Advertisement

Inventiones mathematicae

, Volume 154, Issue 2, pp 223–331 | Cite as

Toroidal varieties and the weak factorization theorem

  • Jarosław Włodarczyk
Article

Abstract

We develop the theory of stratified toroidal varieties, which gives, together with the theory of birational cobordisms [73], a proof of the weak factorization conjecture for birational maps in characteristic zero: a birational map between complete nonsingular varieties over an algebraically closed field K of characteristic zero is a composite of blow-ups and blow-downs with smooth centers.

Keywords

Characteristic Zero Factorization Theorem Weak Factorization Nonsingular Variety Smooth Center 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abhyankar, S.: On the valuations centered in a local domain. Am. J. Math. 78, 321–348 (1956) MathSciNetMATHGoogle Scholar
  2. 2.
    Abramovich, D., de Jong, A.J.: Smoothness, semistability, and toroidal geometry. J. Algebr. Geom. 6, 789–801 (1997) MathSciNetMATHGoogle Scholar
  3. 3.
    Abramovich, D., Karu, K.: Weak semistable reduction in characteristic 0. Invent. Math. 139, 241–273 (2000) CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    Abramovich, D., Karu, K., Matsuki, K., Włodarczyk, J.: Torification and factorization of birational maps. J. Am. Math. Soc. 15, 531–572 (2002) CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Abramovich, D., Matsuki, K., Rashid, S.: A note on the factorization theorem of toric birational maps after Morelli and its toroidal extension. Tohoku Math. J., II. Ser. 51, 489–537 (1999) Google Scholar
  6. 6.
    Abramovich, D., Wang, J.: Equivariant resolution of singularities in characteristic 0. Math. Res. Lett. 4, 427–433 (1997) MathSciNetMATHGoogle Scholar
  7. 7.
    Białynicki-Birula, A., Świecicka, J.: Complete quotients by algebraic torus actions. Lect. Notes Math. 956, pp. 10–21. Springer 1982 Google Scholar
  8. 8.
    Bierstone, E., Milman, D.: Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant. Invent. Math. 128, 207–302 (1997) CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Bittner, F.: The universal Euler characteristic for varieties of characteristic zero. Preprint math. AG/0111062 Google Scholar
  10. 10.
    Borel, A.: Linear Algebraic Groups. New York, Amsterdam: W.A. Benjamin 1969 Google Scholar
  11. 11.
    Bourbaki, N.: Commutative Algebra, Ch. 3. Hermann 1961 Google Scholar
  12. 12.
    Brion, M., Procesi, C.: Action d’un tore dans une variété projective. In: Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989), 509–539, Prog. Math. 92, Boston: Birkhäuser 1990 Google Scholar
  13. 13.
    Christensen, C.: Strong domination/weak factorization of three dimensional regular local rings. J. Indian Math. Soc., New Ser. 45, 21–47 (1981) Google Scholar
  14. 14.
    Corti, A.: Factorizing birational maps of 3-folds after Sarkisov. J. Algebr. Geom. 4, 23–254 (1995) Google Scholar
  15. 15.
    Crauder, B.: Birational morphisms of smooth algebraic threefolds collapsing three surfaces to a point. Duke Math. J. 48, 589–632 (1981) MathSciNetMATHGoogle Scholar
  16. 16.
    Cutkosky, S.D.: Local factorization of birational maps. Adv. Math. 132, 167–315 (1997) CrossRefMathSciNetMATHGoogle Scholar
  17. 17.
    Cutkosky, S.D.: Local factorization and monomialization of morphisms. Astérisque 260 (1999) Google Scholar
  18. 18.
    Cutkosky, S. D.: Monomialization of morphisms from 3-folds to surfaces. Lect. Notes Math. 1786. Berlin: Springer 2002 Google Scholar
  19. 19.
    Cutkosky, S.D., Piltant, O.: Monomial resolutions of morphisms of algebraic surfaces. Special issue in honor of Robin Hartshorne. Commun. Algebra 28, 5935–5959 (2000) MATHGoogle Scholar
  20. 20.
    Danilov, V.I.: The geometry of toric varieties. Russ. Math. 33, 97–154 (1978) MATHGoogle Scholar
  21. 21.
