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arXiv · 2205.05287

Sur la conjecture de Tate pour les diviseurs

Bruno Kahn
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Abstract

We prove that the Tate conjecture in codimension $1$ over a finitely generated field follows from the same conjecture for surfaces over its prime subfield. In positive characteristic, this is due to de Jong--Morrow over $\mathbf{F}_p$ and to Ambrosi for the reduction to $\mathbf{F}_p$. We give a different proof than Ambrosi's, which also works in characteristic $0$; over $\mathbf{Q}$, the reduction to surfaces follows from a simple argument using Lefschetz's $(1,1)$ theorem.

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Bruno Kahn. 2022-05-11. Sur la conjecture de Tate pour les diviseurs. https://doi.org/10.2140/ent.2023.2.83

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