Torification and factorization of birational maps

@article{Abramovich1999TorificationAF,
  title={Torification and factorization of birational maps},
  author={Dan Abramovich and Kalle Karu and Kenji Matsuki and Jarosław Włodarczyk},
  journal={Journal of the American Mathematical Society},
  year={1999},
  volume={15},
  pages={531-572}
}
Building on the work of the fourth author in math.AG/9904074, 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 smooth centers. Such a factorization exists which is functorial with respect to absolute isomorphisms, and compatible with a normal crossings divisor. The same holds for algebraic… 

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