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Termination of log flips for algebraic 3-folds. (English) Zbl 0814.14016
We shall prove that there exists no infinite sequence of successive log flips for algebraic 3-folds. Let $$X$$ be a normal $$\mathbb{Q}$$-factorial variety and $$B$$ a $$\mathbb{Q}$$-divisor such that the pair $$(X,B)$$ has only weak log terminal singularities. A log flip for $$(X,B)$$ is a diagram $$X \overset \varphi {} Y \overset {\varphi^ +} \leftarrow X^ +$$ consisting of projective birational morphisms between normal varieties such that
(1) $$\rho(X/Y) = 1$$, i.e., $$\varphi$$ is not an isomorphism and for any two curves $$C$$ and $$C'$$ which are mapped to points by $$\varphi$$, there exists a positive number $$\alpha$$ such that $$C \sim_{\text{num}} \alpha C'$$,
(2) $$\rho(X^ +/Y) = 1$$,
(3) $$\text{codim Exc}(\varphi) \geq 2$$, where Exc denotes the exceptional locus,
(4) $$\text{codim Exc}(\varphi^ +) \geq 2$$,
(5) $$-(K_ X + B)$$ is $$\varphi$$-ample,
(6) $$K_{X^ +} + B^ +$$ is $$\varphi^ +$$-ample, where $$B^ +$$ is the strict transform of $$B$$ on $$X^ +$$.
The resulting pair $$(X^ +,B^ +)$$ is automatically $$\mathbb{Q}$$-factorial and weak log terminal. The main result of this paper is the following:
Theorem 1. Let $$X$$ be a 3-dimensional normal $$\mathbb{Q}$$-factorial variety and $$B$$ a $$\mathbb{Q}$$-divisor such that the pair $$(X,B)$$ has only log terminal singularities. Then there exists no infinite sequence of successive log flips such as $X = X_ 0 @>\varphi_ 0>> Y_ 0 @<\varphi^ +_ 0<< X_ 1 @>\varphi_ 1>> Y_ 1 @<\varphi^ +_ 2<< X_ 2 @>\varphi_ 2>> Y_ 2 @<\varphi^ +_ 2<< X_ 3 @>\varphi_ 3>> \dots$ where the first pair of arrows is a log flip for $$(X_ 0,B_ 0)$$ with $$B_ 0 = B$$, the second is for $$(X_ 1,B_ 1)$$ with the strict transform $$B_ 1$$ of $$B_ 0$$, the third is for $$(X_ 2,B_ 2)$$ with the strict transform $$B_ 2$$ of $$B_ 1$$, and so on.
By the log minimal model program [see the author, K. Matsuda, K. Matsuki in Algebraic Geometry, Proc. Sympos., Sendai 1985, Adv. Stud. Pure Math. 10, 283-360 (1987; Zbl 0672.14006)], the above results combined yield the following existence theorem of log minimal models:
Theorem 2. Let $$X_ 0$$ be a 3-dimensional nonsingular projective variety and $$B_ 0 = \sum_ i b_ i S_ i$$ a $$\mathbb{Q}$$-divisor such that $$0<b_ i \leq 1$$ and the $$S_ i$$ are mutually distinct nonsingular prime divisors with normal crossings. Then there exist a pair $$(X,B)$$ of a normal projective $$\mathbb{Q}$$-factorial variety and a $$\mathbb{Q}$$-divisor having only weak log terminal singularities, and a birational map $$f:X_ 0 \to X$$ which is surjective in codimension 1, i.e., the image of the domain of $$f$$ contains all the points of codimension 1, such that one of the following holds: $$B$$ being the image of $$B_ 0$$,
(1) there exists a surjective morphism $$\varphi : X \to Y$$ with connected fibers to a normal projective variety $$Y$$ such that $$\dim Y < \dim X$$ and $$-(K_ X + B)$$ is $$\varphi$$-ample,
(2) $$K_ X + B$$ is nef.
 14E30 Minimal model program (Mori theory, extremal rays) 14E05 Rational and birational maps 14J30 $$3$$-folds 14J17 Singularities of surfaces or higher-dimensional varieties