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].

{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].

Reviewer: J.A.Wisniewski (MR 95f:14010)

##### 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 |