Local monomialization of transcendental extensions

Steven Dale CUTKOSKY[1]

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

Annales de l’institut Fourier (2005)

  • Volume: 55, Issue: 5, page 1517-1586
  • ISSN: 0373-0956

Abstract

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Suppose that R ⊂ 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 → R 1 and S → S 1 along V such that R 1 → 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.

How to cite

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Dale CUTKOSKY, Steven. "Local monomialization of transcendental extensions." Annales de l’institut Fourier 55.5 (2005): 1517-1586. <http://eudml.org/doc/116225>.

@article{DaleCUTKOSKY2005,
abstract = {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) $R\rightarrow R_1$ and $S\rightarrow S_1$ along $V$ such that $R_1\rightarrow S_1$ 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.},
affiliation = {University of Missouri, department of mathematics, Columbia, MO 65211 (USA)},
author = {Dale CUTKOSKY, Steven},
journal = {Annales de l’institut Fourier},
keywords = {Monomialization; monoidal transform; valuation ring; Morphism; resolution of singularities; local uniformization theorem; toroidalization},
language = {eng},
number = {5},
pages = {1517-1586},
publisher = {Association des Annales de l'Institut Fourier},
title = {Local monomialization of transcendental extensions},
url = {http://eudml.org/doc/116225},
volume = {55},
year = {2005},
}

TY - JOUR
AU - Dale CUTKOSKY, Steven
TI - Local monomialization of transcendental extensions
JO - Annales de l’institut Fourier
PY - 2005
PB - Association des Annales de l'Institut Fourier
VL - 55
IS - 5
SP - 1517
EP - 1586
AB - 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) $R\rightarrow R_1$ and $S\rightarrow S_1$ along $V$ such that $R_1\rightarrow S_1$ 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.
LA - eng
KW - Monomialization; monoidal transform; valuation ring; Morphism; resolution of singularities; local uniformization theorem; toroidalization
UR - http://eudml.org/doc/116225
ER -

