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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 LociАвторы: 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.
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 1999Количество страниц: 345Добавлена в каталог: 20.11.2009Операции: Положить на полку |
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"Macaulay", symbolic algebra program [BaSone] 96 221 223 312 238 51 117 201 159-176 244 117 97 250 92-99 244 32 250 119 122 124-125 180 214 233 311 225 236 223 48 300 122 124 153 290 291 13 see 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 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 125 153-156 160 172 173 191 Betti number strata for smooth aG curves in 203 Betti number strata 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 20 Catalecticant ideal, prime, in corank one 81 Catalecticant ideal, prime, when 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, 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 125 Dimension formulas, dim 172 Dimension formulas, dim 166 180 Dimension formulas, dim 193 Dimension formulas, dim 121 Dimension formulas,dim 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 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 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 306 Hilbert scheme 310 Hilbert scheme 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, 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 ( 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 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 36 Length of a form, scheme length 135 245 Length of a form, when 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 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 149 Postulation 189-196 Postulation 166 180-181 Postulation 159 Postulation 196 Power sum 10 Power sum, annihilator ideal, Hilbert function of 13 Power sum, parameter space 145 213 238 Power sum, parameter space 200 Power sum, parameter space 102 230- 233 Power sum, parameter space 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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