The Noether-Lefschetz Theorem for the Divisor Class Group

Let X be a normal projective threefold over a field of characteristic zero and |L| be a base-point free, ample linear system on X. Under suitable hypotheses on (X, |L|), we prove that for a very general member Y ∈ |L|, the restriction map on divisor class groups Cl (X) → Cl (Y) is an isomorphism. In particular, we are able to recover the classical Noether-Lefschetz theorem, that a very general hypersurface X ⊂ P^3 C of degree ≥ 4 has Pic (X) ∼= Z.