Genus one fibrations and vertical Brauer elements on del Pezzo surfaces of degree 4

TL;DR

This paper investigates the geometric and arithmetic properties of a specific genus one fibration with reducible fibers on degree four del Pezzo surfaces, focusing on the role of vertical Brauer elements.

Contribution

It introduces a detailed analysis of genus one fibrations with reducible fibers on del Pezzo surfaces of degree four, highlighting the behavior of vertical Brauer elements.

Findings

Identification of conditions for reducible fibers in genus one fibrations

Characterization of vertical Brauer elements on these surfaces

Insights into the arithmetic implications of these structures

Abstract

We consider a family of smooth del Pezzo surfaces of degree four and study the geometry and arithmetic of a genus one fibration with two reducible fibres for which a Brauer element is vertical.