    Danilov, V.I.: Birational geometry of toric 3-folds. Math. USSR-Izv. 21, 269–280 (1983) MATHGoogle Scholar
  22. 22.
    Demushkin, A.S.: Combinatorial invariance of toric singularities. (Russian), Vestn. Mosk. Univ., Ser. I 2, 80–87 (1982) Google Scholar
  23. 23.
    Dolgachev, I.V., Hu, Y.: Variation of geometric invariant theory quotients. Publ. Math., Inst. Hautes Étud. Sci. 87, 5–56 (1998) Google Scholar
  24. 24.
    Ewald, G.: Blow-ups of smooth toric 3-varieties. Abh. Math. Semin. Univ. Hamb. 57, 193–201 (1987) MATHGoogle Scholar
  25. 25.
    Fulton, W.: Introduction to Toric Varieties. Ann. Math. Stud. 131. Princeton: Princeton University Press 1993 Google Scholar
  26. 26.
    Grothendieck, A.: Éléments de Géométrie Algébrique IV. Ibid 20 (1964) Google Scholar
  27. 27.
    Gubeladze, J.: J. Pure Appl. Algebra 129, 35–65 (1998) CrossRefMathSciNetMATHGoogle Scholar
  28. 28.
    Hartshorne, R.: Algebraic Geometry. Grad. Texts Math. 52 (1977) Google Scholar
  29. 29.
    Hironaka, H.: On the theory of birational blowing-up. Harvard University Ph. D. Thesis, 1960 Google Scholar
  30. 30.
    Hironaka, H.: An example of a non-Kählerian complex-analytic deformation of Kählerian complex structure. Ann. Math. (2) 75, 190–208 (1962) Google Scholar
  31. 31.
    Hironaka, H.: Resolution of singularities of an algebraic variety over a field of characteristic zero. Ann. Math. 79, 109–326 (1964) MATHGoogle Scholar
  32. 32.
    Hironaka, H.: Flattening theorem in complex analytic geometry. Am. J. Math. 97, 503–547 (1975) MATHGoogle Scholar
  33. 33.
    Hu, Y.: The geometry and topology of quotient varieties of torus actions. Duke Math. J. 68, 151–184 (1992); Erratum. Duke Math. J. 68, 609 (1992) MathSciNetMATHGoogle Scholar
  34. 34.
    Hu, Y.: Relative geometric invariant theory and universal moduli spaces. Int. J. Math. 7, 151–181 (1996) MathSciNetMATHGoogle Scholar
  35. 35.
    Karu, K.: Boston University dissertation 1999. http://math.bu.edu/people/kllkr/th.ps Google Scholar
  36. 36.
    Kato, K.: Toric singularities. Am. J. Math. 116, 1073–1099 (1994) MathSciNetMATHGoogle Scholar
  37. 37.
    Kawamata, Y.: The cone of curves of algebraic varieties. Ann. Math. (2) 119, 603–633 (1984) Google Scholar
  38. 38.
    Kawamata, Y.: Crepant blowing-ups of three dimensional canonical singularities and its application to degenerations of surfaces. Ann. Math. 127, 93–163 (1988) MathSciNetMATHGoogle Scholar
  39. 39.
    Kempf, G., Knudsen, F., Mumford, D., Saint-Donat, B.: Toroidal embeddings I. Lect. Notes Math. 339. Springer 1973 Google Scholar
  40. 40.
    Kollár, J.: The cone theorem. Note to a paper: The cone of curves of algebraic varieties. [38] Y. Kawamata, Ann. Math. (2) 120, 1–5 (1984) Google Scholar
  41. 41.
    Kulikov, V.S.: Decomposition of a birational map of three-dimensional varieties outside codimension 2. Math. USSR Izvestiya 21, 187–200 (1983) MATHGoogle Scholar
  42. 42.
    Luna, D.: Slices étales. Sur les groupes algébriques, pp. 81–105. Bull. Soc. Math. Fr., Paris, Mémoire 33. Paris: Soc. Math. France 1973 Google Scholar
  43. 43.
    Matsuki, K.: Introduction to the Mori program. Universitext. Berlin: Springer 2002 Google Scholar
  44. 44.
    Matsuki, K.: Lectures on factorization of birational maps. RIMS 1281 (2000) Google Scholar
  45. 45.
    Matsuki, K., Wentworth, R.: Mumford-Thaddeus principle on the moduli space of vector bundles on an algebraic surface. Int. J. Math. 8, 97–148 (1997) MathSciNetMATHGoogle Scholar
  46. 46.
    Milnor, J.: Morse Theory. Ann. Math. Stud. 51. Princeton: Princeton University Press 1963 Google Scholar
  47. 47.
    Moishezon, B.: On n-dimensional compact varieties with n algebraically independent meromorphic functions. Trans. Am. Math. Soc. 63, 51–177 (1967) Google Scholar
  48. 48.
