R \to R_{1}$ et $S \to S_{1}$ le long de $V$ tels que $R_{1} \to S_{1}$ est une application monomiale. Il s’ensuit qu’un morphisme de variétés non singulières peut-être rendu monomial le long d’une valuation après éclatement de sous-variétés non singulières.

Suppose that $R \subset S$ are regular local rings which are essentially of finite type over a field $k$ of characteristic zero. If $V$ is a valuation ring of the quotient field $K$ of $S$ which dominates $S$ , then we show that there are sequences of monoidal transforms (blow ups of regular primes) $R \to R_{1}$ and $S \to S_{1}$ along $V$ such that $R_{1} \to S_{1}$ is a monomial mapping. It follows that a morphism of nonsingular varieties can be made to be a monomial mapping along a valuation, after blow ups of nonsingular subvarieties.

MR Zbl

DOI : 10.5802/aif.2132

Classification : 14E, 13A, 13B
Keywords: Monomialization, monoidal transform, valuation ring, Morphism
Mot clés : monomialisation, transformés monoidaux, anneaux de valuation, morphisme

Affiliations des auteurs :

Dale CUTKOSKY, Steven ¹

¹ University of Missouri, department of mathematics, Columbia, MO 65211 (USA)

@article{AIF_2005__55_5_1517_0,
     author = {Dale CUTKOSKY, Steven},
     title = {Local monomialization of transcendental extensions},
     journal = {Annales de l'Institut Fourier},
     pages = {1517--1586},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {55},
     number = {5},
     year = {2005},
     doi = {10.5802/aif.2132},
     zbl = {1081.14020},
     mrnumber = {2172273},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2132/}
}

TY  - JOUR
AU  - Dale CUTKOSKY, Steven
TI  - Local monomialization of transcendental extensions
JO  - Annales de l'Institut Fourier
PY  - 2005
SP  - 1517
EP  - 1586
VL  - 55
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2132/
DO  - 10.5802/aif.2132
LA  - en
ID  - AIF_2005__55_5_1517_0
ER  -

%0 Journal Article
%A Dale CUTKOSKY, Steven
%T Local monomialization of transcendental extensions
%J Annales de l'Institut Fourier
%D 2005
%P 1517-1586
%V 55
%N 5
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2132/
%R 10.5802/aif.2132
%G en
%F AIF_2005__55_5_1517_0

Dale CUTKOSKY, Steven. Local monomialization of transcendental extensions. Annales de l'Institut Fourier, Tome 55 (2005) no. 5, pp. 1517-1586. doi : 10.5802/aif.2132. https://aif.centre-mersenne.org/articles/10.5802/aif.2132/

Bibliographie
Cité par

[1] S. Abhyankar Local uniformization on algebraic surfaces over ground fields of characteristic $p \neq 0$ , Annals of Math., Volume 63 (1956), pp. 491-526 | DOI | MR | Zbl

[2] S. Abhyankar On the valuations centered in a local domain, Amer. J. Math., Volume 78 (1956), pp. 321-348 | DOI | MR | Zbl

[3] S. Abhyankar Simultaneous resolution for algebraic surfaces, Amer. J. Math., Volume 78 (1956), pp. 761-790 | DOI | MR | Zbl

[4] S. Abhyankar Ramification theoretic methods in algebraic geometry, Princeton University Press (1959) | MR | Zbl

[5] S. Abhyankar Resolution of singularities of embedded algebraic surfaces, second edition, Springer, 1998 | MR | Zbl

[6] S. Abhyankar Resolution of singularities and modular Galois theory, Bulletin of the AMS, Volume 38 (2001), pp. 131-171 | MR | Zbl

[7] S. Abhyankar On the ramification of algebraic functions, American J. Math., Volume 77 (1955), pp. 575-592 | DOI | MR | Zbl

[8] S. Abhyankar Algebraic Geometry for Scientists and Engineers, American Mathematical Society (1990) | MR | Zbl

[9] D. Abramovich; K. Karu; K. Matsuki; J. Wlodarczyk Torification and factorization of birational maps, Journal of the American Mathematical Society, Volume 15 (2002), pp. 351-572 | MR | Zbl

[10] E. Bierstone; P. Milman Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math., Volume 128 (1997), pp. 207-302 | DOI | MR | Zbl

[11] C. Christensen Strong domination, weak factorization or three dimensional regular local local rings, Journal of the Indian Math. Soc., Volume 45 (1981), pp. 21-47 | MR | Zbl

[12] V. Cossart Polyèdre caractéristique d'une singularité (1987) (Thèse, Université de Paris-Sud, Centre d'Orsay)

[13] S. D. Cutkosky Local Factorization of Birational Maps, Advances in Math., Volume 132 (1997), pp. 167-315 | DOI | MR | Zbl

