Torsors, Étale Homotopy and Applications to Rational Points

doi:10.1017/CBO9781139525350

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.

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Contents

Frontmatter
pp i-iv
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Contents
pp v-vi
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List of contributors
pp vii-viii
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Preface
pp ix-x
- By Alexei Skorobogatov, Imperial College London
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PART ONE - LECTURE NOTES
pp 1-2
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1 - Three lectures on Cox rings
pp 3-60
- By J. Hausen, Universität Tübingen
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2 - A very brief introduction to étale homotopy
pp 61-74
- By T. M. Schlank, The University of Jerusalem, A. N. Skorobogatov, Imperial College London
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3 - Torsors and representation theory of reductive groups
pp 75-120
- By V. Serganova, University of California
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PART TWO - CONTRIBUTED PAPERS
pp 121-122
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4 - Torsors over Luna strata
pp 123-137
- By I. V. Arzhantsev, Moscow State University
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5 - Abélianisation des espaces homogènes et applications arithmétiques
pp 138-209
- By C. Demarche, Université Pierre et Marie Curie
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6 - Gaussian rational points on a singular cubic surface
pp 210-230
- By U. Derenthal, Universität München, F. Janda, Zürich, Switzerland
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7 - Actions algébriques de groupes arithmétiques
pp 231-249
- By P. Gille, France, L. Moret-Bailly, Université de Rennes 1
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8 - Descent theory for open varieties
pp 250-279
- By D. Harari, Université Paris-Sud, A. N. Skorobogatov, Imperial College London
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9 - Homotopy obstructions to rational points
pp 280-413
- By Y. Harpaz, The Hebrew University of Jerusalem, T. M. Schlank, The Hebrew University of Jerusalem
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10 - Factorially graded rings of complexity one
pp 414-428
- By J. Hausen, Universität Tübingen, E. Herppich, Universität Tübingen
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11 - Nef and semiample divisors on rational surfaces
pp 429-446
- By A. Laface, Universidad de Concepción, D. Testa, University of Warwick
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12 - Example of a transcendental 3-torsion Brauer-Manin obstruction on a diagonal quartic surface
pp 447-459
- By T. Preu, Universität Zürich
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Torsors, Étale Homotopy and Applications to Rational Points

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Contents

Frontmatter
pp i-iv

Contents
pp v-vi

List of contributors
pp vii-viii

Preface
pp ix-x

PART ONE - LECTURE NOTES
pp 1-2

1 - Three lectures on Cox rings
pp 3-60

2 - A very brief introduction to étale homotopy
pp 61-74

3 - Torsors and representation theory of reductive groups
pp 75-120

PART TWO - CONTRIBUTED PAPERS
pp 121-122

4 - Torsors over Luna strata
pp 123-137

5 - Abélianisation des espaces homogènes et applications arithmétiques
pp 138-209

6 - Gaussian rational points on a singular cubic surface
pp 210-230

7 - Actions algébriques de groupes arithmétiques
pp 231-249

8 - Descent theory for open varieties
pp 250-279

9 - Homotopy obstructions to rational points
pp 280-413

10 - Factorially graded rings of complexity one
pp 414-428

11 - Nef and semiample divisors on rational surfaces
pp 429-446

12 - Example of a transcendental 3-torsion Brauer-Manin obstruction on a diagonal quartic surface
pp 447-459

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