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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. 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.

Stud. , Vol. 1, Kinokuniya and North Holland (1983), pp. 131-180. 20. M. Reid, "Young person's guide to canonical singularities," Proc. Syrup. , 46:1,345-414 (1987). 21. M. Reid, Birational Geometry of 3-Fotds According to Sarkiso~, preprint (1991). 22. V. G. 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.

Proof. 19. C) > 0 is preserved for small perturbations of D. 20 due to the convexity of the Iitaka dimension for divisors. [] Let (X / S, B) be a pair of general type with 3-fold X having a Kawamata log terminal minimal model. Then it has a finite set of the projective weakly log canonical models and their flops. In particular, this holds for the log minimal models and their flops. Each of them is reconstructed from another by a finite chain of elementary flops. 22. C o r o l l a r y . By elementary flops we mean blow-ups and blow-downs X --+ Y / S with relative Picaxd number 1, being numerically trivial with respect to the log divisor K x + B x , but not the usual flops.

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3-Fold log models by Shokurov V. V.


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