Tevelev Projective Duality and Homogeneous Spaces

2005
ISBN: 978-3-540-26957-1
Verlag: Springer Berlin Heidelberg
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 133, 250 Seiten

Reihe: Encyclopaedia of Mathematical Sciences

Projective duality is a very classical notion naturally arising in various areas of mathematics, such as algebraic and differential geometry, combinatorics, topology, analytical mechanics, and invariant theory, and the results in this field were until now scattered across the literature. Thus the appearance of a book specifically devoted to projective duality is a long-awaited and welcome event. Projective Duality and Homogeneous Spaces covers a vast and diverse range of topics in the field of dual varieties, ranging from differential geometry to Mori theory and from topology to the theory of algebras. It gives a very readable and thorough account and the presentation of the material is clear and convincing. For the most part of the book the only prerequisites are basic algebra and algebraic geometry. This book will be of great interest to graduate and postgraduate students as well as professional mathematicians working in algebra, geometry and analysis.

Autoren/Hrsg.

Tevelev, Evgueni A.

Inhaltsverzeichnis

1;Preface;5
2;Contents;10
3;Introduction to Projective Duality;14
3.1;1.1 Projectively Dual Varieties;14
3.2;1.2 Dual Plane Curves;15
3.3;1.3 Reflexivity Theorem;19
3.4;1.4 Projections and Linear Normality;24
4;Actions with Finitely Many Orbits;30
4.1;2.1 Algebraic Groups;30
4.2;2.2 Pyasetskii Pairing and Kashin Examples;45
4.3;2.3 Actions Related to Gradings;48
5;Local Calculations;70
5.1;3.1 Calculations in Coordinates;70
5.2;3.2 Fundamental Forms;77
6;Projective Constructions;86
6.1;4.1 Gauss Map;86
6.2;4.2 Tangents and Secants;87
6.3;4.3 Zak Theorems;93
7;Vector Bundles Methods;102
7.1;5.1 Dual Varieties of Smooth Divisors;102
7.2;5.2 Ample Vector Bundles;107
7.3;5.3 Cayley;110
8;Degree of the Dual Variety;122
8.1;6.1 Katz-Kleiman-Holme Formula;122
8.2;6.2 Formulas Related to the Cayley Method;128
9;Varieties with Positive Defect;131
9.1;7.1 Normal Bundle of the Contact Locus;131
9.2;7.2 Linear Systems of Quadrics of Constant Rank;142
9.3;7.3 Defect and Nef Value;148
9.4;7.4 Flag Varieties with Positive Defect;159
10;Dual Varieties of Homogeneous Spaces;166
10.1;8.1 Calculations of deg X*;166
10.2;8.2 Matsumura-Monsky Theorem;180
10.3;8.3 Discriminants of Commutative Algebras;182
10.4;8.4 Discriminants of Anticommutative Algebras;185
10.5;8.5 Adjoint Varieties;197
10.6;8.6 Homogeneous Vector Bundles;200
11;Self-dual Varieties;217
11.1;9.1 Smooth Self-dual Varieties;217
11.2;9.2 Self-Dual Nilpotent Orbits;223
12;Singularities of Dual Varieties;228
12.1;10.1 Class Formula;228
12.2;10.2 Singularities of X*;231
13;References;241
14;Index;252