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Iarrobino A., Kanev V. — Power Sums, Gorenstein Algebras, and Determinantal Loci
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Название: Power Sums, Gorenstein Algebras, and Determinantal Loci
Авторы: Iarrobino A., Kanev V.
Аннотация: This book treats the theory of representations of homogeneous polynomials as sums of powers of linear forms. The first two chapters are introductory, and focus on binary forms and Waring's problem. Then the author's recent work is presented mainly on the representation of forms in three or more variables as sums of powers of relatively few linear forms. The methods used are drawn from seemingly unrelated areas of commutative algebra and algebraic geometry, including the theories of determinantal varieties, of classifying spaces of Gorenstein-Artin algebras, and of Hilbert schemes of zero-dimensional subschemes. Of the many concrete examples given, some are calculated with the aid of the computer algebra program "Macaulay", illustrating the abstract material. The final chapter considers open problems. This book will be of interest to graduate students, beginning researchers, and seasoned specialists. Prerequisite is a basic knowledge of commutative algebra and algebraic geometry.
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Рубрика: Математика /
Статус предметного указателя: Готов указатель с номерами страниц
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Год издания: 1999
Количество страниц: 345
Добавлена в каталог: 20.11.2009
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Предметный указатель
"Macaulay", symbolic algebra program [BaSone] 96 221 223 312
, intersection with 238
, irreducibility of when 51 117
, morphism to the Hilbert scheme not extending to the closure 201
, morphism to the Hilbert scheme not extending to the closure, when 159-176
, sequence of incomparable 244
, smoothness of, when 117
, variety, closure of 97 250
, variety, closure of , 92-99
, variety, closure ofsequence of incomparable 244
, variety, desingularization of when 32
, variety, desingularization of when 250
, variety, dimension of when 119 122 124-125
, variety, dimension of when , and 180
, variety, having several components 214 233
, variety, having several components, when 311
, variety, having several components, when 225
, variety, having several components, when (Boij) 236
, variety, having several components, when , 223
-regular ideal 48 300
-sequence 122 124 153 290
-sequence, when 291
(parameter space for sums of powers) 13
see
, variety, closure of 97 250
ACM 134
AD see Additive decomposition
Additive decomposition see Decomposition of form
Additive decomposition, generalized 138
Alexander - Hirschowitz Theorem 40-41 46 58 66 96 101
Alignment character, of when 122-124 286-288
Annihilating scheme 54 135-141 202 207-213 308
Annihilating scheme, a self-associated point set 224
Annihilating scheme, and additive decomposition 136
Annihilating scheme, conic 208-211
Annihilating scheme, generalized, sequence of 236
Annihilating scheme, locally Gorenstein 227-234
Annihilating scheme, locally Gorenstein, compressed 222
Annihilating scheme, locally Gorenstein, compressed nonhomogeneous 230 234
Annihilating scheme, nonsmoothable 221-226
Annihilating scheme, nonunique 139
Annihilating scheme, not recoverable from any form 141
Annihilating scheme, of higher dimension 311
Annihilating scheme, points on a RNC 217-220
Annihilating scheme, tight 135 202
Annihilating scheme, uniqueness of 136 208
Apolar polynomials 8
Apolarity 4 12
Aronhold invariant of a plane cubic 247
Artinian Gorenstein algebra see Gorenstein Artin algebra
Associated point 142
Auslander - Buchsbaum Theorem 162
Bertini Theorem 64
Betti number strata for when 125 153-156 160 172 173 191
Betti number strata for smooth aG curves in 203
Betti number strata of , local closure of 151 191
Betti number strata of the Hilbert scheme 172 189-196
Betti number strata, dimension of 125
Buchsbaum-Eisenbud structure theorem 119 126 132 160 199 204 255 277
Canonical curves 215
Canonical form xiv
Canonical form, of a binary form 25
Castelnuovo - Mumford regularity 48 300
Castelnuovo - Mumford regularity, algebraic version 49
Castelnuovo - Mumford regularity, Gotzmann upper bound 301
Castelnuovo - Mumford regularity, Gotzmann upper bound, attained for lex-segment ideal 305
Catalectic binary form xiv
Catalectic line of verse xiv
Catalecticant determinant, irreducibility of 80
Catalecticant determinantal locus 5
Catalecticant determinantal locus, CI component 76 112-116
Catalecticant determinantal locus, connectedness of 239
Catalecticant determinantal locus, high dimensional components, when 121
