The Noether–Lefschetz theorem in arbitrary characteristic

This is the authors' abstract. We don't add key points for this paper.

We show that if X ⊂<!-- ⊂ --> P k N X⊂ P^N_k is a normal variety of dimension ≥<!-- ≥ --> 3 ≥ 3 and H ⊂<!-- ⊂ --> P k N H⊂ P^N_k a very general hypersurface of degree d = 4 d=4 or ≥<!-- ≥ --> 6 ≥ 6 , then the restriction map Cl ⁡<!-- ⁡ --> ( X ) →<!-- → --> Cl ⁡<!-- ⁡ --> ( X ∩<!-- ∩ --> H ) Cl(X)→ Cl(X∩ H) is an isomorphism up to torsion. If dim ⁡<!-- ⁡ --> X ≥<!-- ≥ --> 4 X≥ 4 , the result holds for d ≥<!-- ≥ --> 2 d≥ 2 . The proof uses the relative Jacobian of a curve fibration, together with a specialization argument, and the result holds over fields of arbitrary characteristic.