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Full Description
This book is devoted to arithmetic geometry with special attention given to the unramified Brauer group of algebraic varieties and its most striking applications in birational and Diophantine geometry. The topics include Galois cohomology, Brauer groups, obstructions to stable rationality, Weil restriction of scalars, algebraic tori, the Hasse principle, Brauer-Manin obstruction, and etale cohomology. The book contains a detailed presentation of an example of a stably rational but not rational variety, which is presented as series of exercises with detailed hints. This approach is aimed to help the reader understand crucial ideas without being lost in technical details. The reader will end up with a good working knowledge of the Brauer group and its important geometric applications, including the construction of unirational but not stably rational algebraic varieties, a subject which has become fashionable again in connection with the recent breakthroughs by a number of mathematicians.
Contents
Preliminaries on Galois cohomology: Group Cohomology
Galois Cohomology
Brauer group: Brauer Group of a Field
Residue Map on a Brauer Group
Applications to rationality problems: Example of a Unirational Non-rational Variety
Arithmetic of Two-dimensional Quadratics
Non-rational Double Covers of $\mathbb{P}^3$
Weil Restriction and Algebraic Tori
Example of a Non-rational Stably Rational Variety
Hasse principle and its failure: Minkowski-Hasse Theorem
Brauer-Manin Obstruction
Etale Cohomology
Bibliography
Index
