Tohoku Mathematical Journal

A note on the factorization theorem of toric birational maps after Morelli and its toroidal extension

Dan Abramovich, Kenji Matsuki, and Suliman Rashid

Full-text: Open access

Article information

Tohoku Math. J. (2), Volume 51, Number 4 (1999), 489-537.

First available in Project Euclid: 3 May 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14M25: Toric varieties, Newton polyhedra [See also 52B20]
Secondary: 14E05: Rational and birational maps


Abramovich, Dan; Matsuki, Kenji; Rashid, Suliman. A note on the factorization theorem of toric birational maps after Morelli and its toroidal extension. Tohoku Math. J. (2) 51 (1999), no. 4, 489--537. doi:10.2748/tmj/1178224717.

Export citation


  • [Abramovich-Karu] D ABRAMOVICH AND K. KARU, Weak semistable reduction in characteristic 0, preprint, 1997.
  • [Abramovich-Karu-Matsuki-Wlodarczyk] D ABRAMOVICH, K KARU, K MATSUKI AND J WLODARCZYK, Torification and factorization of birational maps, preprint, 1999
  • [Christensen] C CHRISTENSEN, Strong domination/weak factorization of three dimensional regular local rings, J. Indian Math Soc 45 (1981), 21-47
  • [Corti] A. CORTI, Factorizing birational maps of 3-folds after Sarkisov, J Algebraic Geom. 4 (1995), 23-25
  • [Cutkoskyl] S. D. CUTKOSKY, Local factorization of birational maps, Math 132 (1997), 167-315
  • [Cutkosky2] S D CUTKOSKY, Local factorization and monornialization of morphisms, preprint, 199
  • [CutkoskyS] S. D CUTKOSKY, Local factorization and monomialization of morphisms, math AG/9803078 (1998)
  • [Danilovl] V I. DANILOV, The birational geometry of toric varieties, Russian Math Surveys 33 (1978), 97-154
  • [Danilov2] V. I DANILOV, The birational geometry of toric 3-folds, Math USSR-Izv 21 (1983), 269-28
  • [DeConcini-Procesi] C DE CONCINI AND C. PROCESI, Complete Symmetric Varieties II, in Algebraic Group and Related Topics (Ed. R. Hotta) Adv. Stud Pure Math 6, Kinokuniya, Tokyo and North Holland, Amsterdam, New York, Oxford, 1985, 481-513
  • [DeJong] A J DE JONG, Smoothness, semi-stability, and alterations, Inst Hautes Etudes Sci. Publ Math. 8 (1996), 51-93
  • [Ewald] G. EWALD, Blowups of smooth toric 3-varieties, Abh Math Sem Univ. Hamburg 57 (1987), 193-20
  • [Fulton] W FULTON, Introduction to Toric varieties, Ann. of Math. Stud. 131, Princeton University Press, 199
  • [litaka] S IITAKA, Algebraic Geometry (An Introduction to Birational Geometry of Algebraic Varieties), Gra Texts in Math 76, Springer-Verlag, 1982.
  • [Kawamatal] Y KAWAMATA, On the finiteness of generators of a pluricanonical ring for a 3-fold of general type, Amer. J. Math. 106 (1984), 1503-1512.
  • [Kawamata2] Y KAWAMATA, The cone of curves of algebraic varieties, Ann. of Math 119 (1984), 603-633
  • [Kawamata3] Y. KAWAMATA, Crepant blowing-upsof three dimensional canonical singularities and its applicatio to degenerations of surfaces, Ann of Math. 127 (1988), 93-163
  • [Kempf-Knudsen-Mumford-SaintDonat]G. KEMPF, F. KNUDSEN, D. MUMFORD AND B. SAINT-UONAT, Toroidal Embeddings I, Lecture Notes in Math. 339, Springer-Verlag, 1973
