The Noether--Lefschetz theorem in arbitrary characteristic

Abstract

We show that if $X\subset {\mathbb{P}}_k^N$ is a normal variety of dimension $\ge 3$ and $H\subset {\mathbb{P}}_k^N$ a very general hypersurface of degree $d=4$ or $\ge 6$, then the restriction map $\mathrm{Cl}(X)\to \mathrm{Cl}(X\cap H)$ is an isomorphism up to torsion. If $\dim X\ge 4$, the result holds for $d\ge 2$. The proof uses the relative Jacobian of a curve fibration, together with a specialization argument, and the result holds over fields of arbitrary characteristic.

32 pages. v2: New title and improved exposition. To appear in J. Algebraic Geom

References in corpus (1)

• The Tate conjecture for K3 surfaces over finite fields