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The cone of curves of algebraic varieties. (English) Zbl 0544.14009
In this paper two of the four steps are proved toward the theory of minimal models of algebraic varieties according to the program of M. Reid; we have proved: (1) the cone theorem; (2) the contraction theorem, and there remains: (3) the flip conjecture; (4) the induction conjecture. - J. Kollár [ibid. 120, 1-5 (1984; see the following review)] proved the discreteness of the extremal rays and completed the picture of the cone. We note here that our theorems are easily generalised to the relative case starting from a projective morphism $$f:X\to S$$ instead of a projective variety X.

##### MSC:
 14E30 Minimal model program (Mori theory, extremal rays) 14J30 $$3$$-folds
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