Catalecticant determinantal locus, irreducibility of in corank one 81
Catalecticant determinantal locus, irreducible component of 92 102 105 112 242-244
Catalecticant determinantal locus, reducible 242
Catalecticant determinantal locus, reducible, 76
Catalecticant determinantal locus, tangent space to 74
Catalecticant homomorphism 4
Catalecticant homomorphism, kernel and image of 74
Catalecticant ideal 5
Catalecticant ideal, prime 263
Catalecticant ideal, prime, for (Jacobian) 20
Catalecticant ideal, prime, in corank one 81
Catalecticant ideal, prime, when (binary) 39
Catalecticant matrix 5
Catalecticant matrix, square, when 95
Catalecticant scheme 5 8 73
Catalecticant source of name xiv
Catalecticant variety 5 8
Catalecticant variety, , and quadratic forms 18
Catalecticant variety, 19
Catalecticant variety, 20
Catalecticant variety, 92
Catalecticant variety, 105
Catalecticant variety, 102
Catalecticant variety, and Veronese 21 245-247 264
CI see Complete intersection
Closure of 199
Closure of 238 243
Closure of 97 241 250
Codimension see Height
Codimension of an ideal 276
Complete intersection 76
Complete intersection, not in closure of 140
Complete intersection, not in the closure of 146
Complete intersection, related component of 108- 112
Complete intersection, related component of determinantal locus 76 108-116
Component of A see Irreducible component A
Compressed Gorenstein Artin algebra 80 111
Compressed Gorenstein Artin algebra, nonhomogeneous 229
Compressed Gorenstein Artin algebra, nonhomogeneous, determining a component of , 230
Compressed Gorenstein Artin algebra, nonhomogeneous, dimension of parameter space for 229
Compressed Gorenstein Artin algebra, nonsmoothable 221
Connectedness of catalecticant determinantal locus 239-241
Connectedness of the closure 240
Contraction map 265
Contraction, action 3
Contraction, pairing 11
Curve, canonical 215
Curve, rational normal 215
Decomposition of a form 10 36 62
Decomposition of a form, existence 13
Decomposition of a form, generalized 22
Decomposition of a form, normalized GAD of binary form 22
Decomposition of a form, uniqueness 62
Decomposition of a form, uniqueness, of a prime ideal 205
Decomposition of a form, uniqueness, when 17
Depth of a local ring 47
Derivatives of a form see Inverse system of Gorenstein ideal
Derivatives, dimensions of spaces of higher derivatives of a form 8
Determinantal loci see Catalecticant determinantal
Determinantal scheme 147
Dimension formulas, dim 229
Dimension formulas, dim , the Betti stratum 125
Dimension formulas, dim when 172
Dimension formulas, dim 166 180
Dimension formulas, dim , the Betti stratum 193
Dimension formulas, dim 121
Dimension formulas,dim when 177
Divided power 267
Divided power forms 265
Divided power ring 3 265
DP-forms 265
DP-polynomials 265
Duality, Macaulay 3
Eagon - Northcott complex 40 83
Equations, defining see Catalecticant ideal
Equations, defining 185 199
Equations, defining 10 61 102 255
Equations, defining the Hilbert scheme 305
Equations, of the multisecant varieties of a RNC 31
Equations, of the Veronese variety 16
Extremal Gorenstein algebra 43
Form (homogeneous polynomial), and Gorenstein algebra 8
Form (homogeneous polynomial), annihilating ideal of 14
Form (homogeneous polynomial), annihilating scheme see Annihilating scheme
Form (homogeneous polynomial), annihilating scheme of 135
Form (homogeneous polynomial), annihilating scheme of, a self-associated point set 217
Form (homogeneous polynomial), annihilating scheme of, points on a RNC 217
Form (homogeneous polynomial), cubic 19
Form (homogeneous polynomial), decomposition of see Decomposition of a form
Form (homogeneous polynomial), having no tight annihilating scheme 139-140 245
Form (homogeneous polynomial), inverse system generated by see Inverse system
Form (homogeneous polynomial), quadratic 18
Form (homogeneous polynomial), ternary quartic 6
Form (homogeneous polynomial), uniqueness of power sum representation 42
Frontier property 34
GAD see Decomposition of a form generalized
General position theorem 41 64
Generalized additive decomposition see Decomposition of a form generalized
Generic nonsmoothable 221
Gor(T), scheme 7
Gor(T), scheme, smoothness of, when 116
Gorenstein Artin algebra 8
Gorenstein Artin algebra, compressed 80 229
Gorenstein Artin algebra, compressed, nonhomogeneous 229
Gorenstein Artin algebra, extremal 43
Gorenstein Artin algebra, generic 221
Gorenstein Artin algebra, strong Lefschetz 109 111
Gorenstein Artin algebra, weak Lefschetz 179 235
Gorenstein ideal 8
Gorenstein ideal, a CI when 30 39
Gorenstein ideal, and annihilating ideal of a form 8
Gorenstein ideal, containing a CI 111
Gorenstein ideal, determined by a sum of powers square of, and vanishing ideal 84