[14] S. D. Cutkosky Local Monomialization and Factorization of Morphisms, Astérisque, Volume 260 (1999) | MR | Zbl

[15] S. D. Cutkosky Simultaneous resolution of singularities, Proc. American Math. Soc., Volume 128 (2000), pp. 1905-1910 | DOI | MR | Zbl

[16] S. D. Cutkosky Ramification of valuations and singularities (to appear in Contemporary Math.) | MR | Zbl

[17] S. D. Cutkosky Monomialization of morphisms from 3-folds to surfaces, LNM 1786, Springer-Verlag, 2002 | MR | Zbl

[18] S. D. Cutkosky; J. Herzog; A. Reguera Poincaré series of resolutions of surface singularities, Transactions AMS, Volume 356 (2003), pp. 1833-1874 | MR | Zbl

[19] S. D. Cutkosky; O. Kascheyeva Monomialization of strongly prepared morphisms from nonsingular $n$ -folds to surfaces, J. Algebra, Volume 275 (2004), pp. 275-320 | DOI | MR | Zbl

[20] S. D. Cutkosky; O. Piltant Monomial resolutions of morphisms of algebraic surfaces, Communications in Algebra, Volume 28 (2000), pp. 5935-5959 | DOI | MR | Zbl

[21] S. D. Cutkosky; O. Piltant Ramification of valuations, Advances in Mathematics, Volume 183 (2004), pp. 1-79 | DOI | MR | Zbl

[22] S. D. Cutkosky; L. Ghezzi Completions of valuation rings (to appear in Contemporary Mathematics)

[23] S. D. Cutkosky; H. Srinivasan Factorizations of matrices and birational maps (preprint)

[24] W. Fulton Introduction to Toric Varieties, Princeton University Press (1993) | MR | Zbl

[25] W. Heinzer; C. Rotthaus; S. Wiegand Approximating discrete valuation rings by regular local rings, a remark on local uniformization, Proc. Amer. Math. Soc., Volume 129 (2001), pp. 37-43 | DOI | MR | Zbl

[26] H. Hironaka Resolution of singularities of an algebraic variety over a field of characteristic zero, Annals of Math., Volume 79 (1964), pp. 109-326 | DOI | MR | Zbl

[27] H. Hironaka Desingularization of excellent surfaces, Advanced Science Seminar in Algebraic Geometry, Bowdoin College, Brunswick, Maine, 1967

[28] K. Karu Local strong factorization of birational maps, J. Alg. Geom., Volume 14 (2005), pp. 165-175 | DOI | MR | Zbl

[29] F.V. Kuhlmann Valuation theoretic and model theoretic aspects of local uniformization, in Resolution of Singularities, Springer-Verlag, 2000 | MR | Zbl

[30] J. Lipman Desingularization of two-dimensional schemes, Ann. Math., Volume 107 (1978), pp. 151-207 | DOI | MR | Zbl

[31] H. Matsumura Commutative Ring Theory, Cambridge studies in advanced mathematics, 8, Cambridge University Press, Cambridge, 1986 | MR | Zbl

[32] TT. Moh Quasi-Canonical uniformization of hypersurface singularities of characteristic zero, Comm. Algebra, Volume 20 (1992), pp. 3207-3249 | DOI | MR | Zbl

[33] M. Spivakovsky Valuations in function fields of surfaces, Amer. J. Math., Volume 112 (1990), pp. 107-156 | DOI | MR | Zbl

[34] B. Teissier; Franz-Viktor Kuhlmann, Salma Kuhlmann and Murray Marshall Valuations, Deformations and Toric Geometry, Valuation Theory and its Applications II (Fields Institute Communications), Volume 33 (1990), pp. 441-491 | Zbl

[35] O. Villamayor; S. Encinas A course on constructive desingularization and equivariance, in Resolution of Singularities, Springer-Verlag, 2000 | Zbl

[36] O. Zariski The reduction of the singularities of an algebraic surface, Annals of Math., Volume 40 (1939), pp. 639-689 | DOI | JFM | MR | Zbl

[37] O. Zariski Local uniformization of algebraic varieties, Annals of Math., Volume 41 (1940), pp. 852-896 | DOI | MR | Zbl

[38] O. Zariski Reduction of the singularities of algebraic three dimensional varieties, Annals of Math., Volume 45 (1944), pp. 472-542 | DOI | MR | Zbl

[39] O. Zariski; P. Samuel Commutative Algebra II, Van Nostrand, Princeton, 1960 | MR | Zbl

Cité par Sources :

ANNALES DE L'INSTITUT FOURIER

ARTICLES À PARAÎTRE

FEUILLETER

PRESENTATION

COMITÉ DE RÉDACTION

SOUMETTRE UN ARTICLE

ABONNEMENT