The Noether--Lefschetz theorem in arbitrary characteristic
Abstract
We show that if $X\subset\mathbb P^N_k$ is a normal variety of dimension $\geq 3$ and $H\subset\mathbb P^N_k$ a very general hypersurface of degree $d=4$ or $\geq 6$, then the restriction map $\mathrm{Cl}(X)\to\mathrm{Cl}(X\cap H)$ is an isomorphism up to torsion. If $\dim X\geq 4$, the result holds for $d\geq 2$. The proof uses the relative Jacobian of a curve fibration, together with a specialization argument, and the result holds over fields of arbitrary characteristic.
- Publication:
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arXiv e-prints
- Pub Date:
- July 2021
- DOI:
- 10.48550/arXiv.2107.12962
- arXiv:
- arXiv:2107.12962
- Bibcode:
- 2021arXiv210712962J
- Keywords:
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- Mathematics - Algebraic Geometry;
- 14C20 (Primary);
- 14C22;
- 14K30 (Secondary)
- E-Print:
- 32 pages. v2: New title and improved exposition. To appear in J. Algebraic Geom