The Brauer Group of a Surface over a Finite Field

TL;DR

This paper investigates the structure of the Brauer group of a smooth projective surface over a finite field, proving that a certain 2-primary component has a square order under specific geometric and lifting conditions.

Contribution

It establishes that the order of the 2-primary part of the Brauer group's divisible quotient is a perfect square given certain assumptions on the surface's geometry and liftability.

Findings

The 2-primary component of the Brauer group's divisible quotient has a square order.

No 2-torsion in the Néron-Severi group of the surface.

The result relies on Wu's Theorem and previous constructions by the author.

Abstract

This is an English translation of the author's 1989 note in Russian, published in a collection "Arithmetic and Geometry of Varieties" (V.E. Voskresenski, ed.), Kuibyshev State University, Kuibyshev, 1989, pp. 57--67. Let $X$ be be an absolutely irreducible smooth projective surface over a finite field $k$ of odd characteristic, let $Br(X)$ be the (commutative periodic) Brauer group of $X$ and $DIV Br(X)$ the subgroup of its divisible elements. We write $Br(X)_{DIV}$ for the quotient $Br(X)/DIV Br(X)$ and $Br(X)_{DIV}(2)$ for its (finite) $2$-primary component. We prove that the order of $Br(X)_{DIV}(2)$ is a full square under the following additional assumptions on $\bar{X}=X\times \bar{k}$ where $ \bar{k}$ is an algebraic closure of $k$. There is no 2-torsion in the N\'eron-Severi group of $\bar{X}$. The surface $\bar{X}$ admits a lifting to characteristic 0. The proof is based on constructions of author's paper (Math. USSR Izv. 20 (1983), 203-234) and Wu's Theorem that relates Stiefel-Whitney classes and Steenrod squares.