  • [Kingl] H. KING, Resolving Singularities of Maps, Real algebraic geometry and topology (East Lansing, MI, 1993), Contemp. Math. 182, Amer Math. Soc., Providence, RI (1995), 135-154.
  • [King2] H. KING, A private e-mail to Morelli (1996)
  • [Kollar] J. KOLLAR, The cone theorem Note to a paper: "Thecone of curves of algebraic varieties" by Kawamata, Ann. of Math 120 (1984), 1-5
  • [Matsuki] K. MATSUKI, Introduction to the Mori Program, to appear as a textbook published by Springer-Verlag, 1999.
  • [Morellil] R MORELLI, The birational geometry of toric varieties, J. Algebraic Geom. 5 (1996), 751-78
  • [Morelli2] R MORELLI, Correction to "Thebirational geometry of toric varieties", homepage at the Univ. of Uta (1997), 767-770
  • [Moril] S. MORI, Threefolds whose canonical bundles are not numerically effective, Ann. of Math 116 (1982), 133-176.
  • [Mori2] S MORI, Flip theorem and the existence of minimal models for 3-folds, J Amer. Math. Soc. 1 (1988), 117-253
  • [Odal] T ODA, Lectures on Torus Embeddings and Applications, Based onjoint work with Katsuya Miyake, Tat Inst Fund. Res 58, Springer-Verlag, 1966
  • [Oda2] T ODA, Convex Bodies and Algebraic Geometry (An Introduction to the Theory of Toric Varieties), Ergeb Math. Grenzgeb (3) 15, Springer-Verlag, 1988.
  • [Oda-Park] T. ODA AND H PARK, Linear Gale transforms and Gelfand-Kapranov-Zelvinsky decompositions, Tohoku Math. J. 43 (1991), 375-399
  • [Park] H. PARK, The Chow rings and GKZ decompositions for (^-factorial toric varieties, Tohoku Math. J. 4 (1993), 109-145
  • [Reidl] M. REID, Canonical threefolds, Journees de Geometric Algebrique d'Angers, Juillet 1979/Algebraic Ge ometry Angers, 1979, Sijthoff and Nordhoff, 1980, 273-310.
  • [Reid2] M REID, Minimal models of canonical threefolds, Adv. Stud. Pure Math. 1 (1983), 131-18
  • [Reid3] M. REID, Decomposition of toric morphisms, Arithmetic and Geometry, papers dedicated to I. R. Shafare vich on the occasion of his 60th birthday, vol II (Ed M. Artin and J Tate), Progr Math. 36 (1983), 395-418
  • [Reid4] M REID, Birational geometry of 3-folds according to Sarkisov, preprint, 1991
  • [Sarkisov] V. G. SARKISOV, Birational maps of standard Q-Fano fiberings, I. V Kurchatov Institute for Atomi Energy preprint, 1989.
  • [Shokurov] V. V. SHOKUROV, A non-vanishing theorem, Izv Akad Nauk SSSR Ser.Mat.49 (1985), 635-651
  • [Sumihirol] H. SUMIHIRO, Equivariant Completion I, J Math. Kyoto Univ. 14(1974), 1-2
  • [Sumihiro2] H. SUMIHIRO, Equivariant Completion II, J. Math Kyoto Univ 15 (1975), 573-60
  • [Wlodarczykl] J. WLODARCZYK, Decomposition of birational toric maps in blow-ups and blow-downs, Tran Amer Math Soc. 349 (1997), 373-411
  • [Wlodarczyk2] J. WLODARCZYK, Birational cobordism and factorization of birational maps, math AG/990407 (1999), 23 pp
  • [Wlodarczyk3] J WLODARCZYK, Combinatorial structures on toroidal varieties and a proof of the Weak Factor ization Theorem, math.AG/9904076 (1999), 32 pp

See also

  • See: Kenji Matsuki. Correction: ``A note on the factorization theorem of toric birational maps after Morelli and its toroidal extension''. Tohoku Math. J., Volume 52, Number 4 (2000), pp. 629-631.