On the Brauer group of an arithmetic model of a variety over a global field of positive characteristic

T. V. Prokhorova

Modelirovanie i analiz informacionnyh sistem, Tome 23 (2016) no. 2, pp. 164-172

Résumé

Let $V$ be a smooth projective variety over a global field $k=\kappa(C)$ of rational functions on a smooth projective curve $C$ over a finite field $\Bbb F_q$ of characteristic $p$. Assume that there is a projective flat $\Bbb F_q$-morphism $\pi:X\to C$, where $X$ is a smooth projective variety and the generic scheme fiber of $\pi$ is isomorphic to a variety $V$ (we call $\pi:X\to C$ an arithmetic model of a variety $V$). M. Artin conjectured the finiteness of the Brauer group $\operatorname{Br}(X)$ classifying sheaves of Azumaya algebras on $X$ modulo similitude. It is well known that the group $\operatorname{Br}(X)$ is contained in the cohomological Brauer group $$\operatorname{Br}'(X)=H^2_{et}(X, {\Bbb G}_m).$$ By definition, the $\operatorname{non}-p$ component of the cohomological Brauer group $\operatorname{Br}'(X)$ coincides with the direct sum of the $l$-primary components of the group $\operatorname{Br}'(X)$ for all prime numbers $l$ different from the characteristic $p$. It is known that the structure of $k$-variety on $V$ yields the canonical morphism of the groups $\operatorname{Br}(k)\to \operatorname{Br}'(V)$. The finiteness of the $\operatorname{non}-p$ component of the cohomological Brauer group $\operatorname{Br}'(X)$ of a variety $X$ has been proved if $$[\operatorname{Br}'(V)/\operatorname{Im}[\operatorname{Br}(k)\to\operatorname{Br}'(V)]](\operatorname{non}-p)$$ is finite. In particular, if $V$ is a $\operatorname{K}3$ surface (in other words, $V$ is a smooth projective simply connected surface over a field $k$ and the canonical class of a surface of $V$ is trivial: $\Omega^2_V=\mathcal O_V$) and the characteristic of the ground field $p > 2$, then, by the Skorobogatov–Zarhin theorem, $[\operatorname{Br}'(V)/\operatorname{Im}[\operatorname{Br}(k)\to\operatorname{Br}'(V)]](\operatorname{non}-p)$ is finite, so in this case the groups $\operatorname{Br}'(X)(\operatorname{non}-p)$ and $\operatorname{Br}(X)(\operatorname{non}-p)$ are finite.

Détail

Keywords: Brauer group, arithmetic model, $\operatorname{K}3$ surface.