The Noether–Lefschetz theorem in arbitrary characteristic

Abstract

We show that if X ⊂ P k N X\subset \mathbb P^N_k is a normal variety of dimension ≥ 3 \geq 3 and H ⊂ P k N H\subset \mathbb P^N_k a very general hypersurface of degree d = 4 d=4 or ≥ 6 \geq 6 , then the restriction map Cl ⁡ ( X ) → Cl ⁡ ( X ∩ H ) \operatorname {Cl}(X)\to \operatorname {Cl}(X\cap H) is an isomorphism up to torsion. If dim ⁡ X ≥ 4 \dim X\geq 4 , the result holds for d ≥ 2 d\geq 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.