ISBN: 3-540-66766-0
TITLE: Power Sums, Gorenstein Algebras, and Determinantal Loci
AUTHOR: Iarrobino, Anthony; Kanev, Vassil
TOC:

Introduction: Informal History and Brief Outline xiii
0.1. Canonical forms, and catalecticant matrices of higher partial derivatives of a form xiii
0.2. Apolarity and Artinian Gorenstein algebras xviii
0.3. Families of sets of points xxi
0.4. Brief summary of chapters xxii
Part I. Catalecticant Varieties 1
Chapter 1. Forms and Catalecticant Matrices 3
1.1. Apolarity and catalecticant varieties: the dimensions of the vector spaces of higher partials 3
1.2. Determinantal loci of the first catalecticant, the Jacobian 16
1.3. Binary forms and Hankel matrices 22
1.4. Detailed summary and preparatory results 41
Chapter 2. Sums of Powers of Linear Forms, and Gorenstein Algebras 57
2.1. Waring's problem for general forms 57
2.2. Uniqueness of additive decompositions 62
2.3. The Gorenstein algebra of a homogeneous polynomial 67
Chapter 3. Tangent Spaces to Catalecticant Schemes 73
3.1. The tangent space to the determinantal scheme V_s(u, v; r) of the catalecticant matrix 73
3.2. The tangent space to the scheme Gor(T) parametrizing forms with fixed dimensions of the partials 79
Chapter 4. The Locus P S(s, j; r) of Sums of Powers, and Determinantal Loci of Catalecticant Matrices 91
4.1. The case r = 3 92
4.2. Sets of s points in P^r-1 and Gorenstein ideals 102
4.3. Gorenstein ideals whose lowest degree generators are a complete intersection 108
4.4. The smoothness and dimension of the scheme Gor(T) when r = 3: a survey 116
Part II. Catalecticant Varieties and the Punctual Hilbert Scheme 129
Chapter 5. Forms and Zero-Dimensional Schemes I: Basic Results, and the Case r = 3 131
5.1. Annihilating scheme in P^r-1 of a form 135
5.2. Flat families of zero-dimensional schemes and limit ideals 142
5.3. Existence theorems for annihilating schemes when r = 3 150
5.3.1. The generator and relation strata of the variety Gor(T) parametrizing Gorenstein algebras 151
5.3.2. The morphism from Gor(T): the case T D(s, s, s) 156
5.3.3. Morphism: the case T D(s - a, s, s, s - a) 167
5.3.4. Morphism: the case T D(s - a, s, s - a) 172
5.3.5. Adimension formula for the variety Gor(T) 179
5.4. Power sum representations in three and more variables 182
5.5. Betti strata of the punctual Hilbert scheme 189
5.6. The length of a form, and the closure of the locus PS(s, j; 3) of power sums 197
5.7. Codimension three Gorenstein schemes in P^n 201
Chapter 6. Forms and Zero-Dimensional Schemes, II: Annihilating Schemes and Reducible Gor(T) 207
6.1. Uniqueness of the annihilating scheme; closure of P S(s, J; r) 208
6.2. Varieties Gor(T), T = T (j, r), with several components 214
6.3. Other reducible varieties Gor(T) 224
6.4. Locally Gorenstein annihilating schemes 226
Chapter 7. Connectedness and Components of the Determinantal Locus PV_s(u, v; r) 237
7.1. Connectedness of PV_s(u, v; r) 237
7.2. The irreducible components of V_s(u, v; r) 241
7.3. Multisecant varieties of the Veronese variety 245
Chapter 8. Closures of the Variety Gor(T), and the Parameter Space G(T ) of Graded Algebras 249
Chapter 9. Questions and Problems 255
Appendix A. Divided Power Rings and Polynomial Rings 265
Appendix B. Height Three Gorenstein Ideals 271
B.1. Pfaffian formulas 272
B.2. Resolutions of height 3 Gorenstein ideals and their squares 276
B.3. Resolutions of annihilating ideals of power sums 280
B.4. Maximum Betti numbers, given T 282
Appendix C. The Gotzmann Theorems and the Hilbert Scheme (Anthony Iarrobino and Steven L. Kleiman) 289
C.1. Order sequences and Macaulay's Theorem on Hilbert functions 290
C.2. Macaulay and Gotzmann polynomials 293
C.3. Gotzmann's Persistence Theorem and m-Regularity 297
C.4. The Hilbert scheme Hilb^P(P^r-1) 302
C.5. Gorenstein sequences having a subsequence of maximal growth, and Hilb^P(P^r-1) 307
Appendix D. Examples of "Macaulay" Scripts 313
Appendix E. Concordance with the 1996 Version 317
References 319
Index 335
Index of Names 341
Index of Notation 343
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