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Vector bundles and adjunction. (English) Zbl 0770.14008
Let $$X$$ be a smooth projective variety over $$\mathbb{C}$$ of dimension $$n\geq 4$$ and $$E$$ be an ample vector bundle on $$X$$ of rank $$n-1$$. The authors discuss the isomorphism classes of $$(X,E)$$ in terms of properties of the divisor $$K_ X+\text{det} E$$. The case that $$K_ X+\text{det} E$$ is not numerically effective was investigated already by Y.-G. Ye and Q. Zhang in [Duke Math. J. 60, No. 3, 671-687 (1990; Zbl 0709.14011)]. The second case $$K_ X+\text{det} E$$ is numerical effective but not big contains a main subcase discussed by Peternell, Szurek and J. A. Wisniewski [J. Reine Angew. Math. 417, 141-157 (1991; Zbl 0721.14023)]. The remaining not ample case can be considered as a certain blow-up of the ample case. This was conjectured by M. C. Beltrametti and A. J. Sommese in “Comparing the classical and the adjunction theoretic definition of scrolls”, Proc. Conf. Geometry of Complex Projective Varieties (Cetraro 1990).
In the note under review the case $$n=3$$ is investigated. The case rank $$\geq n$$ was treated by T. Fujita in Complex algebraic varieties, Proc. Conf., Bayreuth 1990, Lect. Notes Math. 1507, 105-112 (1992). The technical tools the authors use is Mori theory: contraction of extremal face and comparing locally adjunction theory of vector bundles with the one of line bundles. An example at the end of paper shows that this is globally impossible.

##### MSC:
 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 14J35 $$4$$-folds 14C20 Divisors, linear systems, invertible sheaves
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