On the Brauer Group of an Arithmetic Model of a Variety over a Global Field of Positive Characteristic

Abstract

Let V be a smooth projective variety over a global field k = κ(C) of rational functions on a smooth projective curve C over a finite field Fq of characteristic p. Assume that there is a projective flat Fq-morphism π : X → C, where X is a smooth projective variety and the generic scheme fiber of π is isomorphic to a variety V (we call π : X → C an arithmetic model of a variety V ). M. Artin conjectured the finiteness of the Brauer group Br(X) classifying sheaves of Azumaya algebras on X modulo similitude. It is well known that the group Br(X) is contained in the cohomological Brauer group Br′(X)=H2(X,G ). et m By definition, the non−p component of the cohomological Brauer group Br′(X) coincides with the direct sum of the l-primary components of the group 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 Br(k) → Br′(V ). The finiteness of the non−p component of the cohomological Brauer group Br′(X) of a variety X has been proved if [Br′(V )/ Im[Br(k) → Br′(V )]](non −p) is finite. In particular, if V is a 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: Ω2V = OV ) and the characteristic of the ground field p > 2, then, by the Skorobogatov – Zarhin theorem, [Br′(V )/ Im[Br(k) → Br′(V )]](non −p) is finite, so in this case the groups Br′(X)(non−p) and Br(X)(non−p) are finite.

Keywords

• brauer group
• arithmetic model
• k 3 surface