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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Torification and factorization of birational maps
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by Dan Abramovich, Kalle Karu, Kenji Matsuki and Jarosław Włodarczyk HTML | PDF
J. Amer. Math. Soc. 15 (2002), 531-572 Request permission


Building on work of the fourth author and Morelli’s work, we prove 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 blowings up and blowings down with nonsingular centers.
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Additional Information
  • Dan Abramovich
  • Affiliation: Department of Mathematics, Boston University, 111 Cummington Street, Boston, Massachusetts 02215
  • MR Author ID: 309312
  • ORCID: 0000-0003-0719-0989
  • Email:
  • Kalle Karu
  • Affiliation: Department of Mathematics, Harvard University, One Oxford Street, Cambridge, Massachusetts 02139
  • Email:
  • Kenji Matsuki
  • Affiliation: Department of Mathematics, Purdue University, 1395 Mathematical Sciences Building, West Lafayette, Indiana 47907-1395
  • Email:
  • Jarosław Włodarczyk
  • Affiliation: Instytut Matematyki UW, Banacha 2, 02-097 Warszawa, Poland
  • Email:
  • Received by editor(s): March 14, 2000
  • Received by editor(s) in revised form: June 1, 2000
  • Published electronically: April 5, 2002
  • Additional Notes: The first author was partially supported by NSF grant DMS-9700520 and by an Alfred P. Sloan research fellowship. In addition, he would like to thank the Institut des Hautes Études Scientifiques, Centre Emile Borel (UMS 839, CNRS/UPMC), and Max Planck Institut für Mathematik for a fruitful visiting period.
    The second author was partially supported by NSF grant DMS-9700520
    The third author has received no financial support from NSF or NSA during the course of this work.
    The fourth author was supported in part by Polish KBN grant 2 P03 A 005 16 and NSF grant DMS-0100598.
  • © Copyright 2002 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 15 (2002), 531-572
  • MSC (2000): Primary 14E05
  • DOI:
  • MathSciNet review: 1896232