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By Shokurov V. V.

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Sarkisov, Birational Maps of Standard Q-Fano Fiberings, I. V. Kurchatov Institute Atomic Energy preprint (1989). 23. V. V. Shokurov, "A nonvanishing theorem," Izv. Akad. Nauk SSSR. Set. , 49,635-651 (1985). 2698 24. V. V. Shokurov, "Problems about Fano varieties," In: Birational Geomet~ of Algebraic Varieties: Open problema. The XXIIIrd International Symposium, Division of Mathematica, The Taniguehi Foundation. Aug. ~-27, 1988, pp. 30-32. 25. V. V. Shokurov, Special 3-Dimensional Flips, preprint, MPI/89-22.

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).

It can be constructed as follows. First, we make a Q-factorialization of X, which:can be decomposed into elementary small blow-ups. 19, the next extremal modifications will be the usual flops and divisorial contractions with respect to K x + B x . The former are decomposable into two elementary flops, and so are the latter themselves. Finally, we decompose an Iitaka contraction into elementary blow-downs. 1. 23. R e m a r k s . 1. The generic points B t in a neighborhood of B are s~ablein the following sense.

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3-Fold log models by Shokurov V. V.


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