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Weak toroidalization over non-closed fields

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Abstract

We prove that any dominant morphism of algebraic varieties over a field k of characteristic zero can be transformed into a toroidal (hence monomial) morphism by projective birational modifications of source and target. This was previously proved by the first and third author when k is algebraically closed. Moreover we show that certain additional requirements can be satisfied.

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References

  1. Abramovich D., de Jong A.J.: Smoothness, semistability, and toroidal geometry. J. Algebraic Geom. 6(4), 789–801 (1997)

    MathSciNet  MATH  Google Scholar 

  2. Abramovich D., Karu K.: Weak semistable reduction in characteristic 0. Invent. Math. 139(2), 241–273 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  3. Abramovich, D., Karu, K., Matsuki, K., Włodarczyk, J.: Torification and factorization of birational maps. J. Am. Math. Soc. 15(3), 531–572 (2002) (electronic)

    Google Scholar 

  4. Bierstone E., Milman P.D.: Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant. Invent. Math. 128(2), 207–302 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cutkosky, S.D.: Local monomialization of transcendental extensions. Ann. Inst. Fourier (Grenoble) 55(5), 1517–1586 (2005)

    Google Scholar 

  6. Cutkosky, S.D.: Toroidalization of dominant morphisms of 3-folds. Mem. Am. Math. Soc. 190(890), vi+222 (2007)

  7. de Jong A.J.: Smoothness, semi-stability and alterations. Inst. Hautes études Sci. Publ. Math. 83, 51–93 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  8. de Jong A.J.: Families of curves and alterations. Ann. Inst. Fourier (Grenoble) 47(2), 599–621 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  9. Denef, J.: Algebraic geometric proof of theorems of Ax-Kochen and Ershov (2012). http://perswww.kuleuven.be/~u0009256/Denef-Mathematics

  10. Denef, J.: Some remarks on toroidal morphisms. Unpublished note (2012). ArXiv:1303.4999

  11. Encinas S., Villamayor O.: Good points and constructive resolution of singularities. Acta Math. 181(1), 109–158 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fulton, W.: Introduction to toric varieties. In: Annals of Mathematics Studies, The William H. Roever Lectures in Geometry, vol. 131. Princeton University Press, Princeton (1993)

  13. Grothendieck, A.: Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes. Inst. Hautes études Sci. Publ. Math. 8, 222 (1961)

    Google Scholar 

  14. Grothendieck, A.: Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I. Inst. Hautes études Sci. Publ. Math. 11, 167 (1961)

    Google Scholar 

  15. Grothendieck, A.; Murre, J.P.: The tame fundamental group of a formal neighbourhood of a divisor with normal crossings on a scheme. In: Lecture Notes in Mathematics, vol. 208. Springer, Berlin (1971)

  16. Hartshorne, R.: Algebraic geometry. In: Graduate Texts in Mathematics, No. 52. Springer, New York (1977)

  17. Hironaka, H.: Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II. Ann. Math. 2(79), 109–203 (1964); ibid. (2), 79, 205–326 (1964)

  18. Illusie, L., Temkin, M.: Gabber’s modification theorem (log smooth case). Exposé 10 In: Illusie, L., Laszlo, Y., Orgogozo, F. (eds.) Travaux de Gabber sur l’uniformisation locale et la cohomologie étale des schémas quasi-excellents (Séminaire à l’ École Polytechnique 2006–2008), arXiv:1207.3648 (2012)

  19. Kato, K.: Logarithmic structures of Fontaine-Illusie. In: Algebraic Analysis, Geometry, and Number Theory (Baltimore, 1988), pp. 191–224. Johns Hopkins University Press, Baltimore (1989)

  20. Kempf, G., Knudsen, F.F., Mumford, D., Saint-Donat, B.: Toroidal embeddings. I. In: Lecture Notes in Mathematics, vol. 339. Springer, Berlin (1973)

  21. Kollár, J.: Lectures on resolution of singularities. In: Annals of Mathematics Studies, vol. 166. Princeton University Press, Princeton (2007)

  22. Nizioł, W.: Toric singularities: log-blow-ups and global resolutions. J. Algebraic Geom. 15(1), 1–29 (2006)

    Google Scholar 

  23. Raynaud, M., Gruson, L.: Critères de platitude et de projectivité. Techniques de “platification” d’un module. Invent. Math. 13, 1–89 (1971)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Brown University, P.O. Box 1917, Providence, RI, 02912, USA

    Dan Abramovich

  2. Department of Mathematics, University of Leuven, Celestijnenlaan 200 B, 3001, Leuven, Heverlee, Belgium

    Jan Denef

  3. Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, V6T 1Z2, Canada

    Kalle Karu

Authors
  1. Dan Abramovich
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  2. Jan Denef
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  3. Kalle Karu
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Correspondence to Jan Denef.

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Abramovich, D., Denef, J. & Karu, K. Weak toroidalization over non-closed fields. manuscripta math. 142, 257–271 (2013). https://doi.org/10.1007/s00229-013-0610-5

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  • DOI: https://doi.org/10.1007/s00229-013-0610-5

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