References

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  1. S. Abhyankar, Local uniformization on algebraic surfaces over ground fields of characteristic p ≠ 0 , Annals of Math. 63 (1956), 491-526 Zbl0108.16803MR78017
  2. S. Abhyankar, On the valuations centered in a local domain, Amer. J. Math. 78 (1956), 321-348 Zbl0074.26301MR82477
  3. S. Abhyankar, Simultaneous resolution for algebraic surfaces, Amer. J. Math. 78 (1956), 761-790 Zbl0073.37902MR82722
  4. S. Abhyankar, Ramification theoretic methods in algebraic geometry, Princeton University Press (1959) Zbl0101.38201MR105416
  5. S. Abhyankar, Resolution of singularities of embedded algebraic surfaces, second edition, (1998), Springer Zbl0914.14006MR1617523
  6. S. Abhyankar, Resolution of singularities and modular Galois theory, Bulletin of the AMS 38 (2001), 131-171 Zbl0999.12003MR1816069
  7. S. Abhyankar, On the ramification of algebraic functions, American J. Math. 77 (1955), 575-592 Zbl0064.27501MR71851
  8. S. Abhyankar, Algebraic Geometry for Scientists and Engineers, American Mathematical Society (1990) Zbl0709.14001MR1075991
  9. D. Abramovich, K. Karu, K. Matsuki, J. Wlodarczyk, Torification and factorization of birational maps, Journal of the American Mathematical Society 15 (2002), 351-572 Zbl1032.14003MR1896232
  10. E. Bierstone, P. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. 128 (1997), 207-302 Zbl0896.14006MR1440306
  11. C. Christensen, Strong domination, weak factorization or three dimensional regular local local rings, Journal of the Indian Math. Soc. 45 (1981), 21-47 Zbl0643.13007MR828858
  12. V. Cossart, Polyèdre caractéristique d'une singularité, (1987) Zbl0333.32008
  13. S. D. Cutkosky, Local Factorization of Birational Maps, Advances in Math. 132 (1997), 167-315 Zbl0934.14006MR1491444
  14. S. D. Cutkosky, Local Monomialization and Factorization of Morphisms, Astérisque 260 (1999) Zbl0941.14001MR1734239
  15. S. D. Cutkosky, Simultaneous resolution of singularities, Proc. American Math. Soc. 128 (2000), 1905-1910 Zbl0974.14010MR1646312
  16. S. D. Cutkosky, Ramification of valuations and singularities Zbl1029.13003MR1986148
  17. S. D. Cutkosky, Monomialization of morphisms from 3-folds to surfaces, (2002), Springer-Verlag Zbl1057.14009MR1927974
  18. S. D. Cutkosky, J. Herzog, A. Reguera, Poincaré series of resolutions of surface singularities, Transactions AMS 356 (2003), 1833-1874 Zbl1076.14006MR2031043
  19. S. D. Cutkosky, O. Kascheyeva, Monomialization of strongly prepared morphisms from nonsingular n -folds to surfaces, J. Algebra 275 (2004), 275-320 Zbl1057.14008MR2047449
  20. S. D. Cutkosky, O. Piltant, Monomial resolutions of morphisms of algebraic surfaces, Communications in Algebra 28 (2000), 5935-5959 Zbl1003.14004MR1808613
  21. S. D. Cutkosky, O. Piltant, Ramification of valuations, Advances in Mathematics 183 (2004), 1-79 Zbl1105.14015MR2038546
  22. S. D. Cutkosky, L. Ghezzi, Completions of valuation rings Zbl1216.13013
  23. S. D. Cutkosky, H. Srinivasan, Factorizations of matrices and birational maps Zbl1132.14013
  24. W. Fulton, Introduction to Toric Varieties, Princeton University Press (1993) Zbl0813.14039MR1234037
  25. W. Heinzer, C. Rotthaus, S. Wiegand, Approximating discrete valuation rings by regular local rings, a remark on local uniformization, Proc. Amer. Math. Soc. 129 (2001), 37-43 Zbl0962.13017MR1694346
  26. H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, Annals of Math. 79 (1964), 109-326 Zbl0122.38603MR199184
  27. H. Hironaka, Desingularization of excellent surfaces, (1967), Bowdoin College, Brunswick, Maine 
  28. K. Karu, Local strong factorization of birational maps, J. Alg. Geom. 14 (2005), 165-175 Zbl1077.14017MR2092130
  29. F.V. Kuhlmann, Valuation theoretic and model theoretic aspects of local uniformization, (2000), Springer-Verlag Zbl1046.14001MR1748629
  30. J. Lipman, Desingularization of two-dimensional schemes, Ann. Math. 107 (1978), 151-207 Zbl0349.14004MR491722
  31. H. Matsumura, Commutative Ring Theory, 8 (1986), Cambridge University Press, Cambridge Zbl0603.13001MR879273
  32. TT. Moh, Quasi-Canonical uniformization of hypersurface singularities of characteristic zero, Comm. Algebra 20 (1992), 3207-3249 Zbl0784.14008MR1186705
  33. M. Spivakovsky, Valuations in function fields of surfaces, Amer. J. Math. 112 (1990), 107-156 Zbl0716.13003MR1037606
  34. B. Teissier, Valuations, Deformations and Toric Geometry, Valuation Theory and its Applications II 33 (1990), 441-491 Zbl1061.14016
  35. O. Villamayor, S. Encinas, A course on constructive desingularization and equivariance, (2000), Springer-Verlag Zbl0969.14007
  36. O. Zariski, The reduction of the singularities of an algebraic surface, Annals of Math. 40 (1939), 639-689 Zbl0021.25303MR159
  37. O. Zariski, Local uniformization of algebraic varieties, Annals of Math. 41 (1940), 852-896 Zbl0025.21601MR2864
  38. O. Zariski, Reduction of the singularities of algebraic three dimensional varieties, Annals of Math. 45 (1944), 472-542 Zbl0063.08361MR11006
  39. O. Zariski, P. Samuel, Commutative Algebra II, (1960), Van Nostrand, Princeton Zbl0121.27801MR120249

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