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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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##### References:
 [1] Fujita, T.: On the structure of polarized manifolds with total deficiency one I, II, III. J. Math. Soc. Japan32, 709-725 (1980);33, 415-434 (1981);36, 75-89 (1984) · Zbl 0474.14017 · doi:10.2969/jmsj/03240709 [2] Fujita, T.: Classification of projective varieties of ?-genus one. Proc. Jap. Acad., Ser A58, 113-116 (1982) · Zbl 0568.14018 · doi:10.3792/pjaa.58.113 [3] Hidaka, F., Watanabe, K.: Normal Gorenstein surface with ample anti-canonical divisor, Tokyo J. Math.4, 319-330 (1981) · Zbl 0496.14023 · doi:10.3836/tjm/1270215157 [4] Iskovskih, V.A.: Fano 3-folds I, II. Math. USSR, Izv.11, 485-527 (1977);12, 469-506 (1978) · Zbl 0382.14013 · doi:10.1070/IM1977v011n03ABEH001733 [5] Koll?r, J.: The cone theorem. Ann. Math.120, 1-5 (1984) · Zbl 0544.14010 · doi:10.2307/2007069 [6] Kawamata, Y.: The cone of curves of algebraic varieties. Ann. Math.119, 603-633 (1984) · Zbl 0544.14009 · doi:10.2307/2007087 [7] Kawamata, Y.: A Generalization of Kodaira-Ramanujam’s Vanishing Theorem. Math. Ann.261, 43-46 (1982) · Zbl 0488.14003 · doi:10.1007/BF01456407 [8] Mori, S.: Threefolds whose canonical bundles are not numerically effective. Ann. Math.116, 133-176 (1982) · Zbl 0557.14021 · doi:10.2307/2007050 [9] Mukai, S.: On Fano manifolds of coindex 3. (Preprint 1983) [10] Nakano, S.: On the inverse of monoidal transformations. Publ. Res. Inst. Math. Sci.6, 483-502 (1971) · Zbl 0234.32017 · doi:10.2977/prims/1195193917 [11] Reid, M.: Projective morphisms according to Kawamata. University of Warwick (Preprint 1983) [12] Reid, M.: Decomposition of torie morphisms. In: Arithmetic and Geometry, II (M. Artin, J. Tata, eds) Prog. Math.36, 395-418. Boston-Basel-Stuttgart: Birkh?user 1983 [13] Shokurov, V.V.: Non vanishing theorem (in Russian). Izv. Akad. Nauk SSSR (To appear) · Zbl 0605.14006 [14] Viehveg, E.: Vanishing theorems. J. Reine Angew. Math.335, 1-8 (1982)
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