Unramified Brauer group and its applications

TL;DR

This textbook explores the unramified Brauer group in arithmetic geometry, covering Galois cohomology, rationality obstructions, and applications to algebraic varieties, with detailed problem-solving guidance.

Contribution

It provides a comprehensive, educational treatment of unramified Brauer groups and their applications, including new insights into rationality problems and explicit examples.

Findings

Analysis of Brauer-Manin obstruction to rationality

Examples of stably rational but non-rational varieties

Detailed methods for computing unramified Brauer groups

Abstract

This is a textbook on arithmetic geometry with special regard to unramified Brauer groups of algebraic varieties. The topics include Galois cohomology, Brauer groups, obstructions to stable rationality, arithmetic and geometry of quadrics, Weil restriction of scalars, algebraic tori, an example of a stably rational non-rational variety, Brauer-Manin obstruction. All material is split into locally trivial problems with detailed hints.