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By Shokurov V. V.

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30-32. 25. V. V. Shokurov, Special 3-Dimensional Flips, preprint, MPI/89-22. 26. V. V. Shokurov, "3-Fold log flips," Izv. Akad. IVauk SSSR. Ser. , 56, No. 1,105-201 (1992). 27. V. V. Shokurov, "Anticanonical boundedness for curves," Appendix to [18]. 28. V. V. Shokurov, "Semi-stable 3-fold flips," Izv. Akad. Nauk SSSR. Set. , 57, No. 2, 162-224 (1993). 29. V. V. c. , preprint. 30. O. Zariski and P. Samuel, Commutative Algebra I, II, Van Nostraad, Princeton (1958, 1960).

Keel, K. Matsuki, and J. McKernan, "Log abundance theorem for threefolds," Duke Math. , 75, No. 1, 99-119 (1994). 13. J. Kolle~', "The Cone theorem: Note to Kawamata's 'The cone of curves of algebraic varieties'," Ann. , 120, 1-5 (1984). J. 14. Kolls and S. Mort, Classification of Three-Dimensional Flips, preprint. 15. J. , "Flips and abundance for algebraic threefolds," A Summer Seminar at the University of Utah, Salt Lake City, 1991, Asterisque, 211 (1992). 16. T. Luo, On the Divisorial Eztremal Contractions of Threefolds: Divisor to a Point, preprint.

Moreover, the LMMP is sufficient for Q-boundaries, except for the termination. REFERENCES 1. V. Alexeev, "Two two-dimensional terminations," Duke Math. , 69, No. 3, 527-545 (1993). 2. A. Borisov, Minimal discrepancies of toric singularities, Algebraic Geometry E-prints. 3: J. W. S. Cassels, An Introduction to Diophantine Approzimatior~, Cambridge University Press (1957). 4. H. Clemens, J. Kolls and S. Mort, "Higher dimensional complex geometry," Astdrisque, 166, Soc. Math. France (1988). 5. A. Corti, Factoring Birational Maps of Threefolds After Sarkisov, preprint.

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3-Fold log models by Shokurov V. V.


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