On the extremal ray of higher dimensional varieties (Q1057330)

Let X be a nonsingular projective variety defined over an algebraically closed field of characteristic zero. An extremal ray is a special edge of the cone generated by effective 1-cycles in the space \(N(X)=(\{1- cycles\quad on X\}/numerical equivalence)\otimes_{{\mathbb{Z}}}{\mathbb{R}},\) which can be contracted by a projective morphism. This notion was introduced by \textit{S. Mori} [Ann. Math., II. Ser. 116, 133-176 (1983; Zbl 0557.14021)], who determined the structure of extremal rays for dim X\(=2\) and 3. The paper under review announces some results in the higher dimensional case; complete proofs have recently appeared in Invent. Math. 81, 347-357 (1985; Zbl 0554.14001). If the canonical divisor of X is not numerically effective, then there exists an extremal ray and a morphism \(f: X\to Y\) onto a projective variety Y, contracting it. The main results are the following. (I) Let f be birational and E be the exceptional set of f. If E is a divisor, then the general fibre F of \(f_ E: E\to f(E)\) is a Gorenstein Fano variety of index \(>1\) (a more complete description is given for dim \(F\leq 4)\). In addition, if dim f(E)\(=\dim E-1\) and \(f_ E\) is equidimensional then both Y and f(E) are nonsingular and f is the blowing-up of Y along f(E). (II) Let dim X\(>\dim Y\). Then the general fibre of F is a Fano variety; moreover, if dim Y\(=\dim X-1\) and f is equidimensional, then Y is nonsingular and \(f: X\to Y\) is a conic bundle. (Reviewer's remark: the same results have been proven independently by M. Beltrametti; his preprint contains also a detailed analysis of the case dim X\(=4)\).