Vertical Brauer groups and del Pezzo surfaces of degree 4

TL;DR

This paper demonstrates that Brauer classes on degree 4 del Pezzo surfaces are vertical with respect to a certain projection, providing a practical algorithm for computing the Brauer group and linking Brauer obstructions to fiber solvability.

Contribution

It proves that all Brauer classes are pullbacks from P^1 and introduces a new, practical algorithm for computing the Brauer group of such surfaces.

Findings

Brauer classes are vertical for some projection from the surface.

A Brauer class obstructs rational points iff a fiber is locally solvable.

Provides a constructive, practical algorithm for Brauer group computation.

Abstract

We show that Brauer classes of a locally solvable degree 4 del Pezzo surface X are vertical for some projection away from a plane f: X ---> P^1, i.e., that every Brauer class is obtained by pullback from an element of Br k(P^1). As a consequence, we prove that a Brauer class does not obstruct the existence of a rational point if and only if there exists a fiber of f that is locally solvable. The proof is constructive and gives a simple and practical algorithm, distinct from that in [BBFL07], for computing all classes in the Brauer group of X (modulo constant algebras).