Néron--Severi groups of proper schemes over finite fields

2026/07/13 by K. V. Shuddhodan, V. Srinivas
Mathematics · #math.AG

paper · pdf

Abstract

Let X be a proper reduced scheme over a finite field k, let ℓ be a prime different from char k, and write X=X×_kk for its base change to an algebraic closure k of k. Call a class in H²_\mathrm\acuteet(X,ℤ_ℓ(1)) Zariski-locally trivial if it vanishes on a Zariski-open cover of X. We prove that the first Chern class map identifies NS(X)⊗ℤ_ℓ with the group of Zariski-locally trivial classes whose image in H²_\mathrm\acuteet(X,ℚ_ℓ(1)) has weight zero. This is the finite-field analogue of a theorem of Barbieri-Viale--Rosenschon--Srinivas for proper seminormal complex varieties. In the finite-field setting neither seminormality nor irreducibility is needed.

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