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Length of extremal rays and generalized adjunction. (English) Zbl 0668.14004
The Mori theory of extremal rays has been successfully applied by several authors, among other things, in the study of the adjunction process for a complex projective manifold X polarized by an ample divisor H. In particular P. Ionescu [Math. Proc. Camb. Philos. Soc. 99, 457-472 (1986; Zbl 0619.14004)] classified pairs (X,H) as above for which \(K_ X+tH\) fails to be numerically effective for t large enough where \(K_ X\) stands for a canonical divisor of X.
In the paper under review Mori’s theory is applied to investigate the numerical effectiveness properties of \(K_ X+c_ 1E\), where E is an ample vector bundle on X. As a first thing the author introduces and discusses the notion of length of an extremal ray R of X. This is defined as the minimum of \(-K_ X\cdot C,\) C running over the rational curves on X, whose numerical equivalence class is in R. Then the author proves some results on manifolds admitting extremal rays of high length. In particular manifolds for which \(\bigwedge^ 2TX\) is ample are considered and it is proved that in dimension 3 these ones are only \({\mathbb{P}}^ 3\) and the quadric threefold.
As to generalized adjunction, the author considers an ample and spanned rank r vector bundle E on a projective n-fold X. He proves that if the divisor \(K_ X+c_ 1E\) is not numerically effective then \(r\leq n\) and provides results for \(r\geq n-2\). In particular, when \(r=n\), the above condition is equivalent to \(c_ n(E)=1\) and is satisfied only by the pair \((X,E)=({\mathbb{P}}^ n;n{\mathcal O}_{{\mathbb{P}}^ n}(1))\). This proves in the smooth case a conjecture due to A. J. Sommese and the reviewer [Abh. Math. Sem. Hamburg 58 (1988)].
Reviewer: A.Lanteri

MSC:
14C20 Divisors, linear systems, invertible sheaves
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