On the algebraic Brauer classes on open degree four del Pezzo surfaces

TL;DR

This paper investigates the structure of algebraic Brauer classes on open degree four del Pezzo surfaces, revealing the generators of 2-torsion and 4-torsion classes and methods for their evaluation over p-adic fields.

Contribution

It characterizes the generators of algebraic Brauer classes on open degree four del Pezzo surfaces and discusses evaluation methods over p-adic fields.

Findings

2-torsion classes are generated by two types.

Two types of 4-torsion classes identified.

Evaluation methods for these classes over p-adic fields are discussed.

Abstract

We study the algebraic Brauer classes on open del Pezzo surfaces of degree $4$. I.e., on the complements of geometrically irreducible hyperplane sections of del Pezzo surfaces of degree $4$. We show that the $2$-torsion part is generated by classes of two different types. Moreover, there are two types of $4$-torsion classes. For each type, we discuss methods for the evaluation of such a class at a rational point over a $p$-adic field.