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On the dimension of the adjoint linear system for quadric fibrations. (English) Zbl 0848.14002
Tikhomirov, Alexander (ed.) et al., Algebraic geometry and its applications. Proceedings of the 8th algebraic geometry conference, Yaroslavl’, Russia, August 10-14, 1992. Braunschweig: Vieweg. Aspects Math. E 25, 9-20 (1994).
The paper under review is set up in the framework of adjunction theory: its main result concerns the dimension of the adjoint linear system $$|K_{\widehat X} + (n - 2) \widehat L |$$ on a smooth $$n$$-fold $$\widehat X$$ polarized by a very ample line bundle $$\widehat L$$. Namely, it is proved that if the first reduction $$(X,L)$$ of the pair $$(\widehat X, \widehat L)$$ has a structure of a quadric fibration associated with the adjoint divisor $$K_X + (n - 2) L$$ then $$\dim H_0 (\widehat X, K_{\widehat X} + (n - 2) \widehat L) \geq 2$$ except in some specific cases described in the paper. The result is obtained by extending theorems and techniques developed by the authors in a series of papers related to adjunction theory and quadric fibrations.
{Reviewer’s remark: The reviewer was informed by the authors about the following misprint in the statement of theorem 2.2. The line just before 2.2.1 should read $$p_g (S) = q(S) = 1,2$$, instead of $$p_g (S) = q(S) = 1\}$$.
For the entire collection see [Zbl 0793.00016].

##### MSC:
 14C20 Divisors, linear systems, invertible sheaves 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli 14D99 Families, fibrations in algebraic geometry