Liaison, Schottky Problem and Invariant Theory (eBook)

Title Page 3
Copyright Page 4
Table of Contents 5
Preface 7
Part I Federico Gaeta 8
Federico Gaeta, Among the Last Classics 10
Federico Gaeta and His Italian Heritage 15
1. Introduction 15
2. The beginnings 15
3. The fellowship in Rome 17
4. The theory of “liaison” 18
5. Elliptic surfaces 22
6. The Istituto Nazionale di Alta Matematica 23
7. The chair in Zaragoza 28
8. Exile and return 32
9. A few personal considerations and memories 34
Acknowledgment 35
References 36
Articles Published by Federico Gaeta 40
Part II Linkage Theory 44
Gaeta’s Work on Liaison Theory:An Appreciation 46
References 52
Symmetric Ladders and G-biliaison 54
Introduction 54
1. Ideals of minors of a symmetric matrix 55
References 66
Liaison Invariants and the Hilbert Schemeof Codimension 2 Subschemes in IPn+2 68
1. Introduction 69
2. Notations and terminology 71
3. The dimension of H(d, g) and biliaison invariants 72
4. The dimension and the smoothness of H(d, p, p) 76
5. The smoothness of the “morphism” . : H.,. . V. 80
6. The tangent space of H.,. 85
7. Linkage of surfaces 87
8. Obstructed surfaces in IP4 91
9. Even liaison of codimension 2 subschemes of IPn+2 95
References 104
Minimal Links and a Result of Gaeta 107
1. Introduction 108
2. Some facts from liaison theory 112
3. Minimal linkage does not necessarily give minimal elements in codimension two 115
4. Hypersurface sections 122
5. A non-arithmetically Cohen-Macaulay extension of Gaeta’s theorem 125
6. Gorenstein ideals of height three 129
References 135
On the Existence of Maximal Rank Curves with Prescribed Hartshorne-Rao Module 137
1. Introduction 137
2. Preliminaries 139
3. Proofs of the main results 141
4. Cyclic Hartshorne-Rao modules and Rao functions of maximal rank curves 145
References 150
Doubling Rational Normal Curves 152
1. Introduction 152
2. Construction of double rational normal curves 155
3. Cohomology estimates 163
4. Arithmetically Gorenstein double rational normal curves 169
5. Double conics 175
References 189
Part IIIThe Schottky Problem 191
Survey on the Schottky Problem 193
1. The statement of the Schottky problem 193
2. Characterization of Jacobians in terms of the existence of trisecants 194
3. The G 00-conjecture 195
References 197
Abelian Solutions of the Soliton Equations and Geometry of Abelian Varieties 199
1. Introduction 199
2. Construction of the wave function 205
3. Commuting difference operators 210
References 223
A Special Case of the G00 Conjecture 225
1. G00 conjecture as a condition on the Kummer variety 226
2. G00 conjecture as a difference-differential equation on the theta divisor 229
References 232
Part IV Computation in Algebraic Geometry 234
Federico Gaeta: His Last Ten Years of Mathematical Activity 236
Covariants Vanishing on Totally Decomposable Forms 238
1. Introduction 238
2. Preliminaries and main results 239
3. Gaeta’s tangential and Gaeta’s covariant 241
3.1. The polars of a form f 242
3.2. Gaeta’s tangential 242
3.3. Gaeta’s covariant 244
4. Brill’s covariant 246
4.1. The apolar covariant 246
4.2. Gordan’s presentation 248
5. Computations in the ring A 251
5.1. Computing sequentially the coefficients from the source 251
5.2. Removing redundancies: expression in the brackets 252
5.3. Method 253
5.4. A toy example: ternary quadratic forms 253
5.5. Still more redundancies 255
References 256
Symmetric Functions and Secant Spaces of Rational Normal Curves 258
1. Historical summary 259
2. Introduction and some prerequisites 260
3. First definitions 265
4. The nth symmetric power of the projective lines P1 and P. 266
5. Orbits, intrinsic definition of the rational normal curves Rn 267
6. Duality 268
7. Osculating spaces to Rn 270
8. Updating Clifford’s theorem in terms of divisors in the line 272
9. Affine properties of the Rn 274
10. Agreement with Macdonald’s approach 275
11. Ubiquity of the Schur functions and the F-coefficients via RN 278
12. The variety S(n N) of n-secant spaces of RN and its dual281
13. The Hook-Schur functions as local Grassmann coordinates in S 283
14. S(n 2n-1) and a new geometrical meaning of the h functions284
15. The jth column entries of S as coefficients of the remainder of xn 1+j mod F and the power sums 285
16. Jacobi-Trudi and Naegelsbach formulas without calculations 286
17. Equations of S(n N) inside G(n - 1N) and the symmetric functions287
18. Rota’s “confluent symmetric functions” 289
References 291