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Abundance theorem for minimal threefolds. (English) Zbl 0777.14011
Let $$X$$ be a complex projective variety which is minimal in Mori’s sense, i.e. $$X$$ is a normal $$\mathbb{Q}$$-factorial variety with only terminal singularities whose canonical divisor is nef. – The abundance conjecture, due to the author, asserts that for a given minimal model, $$X$$, there exists a positive integer $$m$$ such that the pluricanonical system $$| mK_ X|$$ is free, i.e. free from fixed components and base points. – By previous results due to the author it follows that, in the case when $$\dim X=3$$, the abundance conjecture is true if $$\nu(X)>0$$ implies $$\kappa(X)>0$$. Here $$\kappa(X)$$ (respectively $$\nu(X))$$ denotes the Kodaira (respectively numerical Kodaira) dimension of $$X$$. Miyaoka proved that, in the 3-dimensional case, $$\kappa(X)\geq 0$$ and that $$\nu(X)=1$$ implies $$\kappa(X)>0$$. – In this paper the author proves that $$\kappa(X)>0$$ if $$\nu(X)=2$$ and $$\dim X=3$$, and therefore gives the affirmative answer to the abundance conjecture in dimension 3.
Recall that a $$\mathbb{Q}$$-Fano fiber space is uniruled, i.e. covered by a family of rational curves by Miyaoka and Mori. By combining the above results, the author shows in particular the following result:
Theorem. Let $$X$$ be an algebraic variety of dimension 3, defined over the complex field. Then one of the following holds:
1. $$X$$ is birationally equivalent to a $$\mathbb{Q}$$-factorial variety with only terminal singularities whose $$m$$-canonical system is free for a positive integer $$m$$; or
2. $$X$$ is covered by a family of rational curves.

##### MSC:
 14J30 $$3$$-folds 14E30 Minimal model program (Mori theory, extremal rays) 14J26 Rational and ruled surfaces 14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
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