Abstract
AbstractLet S be a smooth surface contained as an ample divisor in a smooth complex projective threefold X, which is a P1 -bundle, and assume that induces OP1 (1) on the fibres of X. The following fact is proven. The restriction to S of the bundle projection of X is exactly the reduction morphism of the pair provided that this one is not a conic bundle. The proof is very simple and does not involve any consideration on the nefness of the adjoint bundle Some applications of the proof are given.
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