    Morelli, R.: The birational geometry of toric varieties. J. Algebr. Geom. 5, 751–782 (1996) Google Scholar
  49. 49.
    Morelli, R.: Correction to “The birational geometry of toric varieties”. 1997. http://www.math.utah.edu/∼morelli/Math/math.html Google Scholar
  50. 50.
    Mori, S.: Threefolds whose canonical bundles are not numerically effective. Ann. Math. (2) 116, 133–176 (1982) Google Scholar
  51. 51.
    Mori, S.: Flip theorem and the existence of minimal models for 3-folds. J. Am. Math. Soc. 1, 117–253 (1988) MATHGoogle Scholar
  52. 52.
    Mumford, D., Fogarty, J., Kirwan, F.: Geometric Invariant Theory (Third Enlarged Edition). Ergeb. Math. Grenzgeb. 34. Berlin: Springer 1992 Google Scholar
  53. 53.
    Nagata M.: On rational surfaces I. Mem. Coll. Sci., Kyoto, A 32, 351–370 (1960) Google Scholar
  54. 54.
    Nagata M.: Imbedding of an abstract variety in a complete variety. J. Math. Kyoto Univ. 2, 1–10 (1962)Google Scholar
  55. 55.
    Oda, T.: Torus embeddings and applications. Based on joint work with Katsuya Miyake. Bombay: Tata Inst. Fund. Res. 1978 Google Scholar
  56. 56.
    Oda, T.: Convex Bodies and Algebraic Geometry. Springer, 15, 1988 Google Scholar
  57. 57.
    Pinkham, H.: Factorization of birational maps in dimension 3. Proc. Symp. Pure Math. 40 (1983) Google Scholar
  58. 58.
    Reid, M.: Canonical threefolds. Géométrie Algébrique Angers 1979, 273–310, ed. by A. Beauville. Sijthoff and Nordhoff 1980 Google Scholar
  59. 59.
    Reid, M.: Minimal models of canonical 3-folds. Adv. Stud. Pure Math. 1, 131–180 (1983) MATHGoogle Scholar
  60. 60.
    Reid, M.: Decomposition of Toric Morphisms. Arithmetic and Geometry, papers dedicated to I. R. Shafarevich on the occasion of his 60th birthday, vol. II, ed. by M. Artin, J. Tate. Prog. Math. 36, 395–418 (1983) Google Scholar
  61. 61.
    Reid, M.: Birational geometry of 3-folds according to Sarkisov. Preprint 1991 Google Scholar
  62. 62.
    Sally, J.: Regular overrings of regular local rings. Trans. Am. Math. Soc. 171, 291–300 (1972)MATHGoogle Scholar
  63. 63.
    Sarkisov, V.G.: Birational maps of standard ℚ-Fano fiberings. I. V. Kurchatov Institute Atomic Energy. Preprint 1989 Google Scholar
  64. 64.
    Schaps, M.: Birational morphisms of smooth threefolds collapsing three surfaces to a curve. Duke Math. J. 48, 401–420 (1981) MathSciNetMATHGoogle Scholar
  65. 65.
    Shannon, D.L.: Monoidal transforms. Am. J. Math. 45, 284–320 (1973) Google Scholar
  66. 66.
    Shokurov, V.V.: A nonvanishing theorem. Izv. Akad. Nauk Az. SSR, Ser. Fiz.-Tekh. Mat. Nauk 49, 635–651 (1985) Google Scholar
  67. 67.
    Sumihiro, H.: Equivariant Completion I, II. J. Math. Kyoto Univ. 14, 1–28 (1974); 15, 573–605 (1975) MATHGoogle Scholar
  68. 68.
    Teicher, M.: Factorization of a birational morphism between 4-folds. Math. Ann. 256, 391–399 (1981) Google Scholar
  69. 69.
    Thaddeus, M.: Stable pairs, linear systems and the Verlinde formula. Invent. Math. 117, 317–353 (1994) MathSciNetMATHGoogle Scholar
  70. 70.
    Thaddeus, M.: Geometric invariant theory and flips. J. Am. Math. Soc. 9, 691–723 (1996) CrossRefMathSciNetMATHGoogle Scholar
  71. 71.
    Villamayor, O.: Constructiveness of Hironaka’s resolution. Ann. Sci. Éc. Norm. Supér., IV Sér. 22, 1–32 (1989) Google Scholar
  72. 72.
    Włodarczyk, J.: Decomposition of birational toric maps in blow-ups and blow-downs. A proof of the Weak Oda Conjecture. Trans. Am. Math. Soc. 349, 373–411 (1997) Google Scholar
  73. 73.
    Włodarczyk, J.: Birational cobordisms and factorization of birational maps. J. Algebr. Geom. 9, 425–449 (2000) Google Scholar
  74. 74.
    Zariski, O.: Algebraic Surfaces. Springer 1934Google Scholar

Copyright information

© Springer-Verlag 2003

Authors and Affiliations

  1. 1.Department of MathematicsPurdue UniversityWest LafayetteUSA

Personalised recommendations