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Iarrobino A., Kanev V. — Power Sums, Gorenstein Algebras, and Determinantal Loci
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.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1999

Количество страниц: 345

Добавлена в каталог: 20.11.2009

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
"Macaulay", symbolic algebra program [BaSone]      96 221 223 312
$Gor(T)$, intersection with $\overline{PS(s,j; r)}$      238
$Gor(T)$, irreducibility of when $r = 3$      51 117
$Gor(T)$, morphism to the Hilbert scheme not extending to the closure      201
$Gor(T)$, morphism to the Hilbert scheme not extending to the closure, when $r = 3$      159-176
$Gor(T)$, sequence of incomparable      244
$Gor(T)$, smoothness of, when $r = 3$      117
$Gor(T)$, variety, closure of $r = 3$      97 250
$Gor(T)$, variety, closure of $r = 3$, $T = H(s,j, 3)$      92-99
$Gor(T)$, variety, closure ofsequence of incomparable      244
$Gor(T)$, variety, desingularization of when $r = 2$      32
$Gor(T)$, variety, desingularization of when $r = 3$      250
$Gor(T)$, variety, dimension of when $r = 3$      119 122 124-125
$Gor(T)$, variety, dimension of when $r = 3$, and $T\supset (s, s, s)$      180
$Gor(T)$, variety, having several components      214 233
$Gor(T)$, variety, having several components, when $r = 4$      311
$Gor(T)$, variety, having several components, when $r = 5, 6$      225
$Gor(T)$, variety, having several components, when $r\geq 4$ (Boij)      236
$Gor(T)$, variety, having several components, when $r\geq 7$, $r\neq 8$      223
$m$-regular ideal      48 300
$O$-sequence      122 124 153 290
$O$-sequence, when $r = 2$      291
$PS(s,j; r)$ (parameter space for sums of powers)      13
$\mathbb{P}Gor(T)$      see $Gor(T)$
$\mathbb{P}Gor(T)$, 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 $T$ when $r = 3$      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 $f\in Gor(T)$ when $r = 3$      125 153-156 160 172 173 191
Betti number strata for smooth aG curves in $\mathbb{P}^{4}$      203
Betti number strata of $Gor(T)$, local closure of      151 191
Betti number strata of the Hilbert scheme $Hilb^{s}(\mathbb{P}^{2})$      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 $r = 3$      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, $r = 3$      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 $V_{s}(1,j - 1;r)$ (Jacobian)      20
Catalecticant ideal, prime, in corank one      81
Catalecticant ideal, prime, when $r = 2$ (binary)      39
Catalecticant matrix      5
Catalecticant matrix, square, when $r = 3$      95
Catalecticant scheme      5 8 73
Catalecticant source of name      xiv
Catalecticant variety      5 8
Catalecticant variety, $V_{s}(1, 1; r)$, and quadratic forms      18
Catalecticant variety, $V_{s}(1, 2; r)$      19
Catalecticant variety, $V_{s}(1, j - 1; r)$      20
Catalecticant variety, $V_{s}(t, t; 3)$      92
Catalecticant variety, $V_{s}(t, t; r)$      105
Catalecticant variety, $V_{s}(u, v; r)$      102
Catalecticant variety, and Veronese      21 245-247 264
CI      see Complete intersection
Closure of $PS(s,j;3)$      199
Closure of $PS(s,j;r)$      238 243
Closure of $\mathbb{P}Gor(T)$      97 241 250
Codimension      see Height
Codimension of an ideal      276
Complete intersection      76
Complete intersection, not in closure of $PS(12,9; 3)$      140
Complete intersection, not in the closure of $PS(s,j;r)$      146
Complete intersection, related component of $Gor(T)$      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 $Gor(T),T = H(s,j,r)$,$r > 5$      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 $\overline{\mathbb{P}(Gor(T))}$      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 $P (= grade(P))$      205
Decomposition of a form, uniqueness, when $s\leq r$      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 $CompAlg(H(t, n))$      229
