Lectures on Invariant Theory,9780521525480

Lectures on Invariant Theory

by
ISBN13: 9780521525480
ISBN10: 0521525489
Format: Paperback
Pub. Date: 2003-08-25
Publisher(s): Cambridge University Press
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Summary

This introduction to the main ideas of algebraic and geometric invariant theory assumes only a minimal background in algebraic geometry, algebra and representation theory. Topics covered include the symbolic method for computation of invariants on the space of homogeneous forms, the problem of finite-generatedness of the algebra of invariants, and the theory of covariants and constructions of categorical and geometric quotients. Throughout, the emphasis is on concrete examples that originate in classical algebraic geometry. Written in an accessible style with many examples and exercises, the book offers a novel discussion of possible linearizations of actions and the variation of quotients under the change of linearization.

Table of Contents

Preface vii
Introduction xiii
1 The symbolic method 1(16)
1.1 First examples
1(3)
1.2 Polarization and restitution
4(6)
1.3 Bracket functions
10(3)
Bibliographical notes
13(1)
Exercises
14(3)
2 The First Fundamental Theorem 17(12)
2.1 The omega-operator
17(3)
2.2 The proof
20(1)
2.3 Grassmann varieties
21(1)
2.4 The straightening algorithm
22(5)
Bibliographical notes
27(1)
Exercises
27(2)
3 Reductive algebraic groups 29(18)
3.1 The Gordan Hilbert Theorem
29(3)
3.2 The unitary trick
32(3)
3.3 Affine algebraic groups
35(6)
3.4 Nagata's Theorem
41(4)
Bibliographical notes
45(1)
Exercises
46(1)
4 Hilbert's Fourteenth Problem 47(18)
4.1 The problem
47(2)
4.2 The Weitzenböck Theorem
49(3)
4.3 Nagata's counterexample
52(10)
Bibliographical notes
62(1)
Exercises
62(3)
5 Algebra of covariants 65(26)
5.1 Examples of covariants
65(4)
5.2 Covariants of an action
69(3)
5.3 Linear representations of reductive groups
72(5)
5.4 Dominant weights
77(2)
5.5 The Cayley-Sylvester formula
79(5)
5.6 Standard tableaux again
84(3)
Bibliographical notes
87(1)
Exercises
88(3)
6 Quotients 91(12)
6.1 Categorical and geometric quotients
91(4)
6.2 Examples
95(3)
6.3 Rational quotients
98(2)
Bibliographical notes
100(1)
Exercises
100(3)
7 Linearization of actions 103(12)
7.1 Linearized line bundles
103(4)
7.2 The existence of linearization
107(3)
7.3 Linearization of an action
110(2)
Bibliographical notes
112(1)
Exercises
113(2)
8 Stability 115(14)
8.1 Stable points
115(2)
8.2 The existence of a quotient
117(4)
8.3 Examples
121(6)
Bibliographical notes
127(1)
Exercises
127(2)
9 Numerical criterion of stability 129(16)
9.1 The function μ (ρ λ)
129(3)
9.2 The numerical criterion
132(1)
9.3 The proof
133(2)
9.4 The weight polytope
135(3)
9.5 Kempf-stability
138(4)
Bibliographical notes
142(1)
Exercises
143(2)
10 Projective hypersurfaces 145(20)
10.1 Nonsingular hypersurfaces
145(2)
10.2 Binary forms
147(6)
10.3 Plane cubics
153(8)
10.4 Cubic surfaces
161(1)
Bibliographical notes
162(1)
Exercises
162(3)
11 Configurations of linear subspaces 165(22)
11.1 Stable configurations
165(6)
11.2 Points in Pn
171(10)
11.3 Lines in P3
181(2)
Bibliographical notes
183(1)
Exercises
184(3)
12 Toric varieties 187(18)
12.1 Actions of a torus on an affine space
187(3)
12.2 Fans
190(6)
12.3 Examples
196(6)
Bibliographical notes
202(1)
Exercises
202(3)
Bibliography 205(10)
Index of Notation 215(2)
Index 217

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