Quartic del Pezzo surfaces with a Brauer group of order 4

TL;DR

This paper investigates degree 4 del Pezzo surfaces with maximal Brauer group order, demonstrating conditions for rational points and analyzing Brauer group structures related to the Hasse principle.

Contribution

It provides new insights into the arithmetic of quartic del Pezzo surfaces with large Brauer groups, including examples where the Brauer-Manin obstruction explains failures of the Hasse principle.

Findings

Surfaces with conic fibrations always have rational points.

Brauer groups are not vertical with respect to any projection from a plane.

First examples where Brauer-Manin obstruction explains Hasse principle failure.

Abstract

We study arithmetic properties of del Pezzo surfaces of degree 4 for which the Brauer group has the largest possible order using different fibrations into curves. We show that if such a surface admits a conic fibration, then it always has a rational point. We also answer a question of V\'arilly-Alvarado and Viray by showing that the Brauer groups these surfaces cannot be vertical with respect to any projection away from a plane. We conclude that the available techniques for proving existence of rational points or even Zariski density do not directly apply if there is no Brauer-Manin obstruction to the Hasse principle. In passing we pick up the first examples of quartic del Pezzo surfaces with a Brauer group of order 4 for which the failure of the Hasse principle is explained by a Brauer-Manin obstruction.