Dimension formulas, dim $Gor_{D}(T)$, the Betti stratum      125
Dimension formulas, dim $Gor_{sch}(T)$ when $T\supset (s,s)$      172
Dimension formulas, dim $Hilb^{H}\mathbb{P}^{2}$      166 180
Dimension formulas, dim $Hilb^{H}_{D}\mathbb{P}^{2}$, the Betti stratum      193
Dimension formulas, dim $\mathbb{P}Gor(T)$      121
Dimension formulas,dim $Gor_{sch}(T)$ when $T\supset (s-a,s, s-a)$      177
Divided power $L^{[j]}$      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 $V_{s}(u,v,r)$      see Catalecticant ideal
Equations, defining $\overline{PS(s,j; 3)}$      185 199
Equations, defining $\overline{PS(s,j;r)}$      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 $r = 3$      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 $r = 2$      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 $r = 3$      126
Gorenstein ideal, square of, and tangent space to $Gor(T)$      79
Gorenstein ideal, square of, minimal resolution      97
Gorenstein ideal, square of, minimal resolution, when $r = 3$      276
Gorenstein ideal, square of, when $r = 3$, s special      93
Gorenstein sequence      8 240
Gorenstein sequence, bounds on growth of      308
Gorenstein sequence, non-unimodal      203 239
Gorenstein sequence, occuring for $f\in \overline{PS(s, j; r)}$      238
Gorenstein sequence, unimodal      153
Gorenstein sequence, when $r = 3$      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 $grade (I, R)$      276
Graded algebras of Hilbert function $T$      249
Graded algebras of Hilbert function $T$, singularity of family of, when $r = 3$      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 $Ann(f)$      93
Hilbert polynomial      298
Hilbert polynomial, identical with Macaulay polynomial      299
Hilbert scheme      42 159 212 226
Hilbert scheme $Hilb^{P}(\mathbb{P}^{r - 1})$      302-307
Hilbert scheme $Hilb^{P}(\mathbb{P}^{r - 1})$, equations of      306
Hilbert scheme $Hilb^{P}(\mathbb{P}^{r - 1})$, Gotzmann's structure theorem, for certain $P$      310
Hilbert scheme $Hilb^{P}(\mathbb{P}^{r - 1})$, Gotzmann's theorem      305
Hilbert scheme of 0-dimensional schemes      159 308
Hilbert scheme, and $Gor(T)$      212
Hilbert scheme, Betti number strata      see Betti number strata
Hilbert scheme, postulation strata      see Postulation H punctual Hilbert scheme
Hilbert scheme, subscheme of $Grass^{p}(R_{d})$      302
Ideal, $m$-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 ($O$-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 $Gor{T)$ when $r = 3$      51
Irreducibility, of the catalecticant determinant      80
Irreducible components of $Gor(T)$      53 214-236
Irreducible components of $V_{s}(u,v; r)$      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 $f$ a binary DP-form      36
Length of a form, scheme length      135 245
Length of a form, when $f$ 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 $Gor(T)$, when $r = 3$, 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 $H$ punctual Hilbert scheme      149
Postulation $H$ punctual Hilbert scheme, dimension of Betti strata, when $r = 3$      189-196
Postulation $H$ punctual Hilbert scheme, dimension of, when $r = 3$      166 180-181
Postulation $H$ punctual Hilbert scheme, morphism to, from $Gor(T)$, when $r = 3$ and $T\supset (s, s, s)$      159
Postulation $H$ punctual Hilbert scheme, reducibility of Betti strata of, for $r$ large      196
Power sum      10
Power sum, annihilator ideal, Hilbert function of      13
Power sum, parameter space $PS(s,j; r)$, closure of      145 213 238
Power sum, parameter space $PS(s,j; r)$, closure of, and $Gor(T)$ when $r = 3$      200
Power sum, parameter space $PS(s,j; r)$, component of $Gor(T)$      102 230- 233
Power sum, parameter space $PS(s,j; r)$, 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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