The Noether–Lefschetz theorem for the divisor class group

Abstract

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 C 3 of degree ⩾4 has Pic ( X ) ≅ Z .