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 of degree ⩾4 has Pic(X)≅Z.