A simpler proof of toroidalization of morphisms from 3-folds to surfaces

Steven Dale Cutkosky[1]

  • [1] Department of Mathematics University of Missouri, Columbia, MO 65211, USA

Annales de l’institut Fourier (2013)

  • Volume: 63, Issue: 3, page 865-922
  • ISSN: 0373-0956

Abstract

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We give a simpler and more conceptual proof of toroidalization of morphisms of 3-folds to surfaces, over an algebraically closed field of characteristic zero. A toroidalization is obtained by performing sequences of blow ups of nonsingular subvarieties above the domain and range, to make a morphism toroidal. The original proof of toroidalization of morphisms of 3-folds to surfaces is much more complicated.

How to cite

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Cutkosky, Steven Dale. "A simpler proof of toroidalization of morphisms from 3-folds to surfaces." Annales de l’institut Fourier 63.3 (2013): 865-922. <http://eudml.org/doc/275453>.

@article{Cutkosky2013,
abstract = {We give a simpler and more conceptual proof of toroidalization of morphisms of 3-folds to surfaces, over an algebraically closed field of characteristic zero. A toroidalization is obtained by performing sequences of blow ups of nonsingular subvarieties above the domain and range, to make a morphism toroidal. The original proof of toroidalization of morphisms of 3-folds to surfaces is much more complicated.},
affiliation = {Department of Mathematics University of Missouri, Columbia, MO 65211, USA},
author = {Cutkosky, Steven Dale},
journal = {Annales de l’institut Fourier},
keywords = {Morphism; toroidalization; monomialization; morphism},
language = {eng},
number = {3},
pages = {865-922},
publisher = {Association des Annales de l’institut Fourier},
title = {A simpler proof of toroidalization of morphisms from 3-folds to surfaces},
url = {http://eudml.org/doc/275453},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Cutkosky, Steven Dale
TI - A simpler proof of toroidalization of morphisms from 3-folds to surfaces
JO - Annales de l’institut Fourier
PY - 2013
PB - Association des Annales de l’institut Fourier
VL - 63
IS - 3
SP - 865
EP - 922
AB - We give a simpler and more conceptual proof of toroidalization of morphisms of 3-folds to surfaces, over an algebraically closed field of characteristic zero. A toroidalization is obtained by performing sequences of blow ups of nonsingular subvarieties above the domain and range, to make a morphism toroidal. The original proof of toroidalization of morphisms of 3-folds to surfaces is much more complicated.
LA - eng
KW - Morphism; toroidalization; monomialization; morphism
UR - http://eudml.org/doc/275453
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, Resolution of singularities of embedded algebraic surfaces, (1998), Springer Verlag, New York, Berlin, Heidelberg Zbl0914.14006MR1617523
  3. A. Benito, O. Villamayor, Monoidal transforms and invariants of singularities in positive characteristic Zbl1278.14019
  4. E. Bierstone, P. Millman, Canonical desingularization in characteristic zero by blowing up the maximal strata of a local invariant, Inv. Math 128 (1997), 207-302 Zbl0896.14006MR1440306
  5. A. Bravo, S. Encinas, O. Villamayor, A simplified proof of desingularization and applications Zbl1086.14012
  6. F. Cano, Reduction of the singularities of codimension one singular foliations in dimension three, Ann. of Math. 160 (2004), 907-1011 Zbl1088.32019MR2144971
  7. V. Cossart, Desingularization of Embedded Excellent Surfaces, Tohoku Math. Journ. 33 (1981), 25-33 Zbl0472.14019MR613106