Gorenstein ideal, naming of xviii 8
Gorenstein ideal, nonhomogeneous, dimension of parameter space, when 126
Gorenstein ideal, square of, and tangent space to 79
Gorenstein ideal, square of, minimal resolution 97
Gorenstein ideal, square of, minimal resolution, when 276
Gorenstein ideal, square of, when , s special 93
Gorenstein sequence 8 240
Gorenstein sequence, bounds on growth of 308
Gorenstein sequence, non-unimodal 203 239
Gorenstein sequence, occuring for 238
Gorenstein sequence, unimodal 153
Gorenstein sequence, when 153 239
Gotzmann number of a polynomial 295
Gotzmann number of a polynomial, equality with Castelnuovo - Mumford regularity 301
Gotzmann polynomial 293
Gotzmann's persistence theorem 297
Grade of an ideal 276
Graded algebras of Hilbert function 249
Graded algebras of Hilbert function , singularity of family of, when 252
Grassmann variety 15 249 302
Hankel matrix 6 28 74
Height of an ideal 202 276
Hilbert - Burch Theorem 39 163
Hilbert function 9
Hilbert function, global, of local Gorenstein compressed scheme 234
Hilbert function, Macaulay's characterization of 290
Hilbert function, of Gorenstein algebra see Gorenstein sequence
Hilbert function, of Gorenstein algebra, and dimension of higher partial derivatives 8
Hilbert function, of square of Gorenstein ideal 93
Hilbert polynomial 298
Hilbert polynomial, identical with Macaulay polynomial 299
Hilbert scheme 42 159 212 226
Hilbert scheme 302-307
Hilbert scheme , equations of 306
Hilbert scheme , Gotzmann's structure theorem, for certain 310
Hilbert scheme , Gotzmann's theorem 305
Hilbert scheme of 0-dimensional schemes 159 308
Hilbert scheme, and 212
Hilbert scheme, Betti number strata see Betti number strata
Hilbert scheme, postulation strata see Postulation H punctual Hilbert scheme
Hilbert scheme, subscheme of 302
Ideal, -regular 48 300
Ideal, ancestor, of a vector space of forms 68 299
Ideal, defining of scheme see Name of scheme
Ideal, limit 142 143
Ideal, minimal growth of 290 291
Ideal, order ideal ( -ideal) 290
Ideal, order of 43
Ideal, primitive, of a singularity 101
Ideal, saturated 46
Ideal, saturation 298
Initial ideal see Lex-initial ideal
Inverse system of Gorenstein ideal 4 8
Inverse system of Gorenstein ideal, and Hilbert function 8
Inverse system of Gorenstein ideal, dimension of graded components of 8
Inverse system of locally Gorenstein ideal 228
Irreducibility see Catalecticant determinantal and
Irreducibility, of when 51
Irreducibility, of the catalecticant determinant 80
Irreducible components of 53 214-236
Irreducible components of 53 241-245
Jacobian ideal 15-21
Jordan lemma 24
Lattice of height three Gorenstein sequences 153 286
Length of a form 197-198
Length of a form, differential length 135
Length of a form, for a binary DP-form 36
Length of a form, scheme length 135 245
Length of a form, when is binary 23
Length of GAD 22
Lex-initial ideal 290
liaison 108 205
Limit ideal 142 143
Limit scheme 144
Macaulay or Matlis duality 3 4
Macaulay polynomial 293 299
Macaulay polynomial, conditions defining 295
Macaulay representations of an integer 294
Minimal resolution of square of Pfaffain ideal 97
Minimal resolution strata of , when , dimension of 125
Morphism, finite 142
Morphism, proper 142
Morphism, quasifinite 142
Multiplication map 94 107 113
Multiplication map, kernel of 104
Multisecant varieties 10
Multisecant varieties, of the Veronese 245-247
Multisecant varieties, of the Veronese, degree of 247
Nonsmoothable scheme 221
Nonsmoothable scheme, conic, Gorenstein 54
Normalized GAD 22
Order ideal 290
Order of an ideal 43
Partition, asssociated to a height three Gorenstein sequence 122-124 286- 288
Partition, self-complementary, and height three Gorenstein sequence 286
Persistence, Gotzmann's Theorem 297 305
Pfaffians of an alternating matrix 160 204 272-277
Points on RNC 313
Points, imposing independent conditions on a linear system 12
Points, vanishing ideal of see Vanishing ideal
Polar polyhedron 11 62
Postulation punctual Hilbert scheme 149
Postulation punctual Hilbert scheme, dimension of Betti strata, when 189-196
Postulation punctual Hilbert scheme, dimension of, when 166 180-181
Postulation punctual Hilbert scheme, morphism to, from , when and 159
Postulation punctual Hilbert scheme, reducibility of Betti strata of, for large 196
Power sum 10
Power sum, annihilator ideal, Hilbert function of 13
Power sum, parameter space , closure of 145 213 238
Power sum, parameter space , closure of, and when 200
Power sum, parameter space , component of 102 230- 233
Power sum, parameter space , dimension of 40 59-61
Power sum, representation 17 22 29 38 40 182-189
Power sum, uniqueness of, for a given form 62
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