By Shokurov V. V.
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12. S. 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.
Category of flops. The objects of it are projective weakly log canonical models of (X/S, B) and morphisms are their regular flops. It may be considered as an order. ) If (X/S, B) is of general type, it will be finite with the log canonical model as the least element. It is always connected and the nearest elements are related by elementary flops. The maximal elements are the log minimal models. In Nikulin's case, we have only one maximal element, which is the greatest. 5. Everything in this section should work in any dimension if the LMMP holds.
8. Y. Kawamata, "The minimal discrepancy coefficients of terminal singularities of dimension 3," Appendix to , Izv. Akad. Nauk SSSR. Set. , 56, No. 1,201-203 (1992). Y. Kawamata, "Abundance theorem for minimal threefolds," Inv. , 108, 229-246 (1992). 10. Y. Kawamata, "Termination of log flips for algebraic 3-folds," Int. J. , 3, No. 5, 653-659 (1992). 11. Y. Kawamata, K. Matsuda, and K. Matsuki, "Minimal model problem," In: Adv. Stud. , Vol. 10, Kinokuniya Company (1987), pp. 283-360. 12. S.
3-Fold log models by Shokurov V. V.