The cone of curves of algebraic varieties. (English) Zbl 0544.14009

Kawamata, Yujiro

Ann. Math. (2) 119, 603-633 (1984).

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

Keywords:

cone of curves; vanishing theorem; minimal models of algebraic varieties; contraction theorem

Citations:

Zbl 0544.14008; Zbl 0544.14010