Néron--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 $\mathrm{char}k$, 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 ${\mathrm{H}}_{\acute{\mathrm{e}}\mathrm{t}}^2(\overline{X},{\mathbb{Z}}_{\ell }(1))$ Zariski-locally trivial if it vanishes on a Zariski-open cover of $\overline{X}$. We prove that the first Chern class map identifies $\mathrm{NS}(\overline{X})\otimes {\mathbb{Z}}_{\ell }$ with the group of Zariski-locally trivial classes whose image in ${\mathrm{H}}_{\acute{\mathrm{e}}\mathrm{t}}^2(\overline{X},{\mathbb{Q}}_{\ell }(1))$ 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.

27 pp, v2: Minor corrections, added Acknowledgment, Abstract renders correctly