# 3-Fold log models by Shokurov V. V.

By Shokurov V. V.

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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.

Then the stability of B implies that B is of general type. 22 also implies that it has only one weakly log minimal model which is simultaneously log canonical and log minimal. , for all exceptional divisors Di of such models X / S , a(Di, B, X) > 1 - bi. Every B has such an approximation whenever it has a weakly log canonical model X/S. Indeed, after the replacement of X / S by a log minimal model, we can replace B by B + ell, where H is very ample on X/S. , stable for a small variation of B. 2.

So, it is a canonical model for B + CH'. The latter makes sense whenever we consider H ' as a 2696 bi-divisor and define its multiplicities as for the complete inverse image for the chosen divisors and 0 on the other exceptional divisors of Y. Then we may find H ' = Y~ aiDi, where the sum runs over chosen divisors nonexceptional in Y and ai > 0. Indeed, by the construction we have the same presentation with integer cxi. Now note that there exists a projection g: Y -+ X / S , and we may replace H ~ by H' + Ng'H, where N is any natural number, and g*H = ~ hiDi in Y is an effective divisor with positive multiplicities on each Di.