Projective Duality and Homogeneous Spaces (eBook)

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