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Global generation of pluricanonical and adjoint linear series on smooth projective threefolds. (English) Zbl 0803.14004
The purpose of this paper is to show how the cohomological techniques developed by Kawamata, Reid, Shokurov, etc., lead to some effective and practical results of Reider-type on freeness of linear series on smooth complex projective threefolds. Let $$X$$ be such a threefold, let $$h:X \to X_ 0$$ be a surjective birational morphism to the normal projective threefold $$X_ 0$$, and let $$B$$ be a big and nef line bundle on $$X$$ such that $$K_ X+B$$ is a preimage of a line bundle on $$X_ 0$$.
The main theorem 3.2 formulates sufficient numerical conditions under which $${\mathcal O}_ X (K_ X + B)$$ is free at any point of a fixed fiber of $$h$$. Theorem 3.2 implies a series of effective conditions for freeness and globally generatedness of small multiples of $$K_ X$$ and adjoint series. In particular, if $$X$$ is minimal and $$K_ X$$ is nef and big, 3.2 implies that $${\mathcal O}_ X (m,K_ X)$$ is globally generated for $$m \geq 7$$, $${\mathcal O}_ X (6.K_ X)$$ is free if $$K^ 3_ X \geq 2$$, and $${\mathcal O}_ X (5.K_ X)$$ is free if $$K^ 3_ X \geq 26$$ – see theorem 3 [the Fujita’s conjecture cited in the paper implies in particular that $${\mathcal O}_ X(5.K_ X)$$ should be globally generated].
In the end, after involving of some additional argument, the authors prove the following result (in the spirit of the Fujita conjecture): If $$L$$ is ample, then $$K_ X + 4.L$$ is globally generated (corollary $$2^*)$$.
Reviewer: A.Iliev (Sofia)

##### MSC:
 14C20 Divisors, linear systems, invertible sheaves 14J30 $$3$$-folds
##### Keywords:
adjoint divisor; freeness of linear series; threefolds
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##### References:
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