Torsors, Étale Homotopy and Applications to Rational Points London Mathematical Society Lecture Note Series

Torsors, also known as principal bundles or principal homogeneous spaces, are ubiquitous in mathematics. The purpose of this book is to present expository lecture notes and cutting-edge research papers on the theory and applications of torsors and étale homotopy, all written from different perspectives by leading experts. Part one of the book contains lecture notes on recent uses of torsors in geometric invariant theory and representation theory, plus an introduction to the étale homotopy theory of Artin and Mazur. Part two of the book features a milestone paper on the étale homotopy approach to the arithmetic of rational points. Furthermore, the reader will find a collection of research articles on algebraic groups and homogeneous spaces, rational and K3 surfaces, geometric invariant theory, rational points, descent and the Brauer?Manin obstruction. Together, these give a state-of-the-art view of a broad area at the crossroads of number theory and algebraic geometry.

List of contributors; Preface; Part I. Lecture Notes: 1. Three lectures on Cox rings Jürgen Hausen; 2. A very brief introduction to étale homotopy Tomer M. Schlank and Alexei N. Skorobogatov; 3. Torsors and representation theory of reductive groups Vera Serganova; Part II. Contributed Papers: 4. Torsors over luna strata Ivan V. Arzhantsev; 5. Abélianisation des espaces homogènes et applications arithmétiques Cyril Demarche; 6. Gaussian rational points on a singular cubic surface Ulrich Derenthal and Felix Janda; 7. Actions algébriques de groupes arithmétiques Philippe Gille and Laurent Moret-Bailly; 8. Descent theory for open varieties David Harari and Alexei N. Skorobogatov; 9. Factorially graded rings of complexity one Jürgen Hausen and Elaine Herppich; 10. Nef and semiample divisors on rational surfaces Antonio Laface and Damiano Testa; 11. Example of a transcendental 3-torsion Brauer–Manin obstruction on a diagonal quartic surface Thomas Preu; 12. Homotopy obstructions to rational points Yonatan Harpaz and Tomer M. Schlank.

Alexei Skorobogatov is a Professor of Pure Mathematics at Imperial College London. He is the author of Torsors and Rational Points (Cambridge University Press, 2001) and about 60 research papers on arithmetic and algebraic geometry and error-correcting codes. In 2001 he was awarded the Whitehead Prize by the London Mathematical Society.