Cubic forms: algebra, geometry, arithmetic

Medientyp: E-Book

Titel: Cubic forms : algebra, geometry, arithmetic

Weitere Titel: Kubičeskie formy <engl.>

Enthält: Front Cover; Cubic Forms: Algebra, Geometry, Arithmetic; Copyright Page; CONTENTS; PREFACE TO THE SECOND EDITION; INSTRUCTIONS TO THE READER; INTRODUCTION; CHAPTER I. CH-QUASIGROUPS AND MOUFANG LOOPS; 1. A survey of definitions and results; 2. Symmetric Abelian quasigroups; 3. CH-quasigroups; 4. Commutative Moufang loops; 5. The connection between CH-quasigroups and Moufang loops; 6. Morphisms of CH-quasigroups and Moufang loops; 7. The first structure theorem; 8. The second structure theorem; 9. Finite Fischer groups; 10. Unsolved problems and bibliographical remarks
CHAPTER II. CLASSES OF POINTS ON CUBIC HYPERSURFACES11. Admissible equivalence relations: a survey; 12. Unirationality; 13. Universal equivalence; 14. R-equivalence: the basic properties; 15. R-equivalence and quadratic extensions; 16. Universal equivalence over local fields. Examples; 17. Bibliographical remarks; CHAPTER III. TWO-DIMENSIONAL BIRATIONAL GEOMETRY; 18. The main results; 19. Monoidal transformations; 20. Monoidal transformations and divisors; 21. The main theorems on birational maps; 22. Bibliographical remarks; CHAPTER IV. THE TWENTY-SEVEN LINES; 23. A survey of the results
24. Del Pezzo surfaces25. The Picard group and root systems; 26. Exceptional curves and Weyl groups; 27. The zeta function; 28. Minimality and classes of conjugate elements in Weyl groups; 29. A cohomological invariant and the degree of unirationality; 30. Rational points; 31. Tables and comments. Calculation of H1. The theorem of Artin and Tate; 32. Bibliographical remarks; CHAPTER V. MINIMAL CUBIC SURFACES; 33. A survey of the results; 34. The fundamental birational invariant; 35. A bubble space; 36. Calculations on cubic surfaces; 37. Birational non-triviality
38. Birational classification39. Relations between the generators; 40. Bibliographical remarks; CHAPTER VI. THE BRAUER-GROTHENDIECK GROUP; 41. A survey of the results. Obstructions to the Hasse principle; 42. The construction of Azumaya algebras; 43. Brauer equivalence; 44. The finiteness theorem; 45. Calculations for Brauer equivalence. Examples; 46. A negative result; 47. Counter-examples to the Hasse principle; 48. Bibliographical remarks; APPENDIX. ALGEBRAIC VARIETIES CLOSE TO THE RATIONAL ONES. ALGEBRA, GEOMETRY, ARITHMETIC; Introduction
1. Galois cohomology, Picard groups and birational geometry2. The Hasse principle and descent on rational varieties; 3. Geometry of rational surfaces. Complements; 4.The Lüroth problem and the Zariski problem in dimension G 3; 5. Rational points and equivalence relations; 6. Cubic surfaces and commutative Moufang loops (CML); References (for the Appendix); REFERENCES; AUTHOR INDEX; LIST OF SYMBOLS; SUBJECT INDEX;

Beteiligte: Manin, Jurij I. [Sonstige Person, Familie und Körperschaft]

Erschienen: Amsterdam; New York; New York, NY, U.S.A: North-Holland, 1986

Ausgabe: 2nd ed.

Umfang: Online Ressource (x, 326 p.)

Sprache: Englisch; Russisch

ISBN: 9780444878236; 9780080963167; 0080963161

RVK-Notation: SK 180 : Zahlentheorie

Schlagwörter: Algebraische Fläche
Kubische Hyperfläche

Entstehung:

Anmerkungen: Translation of: Kubicheskie formy. - Includes indexes. - Includes bibliographical references (p. 313-322). - Translation of. - Description based on print version record

Beschreibung: Front Cover; Cubic Forms: Algebra, Geometry, Arithmetic; Copyright Page; PREFACE TO THE SECOND EDITION; CONTENTS; INSTRUCTIONS TO THE READER; INTRODUCTION; CHAPTER I. CH-QUASIGROUPS AND MOUFANG LOOPS; CHAPTER II. CLASSES OF POINTS ON CUBIC HYPERSURFACES; CHAPTER III. TWO-DIMENSIONAL BIRATIONAL GEOMETRY; CHAPTER IV. THE TWENTY-SEVEN LINES; CHAPTER V. MINIMAL CUBIC SURFACES; CHAPTER VI. THE BRAUER-GROTHENDIECK GROUP; APPENDIX. ALGEBRAIC VARIETIES CLOSE TO THE RATIONAL ONES. ALGEBRA, GEOMETRY, ARITHMETIC; REFERENCES; AUTHOR INDEX; LIST OF SYMBOLS; SUBJECT INDEX

Since this book was first published in English, there has been important progress in a number of related topics. The class of algebraic varieties close to the rational ones has crystallized as a natural domain for the methods developed and expounded in this volume. For this revised edition, the original text has been left intact (except for a few corrections) and has been brought up to date by the addition of an Appendix and recent references. The Appendix sketches some of the most essential new results, constructions and ideas, including the solutions of the Luroth and Zariski problems, the th

Cubic forms : algebra, geometry, arithmetic - [2nd ed.]

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