Moishezon's theorem and degeneration

Abstract

Let L be a tensor product of two very ample line bundles on the smooth projective complex threefold X. Under the hypothesis that H0(X,KX(L))≠0, we show that the restriction map r:PicX→PicY is an isomorphism for very general Y in the linear system |L|. For such L this result recovers the Noether-Lefschetz theorem of Moishezon, who extended the original topological and Hodge-theoretic arguments of Lefschetz. We give a degeneration argument which is almost completely algebraic in nature.