Brauer Groups and Obstruction Problems

The contributions in this book explore various contexts in which the derived category of coherent sheaves on a variety determines some of its arithmetic. This setting provides new geometric tools for interpreting elements of the Brauer group. With a view towards future arithmetic applications, the book extends a number of powerful tools for analyzing rational points on elliptic curves, e. g. , isogenies among curves, torsion points, modular curves, and the resulting descent techniques, as well as higher-dimensional varieties like K3 surfaces. Inspired by the rapid recent advances in our understanding of K3 surfaces, the book is intended to foster cross-pollination between the fields of complex algebraic geometry and number theory.

Contributors:

· Nicolas Addington

· Kenneth Ascher

· Fedor Bogomolov

· Jean-Louis Colliot-Thélène

· Krishna Dasaratha

· Brendan Hassett

· Colin Ingalls

· Emanuele Macrì

· Kelly McKinnie

· Andrew Obus

· Ekin Ozman

· Raman Parimala

· Alexander Perry

· Alena Pirutka

· Justin Sawon

· Alexei N. Skorobogatov

· Paolo Stellari

· Hugh Thomas

· Yuri Tschinkel

· Anthony Várilly-Alvarado

· Bianca Viray

· Rong Zhou

The Brauer group is not a derived invariant. - Twisted derived equivalences for affine schemes. - Rational points on twisted K3 surfaces and derived equivalences. - Universal unramified cohomology of cubic fourfolds containing a plane. - Universal spaces for unramified Galois cohomology. - Rational points on K3 surfaces and derived equivalence. - Unramified Brauer classes on cyclic covers of the projective plane. - Arithmetically Cohen-Macaulay bundles on cubic fourfolds containing a plane. - Brauer groups on K3 surfaces and arithmetic applications. - On a local-global principle for
H³
of function fields of surfaces over a finite field. - Cohomology and the Brauer group of double covers.