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On the relative adjunction mapping. (English) Zbl 0717.14003

Let \(\phi: X\to Y\) be a proper algebraic map with connected fibres from a connected quasi-projective n-dimensional complex manifold X, \(n\geq 2\), onto a quasi-projective variety Y and let L be an algebraic line bundle on X, which is very ample relatively to \(\phi\). The authors use Reider’s technique [I. Reider, Ann. Math., II. Ser. 127, No.2, 309-316 (1988; Zbl 0663.14010)] in a local setting to provide results about the adjoint bundle \(K_ X\otimes L^{n-1}\), which generalize those obtained by A. J. Sommese and A. Van de Ven [Math. Ann. 278, 593-603 (1987; Zbl 0655.14001)] in the absolute case, i.e. when Y is a point. The authors prove that, unless \(\phi\) exhibits (X,L) as a scroll over a smooth curve, the natural morphism \(\phi^*\phi_*(K_ X\otimes L^{n-1})\to K_ X\otimes L^{n-1}\) is onto. This allows them to construct a normal quasi-projective space \(X'\) and algebraic morphisms with connected fibres \(\Phi: X\to X'\), \(\phi ': X'\to Y\) such that \(\phi =\phi '\circ \Phi\) and \(K_ X\otimes L^{n-1}=\Phi^*{\mathcal L}\), where \({\mathcal L}\) is a line bundle on \(X'\), which is ample and spanned relatively to \(\phi '\). If \(\dim (X')<\dim (X)\) then there is a precise description of \(\phi\), while if \(\dim (X')=\dim (X)\) then \(\Phi\) defines a sort of relative reduction \((X',L')\) of (X,L), up to which, the authors prove that \(K_ X\otimes L^{n-1}\) is very ample relatively to \(\phi\).
Reviewer: A.Lanteri

MSC:

14C20 Divisors, linear systems, invertible sheaves
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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