N\'eron--Severi groups of proper schemes over finite fields

Abstract

Let $X$ be a proper reduced scheme over a finite field $k$, let $\ell$ be a prime different from $\operatorname{char} k$, and write $\overline{X}=X\times_{k}\overline{k}$ for its base change to an algebraic closure $\overline{k}$ of $k$. Call a class in $\mathrm{H}^{2}_{\mathrm{\acute{e}t}}(\overline{X},\mathbb{Z}_{\ell}(1))$ Zariski-locally trivial if it vanishes on a Zariski-open cover of $\overline{X}$. We prove that the first Chern class map identifies $\operatorname{NS}(\overline{X})\otimes\mathbb{Z}_{\ell}$ with the group of Zariski-locally trivial classes whose image in $\mathrm{H}^{2}_{\mathrm{\acute{e}t}}(\overline{X},\mathbb{Q}_{\ell}(1))$ 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.