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On extremal rays of the higher dimensional varieties. (English) Zbl 0554.14001
The notion of extremal rays was introduced by S. Mori [”Threefolds whose canonical bundles are not numerically effective.” Ann. Math., II. Ser. 116, 133-176 (1982)]. He also determined the structure of extremal rays in the case of dimension three. In this article, the case of dimension greater than three is treated. The main results are stated as follows:
Theorem. Let X be a non-singular projective variety over an algebraically closed field of characteristic zero. Assuming that the canonical divisor \(K_ X\) is not nef, there exist an extremal ray \({\mathbb{R}}_+[C]\) and the contraction \(f: X\to Y\) of \({\mathbb{R}}_+[C]\). (i) If f is birational and if the exceptional set of f is a divisor D, then the general fiber F of \(f_ D: D\to f(D)\) is a Gorenstein Fano variety with index greater than 1. (ii) In addition, if \(\dim f(D)=\dim D-1\) and if \(f_ D\) is equidimensional, then Y and f(D) are non-singular and f is the blowing up of Y along the non-singular center f(D). (iii) If \(\dim Y<\dim X,\) then the general fiber of f are Fano manifolds. (iv) Moreover if \(\dim Y=\dim X-1\) and if f is equi-dimensional, then f induces a conic bundle structure.

MSC:
14C20 Divisors, linear systems, invertible sheaves
14J40 \(n\)-folds (\(n>4\))
14E05 Rational and birational maps
14J10 Families, moduli, classification: algebraic theory
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