Liaison, Schottky Problem and Invariant Theory: Remembering Federico Gaeta

Cover......Page 1
Progress in Mathematics\rVolume 280......Page 3
Liaison,\rSchottky Problem\rand Invariant Theory......Page 4
ISBN 9783034602006......Page 5
Table of Contents \r......Page 6
Preface......Page 8
Part I\rFederico Gaeta......Page 10
Federico Gaeta, Among the Last Classics......Page 12
2. The beginnings......Page 18
3. The fellowship in Rome......Page 20
4. The theory of “liaison”......Page 21
5. Elliptic surfaces......Page 25
6. The Istituto Nazionale di Alta Matematica......Page 26
7. The chair in Zaragoza......Page 31
8. Exile and return......Page 35
9. A few personal considerations and memories......Page 37
Acknowledgment......Page 38
References......Page 39
Articles Published by Federico Gaeta......Page 44
Part II: Linkage Theory......Page 48
Gaeta’s Work on Liaison Theory:An Appreciation......Page 50
References......Page 56
Introduction......Page 58
1. Ideals of minors of a symmetric matrix......Page 59
References......Page 70
Liaison Invariants and the Hilbert Schemeof Codimension 2 Subschemes in IPn+2......Page 72
1. Introduction......Page 73
2. Notations and terminology......Page 75
3. The dimension of H(d, g) and biliaison invariants......Page 76
4. The dimension and the smoothness of H(d, p, π)......Page 80
5. The smoothness of the “morphism” ϕ : Hγ,ρ → Vρ......Page 84
6. The tangent space of Hγ,ρ......Page 89
7. Linkage of surfaces......Page 91
8. Obstructed surfaces in IP4......Page 95
9. Even liaison of codimension 2 subschemes of IPn+2......Page 99
References......Page 108
Minimal Links and a Result of Gaeta......Page 112
1. Introduction......Page 113
2. Some facts from liaison theory......Page 117
3. Minimal linkage does not necessarily give minimal elements in codimension two......Page 120
4. Hypersurface sections......Page 127
5. A non-arithmetically Cohen-Macaulay extension of Gaeta’s theorem......Page 130
6. Gorenstein ideals of height three......Page 134
References......Page 140
1. Introduction......Page 142
2. Preliminaries......Page 144
3. Proofs of the main results......Page 146
4. Cyclic Hartshorne-Rao modules and Rao functions of maximal rank curves......Page 150
References......Page 155
1. Introduction......Page 158
2. Construction of double rational normal curves......Page 161
3. Cohomology estimates......Page 169
4. Arithmetically Gorenstein double rational normal curves......Page 175
5. Double conics......Page 181
References......Page 195
Part III: The Schottky Problem......Page 197
1. The statement of the Schottky problem......Page 200
2. Characterization of Jacobians in terms of the existence of trisecants......Page 201
3. The Γ 00-conjecture......Page 202
References......Page 204
1. Introduction......Page 206
2. Construction of the wave function......Page 212
3. Commuting difference operators......Page 217
References......Page 230
A Special Case of the Γ00 Conjecture......Page 232
1. Γ00 conjecture as a condition on the Kummer variety......Page 233
2. Γ00 conjecture as a difference-differential equation on the theta divisor......Page 236
References......Page 239
Part IV: Computation in Algebraic Geometry......Page 241
Federico Gaeta: His Last Ten Years of Mathematical Activity......Page 244
1. Introduction......Page 246
2. Preliminaries and main results......Page 247
3. Gaeta’s tangential and Gaeta’s covariant......Page 249
3.2. Gaeta’s tangential......Page 250
3.3. Gaeta’s covariant......Page 252
4.1. The apolar covariant......Page 254
4.2. Gordan’s presentation......Page 256
5.1. Computing sequentially the coefficients from the source......Page 259
5.2. Removing redundancies: expression in the brackets......Page 260
5.4. A toy example: ternary quadratic forms......Page 261
5.5. Still more redundancies......Page 263
References......Page 264
Symmetric Functions and Secant Spaces of Rational Normal Curves......Page 266
1. Historical summary......Page 267
2. Introduction and some prerequisites......Page 268
3. First definitions......Page 273
4. The nth symmetric power of the projective lines P1 and P∨......Page 274
5. Orbits, intrinsic definition of the rational normal curves Rn......Page 275
6. Duality......Page 276
7. Osculating spaces to Rn......Page 278
8. Updating Clifford’s theorem in terms of divisors in the line......Page 280
9. Affine properties of the Rn......Page 282
10. Agreement with Macdonald’s approach......Page 283
11. Ubiquity of the Schur functions and the F-coefficients via RN......Page 286
12. The variety Σ(n;N) of n-secant spaces of RN and its dual......Page 289
13. The Hook-Schur functions as local Grassmann coordinates in Σ......Page 291
14. Σ(n;2n−1) and a new geometrical meaning of the h functions......Page 292
15. The jth column entries of S as coefficients of the remainder of xn 1+j mod F and the power sums......Page 293
16. Jacobi-Trudi and Naegelsbach formulas without calculations......Page 294
17. Equations of Σ(n;N) inside G(n − 1;N) and the symmetric functions......Page 295
18. Rota’s “confluent symmetric functions”......Page 297
References......Page 299