  8. V. Cossart, O. Piltant, Resolution of singularities of threefolds in positive characteristic I, Journal of Algebra 320 (2008), 1051-1082 Zbl1159.14009MR2427629
  9. V. Cossart, O. Piltant, Resolution of singularities of threefolds in positive characteristic II, Journal of Algebra 321 (2009), 1336-1976 Zbl1173.14012MR2494751
  10. S.D. Cutkosky, Local monomialization and factorization of Morphisms, 260 (1999) Zbl0941.14001MR1734239
  11. S.D. Cutkosky, Monomialization of morphisms from 3-folds to surfaces, 1786 (2002), Springer Verlag, Berlin, Heidelberg, New York Zbl1057.14009MR1927974
  12. S.D. Cutkosky, Resolution of Singularities, (2004), American Mathematical Society Zbl1076.14005MR2058431
  13. S.D. Cutkosky, Local monomialization of trancendental extensions, Annales de L’Institut Fourier 55 (2005), 1517-1586 Zbl1081.14020MR2172273
  14. S.D. Cutkosky, Toroidalization of birational morphisms of 3-folds, 190 (2007) Zbl1133.14013
  15. S.D. Cutkosky, Resolution of Singularities for 3-Folds in Positive Characteristic, American Journal of Math. 131 (2009), 59-128 Zbl1170.14011MR2488485
  16. S.D. Cutkosky, O. Kascheyeva, Monomialization of strongly prepared morphisms from nonsingular n -folds to surfaces, J. Algebra 275 (2004), 275-320 Zbl1057.14008MR2047449
  17. S.D. Cutkosky, O. Piltant, Monomial resolutions of morphisms of algebraic surfaces, Communications in Algebra 28 (2000), 5935-5960 Zbl1003.14004MR1808613
  18. A.J. de Jong, Smoothness, semistability and Alterations, Publ. Math. I.H.E.S. 83 (1996), 51-93 Zbl0916.14005
  19. S. Encinas, H. Hauser, Strong resolution of singularities in characteristic zero, Comment Math. Helv. 77 (2002), 821-845 Zbl1059.14022MR1949115
  20. H. Hauser, Kangaroo Points and Oblique Polynomials in Resolution of Positive Characteristic 
  21. H. Hauser, Excellent surfaces and their taught resolution, Resolution of Singularities, (Obergurgl, 1997) 181 (2000), 341-373, Birkhäuser, Basel Zbl0979.14007MR1748627
  22. H. Hauser, The Hironaka theorem on resolution of singularities (or: A proof we always wanted to understand), Bull. Amer. Math. Soc. 40 (2003), 323-348 Zbl1030.14007MR1978567
  23. H. Hauser, On the problem of resolution of singularites in positive characteristic (Or: a proof we are still waiting for), Bull. Amer. Math. Soc. 47 (2010), 1-30 Zbl1185.14011MR2566444
  24. H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, Annals of Math 79 (1964), 109-326 Zbl0122.38603MR199184
  25. H. Hironaka, Desingularization of excellent surfaces, Cossart V., Giraud J. and Orbanz U., Resolution of singularities 1101 (1980), Springer Verlag, Heidelberg, Berlin, New York 
  26. H. Hironaka, A program for resolution of singularities, in all characteristics p &gt; 0 and in all dimensions, (Trieste, 2006; Clay Math Insititute Workshop, 2008; and RIMS Workshop, 2008) 
  27. H. Knaf, F.-V. Kuhlmann, Every place admits local uniformization in a finite extension of the function field, Advances in Math. 221 (2009), 428-453 Zbl1221.14016MR2508927
  28. D. Panazzolo, Resolution of singularities of real-analytic vector fields in dimension three, Acta Math. 197 (2006), 167-289 Zbl1112.37016MR2296055
  29. G. Rond, Homomorphisms of local algebras in positive characteristic, J. Algebra 322 (2009), 4382-4407 Zbl1189.13022MR2558869
  30. A. Seidenberg, Reduction of the singlarities of the differential equation A d y = B d x , Amer. J. Math. 90 (1968), 248-269 Zbl0159.33303MR220710
  31. B. Teissier, Valuations, deformations and toric geometry, Valuation Theory and its Applications II, Franz-Viktor KuhlmannSalma KuhlmannS. K. Zbl1061.14016

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