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) |