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On the closed cone of curves of algebraic 3-folds. (English. Russian original) Zbl 0565.14025
Math. USSR, Izv. 24, 193-198 (1985); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 48, No. 1, 203-208 (1984).
Let X be a normal projective 3-fold defined over an algebraically closed field of characteristic 0. In the real vector space $$N(X)=(\{1-cycles on X\})/num. equivalence)\otimes {\mathbb{R}},$$ consider the closure (with respect to any euclidean norm) $$\overline{NE}(X)$$ of the cone spanned by effective 1-cycles. If X is $${\mathbb{Q}}$$-factorial (i.e. every Weil divisor has an integral multiple which is Cartier), one can take intersections of Weil divisors with 1-cycles. Let $$K_ X$$ be the $${\mathbb{Q}}$$-Cartier divisor defined by the canonical Weil divisor of X, and let $$\overline{NE}(X)^-=\{Z\in \overline{NE}(X)| Z\cdot K_ X<0\}$$. The paper gives a weakened version of Mori’s cone theorem [S. Mori, Ann. Math., II. Ser. 116, 133-176 (1982; Zbl 0557.14021)] for projective normal $${\mathbb{Q}}$$-factorial 3-folds having canonical singularities. In particular, this implies that if $$K_ X$$ is not numerically effective, then $$\overline{NE}(X)^-$$ always contains a good extremal ray, i.e. a ray R, which is extremal in the sense of convexity and satisfies: (i) R$$={\mathbb{R}}_+(C)$$ is generated by some curve $$C\subset X$$, and (ii) there exists a $${\mathbb{Q}}$$-Cartier divisor D on X which is numerically effective and such that $$D^{\perp}\cap \overline{NE}(X)=R$$. This has been proved independently by M. Reid [Proc. Lond. Math. Soc., III. Ser. (to appear)]. The weakened version of the cone theorem can be stated as follows. Suppose that $$\overline{NE}(X)^-$$ contains at most a finite number of good extremal rays such that the corresponding contraction morphism contracts a finite set of curves but no surface. Then for any ample divisor A and for any $$\epsilon >0$$, there exists a finite set of curves $$C_ 1,...,C_ r$$, such that $$\overline{NE}(X)$$ is the convex hull of the cone $$\overline{NE}_{\epsilon}(X,A)=\{Z\in \overline{NE}(X)| (K_ X+\epsilon A)\cdot Z\geq 0\}$$ and of the rays $${\mathbb{R}}_+(C_ i)$$.
Reviewer: A.Lanteri

##### MSC:
 14J30 $$3$$-folds 14C20 Divisors, linear systems, invertible sheaves 14E30 Minimal model program (Mori theory, extremal rays)
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