3-Fold log models by Shokurov V. V.

By Shokurov V. V.

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By the termination we get simultaneously either a negative Kodaira dimension for the given boundaries B, or a log minimal model. 6. Thus, in an ~-neighborhood of B ~ all boundaries with a nonnegative Kodaira dimension form a cone, as well as the equivalent classes. 6. it is rational and polyhedral. [] The numerical dimension of (X/S, B) is a step upper convex function, and it is lower semi-continuous with respecg to B on M . 21. C o r o l l a r y . Proof. 19. C) > 0 is preserved for small perturbations of D.

2), 127, 93-163 (1988). 7. Y. Kawamata, "On the length of an extremal rational curve," Invent. , 105, No. 3, 609-611 (1991). 8. Y. Kawamata, "The minimal discrepancy coefficients of terminal singularities of dimension 3," Appendix to [26], 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.

There exists a more direct approach. Let X / B be the initial model and D be the log birational transform of a very ample divisor from the reconstructed model. Then the reconstructed model will be a model for B + e H a n d 0 < e << 1. 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.

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3-Fold log models by Shokurov V. V.
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