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Harris J. — Algebraic Geometry: a first course
Harris J. — Algebraic Geometry: a first course

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Название: Algebraic Geometry: a first course

Автор: Harris J.

Аннотация:

This book is intended to introduce students to algebraic geometry; to give them a sense of the basic objects considered, the questions asked about them, and the sort of answers one can expect to obtain. It thus emplasizes the classical roots of the subject. For readers interested in simply seeing what the subject is about, this avoids the more technical details better treated with the most recent methods. For readers interested in pursuing the subject further, this book will provide a basis for understanding the developments of the last half century, which have put the subject on a radically new footing. Based on lectures given at Brown and Harvard Universities, this book retains the informal style of the lectures and stresses examples throughout; the theory is developed as needed. The first part is concerned with introducing basic varieties and constructions; it describes, for example, affine and projective varieties, regular and rational maps, and particular classes of varieties such as determinantal varieties and algebraic groups. The second part discusses attributes of varieties, including dimension, smoothness, tangent spaces and cones, degree, and parameter and moduli spaces.


Язык: en

Рубрика: Математика/Алгебра/Алгебраическая геометрия/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2000

Количество страниц: 328

Добавлена в каталог: 18.11.2004

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
1—generic matrix      102
Action (of a group)      116
Action, examples of      100 116—123 125—129 161 280
Affine coordinates      4
Affine space      3
Affine tangent space      175
Affine variety      3
Algebraic group      114
Algebraic, dimensions of      138
Algebraic, examples of      114 115 116
Analytic coordinates      187
Analytic topology      17
Analytic variety      8 77 175
Area of a variety      227
Arithmetic genus      167 173
Associated curve      214
Associated curve, tangent line to      214
Associated map      214
Automorphism group      116
Automorphism group, of Grassmannians      122
Automorphism group, of projective space      228
Base (of a family)      41
base locus (of a pencil)      301
Bertini's theorem      216 218 231 234 244
Betti numbers      173
Bezout theorem      173 227 228 237 248
Bidegree      27
Bihomogeneous polynomial      27
Binary quantic      120
Birational      77
Birational isomorphism      77
Birational isomorphism with hypersurface      79
Birational isomorphism with smooth variety      219 264
Birational map      77
Birational map, examples      157 288;
Bitangent lines      195
Blowing down      84; see also Blowing down
Blowing up      81 82
Blowing up, examples of      84 85 86 92 192 220—221 254 259 277 288 298
Blowing up, Nash      221
Blowing up, resolving rational maps by      84
blowing up, resolving singularities by      219
Borel — Moore homology      259
Canonical embedding      280
Cardioids      263
Catalecticant matrix      108
Categorycal product      28 30
Characteristic (of ground field)      186
Chordal variety      see secant variety
Chow point of a variety      269 272
Chow variety      268 270 271 272
Chow variety of curves of degree      2 276
Chow variety, is connected      272
Chow's theorem      8 77
Circuits (of real plane curves)      247
Circuits, maximum number of      248
Classical topology      17
Coarse moduli space      278
Coarse moduli space, of plane cubics      279
Coarse moduli space, of stable curves      281
Codimension subadditivity in intersections      222
Cohomology      226 232 233 239 259
Complementary section (of a scroll)      93
Complete conies      297
Complete intersection      136
Complete intersection, dimension of family      157
Complete intersection, general hypersurfaces cut out      218
Complete intersection, local      138
Complete intersection, resolution of      172
Complete intersection, set-theoretic      137
Complex analytic variety/submanifold      8 77 174 175 226
complex numbers      3 187
Cone      32 37 259
Cone, degree of      234
Cone, dimension of      138
Cone, dual variety of      197
Conics      12 13 34 117 284
Conics, complete conies      297
Conics, on a general surface      157
Conics, pencils of      302
Conics, universal family of      12 45 46 120
Conics, universal hyerplane section of      43
Conics, variety of incident planes of      70; see also rational normal curve
Conjugate (matrices of linear forms)      100 102 106
Constructible sets      39
Contact of order m      214 238
Coordinate ring      18 49
Coordinate ring homogeneous      20
Coordinates affine      4
Coordinates Euclidean      4
Coordinates homogeneous      4
Coordinates local analytic      187
Cross-ratio      7 12 119 304
Cubic hypersurfaces      238
Cubic plane curves      121 279;
Cubic scroll      85 92;
Cubic surface contains a twisted cubic      157
Cubic surface, Fano variety of      153
Cubic surface, is rational      157
Cubic threefold      237
Cubic, nodal      15
Curves, arithmetic genus of      167 173
Curves, associated curves of      214
Curves, Gauss map on      189 211
Curves, in Grassmannians      212
Curves, inflectionary points of      214
Curves, real plane      247
Curves, resolution of singularities of      264
Curves, secant variety of      144 206
Curves, tangential surface of      118 119 212
Cusp      221 261
Cuspidal cubic      15 36—37 54—55 122 221 222
Cut out a variety      48
Cut out a variety, ideal-theoretically      49
Cut out a variety, locally      50
Cut out a variety, scheme-theoretically      49 51
Cut out a variety, set-theoretically      49 51
Cycle parameter space      268
Cycles      272
Decomposablee multivectors      64
Deficient dual variety      197 199
Deficient secant variety      145 195
Deficient tangent variety      190
Degenerate variety      144
Degree      16 166 225 226
Degree minimal      229 231
Degree of a finite set      6
Degree of a general projection      225
Degree of a hypersurface      8
Degree of a join      235—236
Degree of a rational map      79
Degree of cones      234
Degree of determinantal varieties      243
Degree of images under Segre maps      240
Degree of images under Veronese maps      232
Degree of joins of corresponding points      241
Degree of projections      234—235 259
Degree of rational normal curves      229
Degree of rational normal scrolls      241 245
Degree of Segre varieties      232
Degree of some Grassmannians      245
Degree of tangential surfaces      245
Degree of unions of planes      244
Degree of Veronese varieties      231
Degree projective degree of a map      240
Determinantal varieties      98 99 111 112
Determinantal varieties, degrees of      243
Determinantal varieties, dimension of      151 223
Determinantal varieties, dimension of family of      159
Determinantal varieties, examples of      11 14 24 26 98 99 111 159—160 192 274
Determinantal varieties, Fano varieties of      112
Determinantal varieties, general quartic surface is not      160
Determinantal varieties, projective tangent space to      184
Determinantal varieties, proper      223
Determinantal varieties, resolution of      206
Determinantal varieties, tangent spaces to      184 207
Differential of a map      175 176
Differential of a map, is generically surjective      176
DIMENSION      16 133 134 135
Dimension local      136
Dimension of a hypersurface section      136
Dimension of a join of varieties      148
Dimension of determinantal varieties      151 223
Dimension of dual varieties      197
Dimension of Fano varieties of hypersurfaces      152
Dimension of fiber products      140
Dimension of fibers of maps      138
Dimension of flag manifolds      148
Dimension of intersections is subadditive      222
Dimension of parameter spaces of complete intersections      157
Dimension of parameter spaces of curves on a quadric      158
Dimension of parameter spaces of determinantal varieties      159
Dimension of parameter spaces of rational normal curves      156
Dimension of parameter spaces of twisted cubics      155
Dimension of parameter spaces of twisted cubics on a surface      156
Dimension of secant varieties      144 146
Dimension of some Schubert cycles      149
Dimension of tangential variety      190
Dimension of variety of secant lines      144
Dimension of variety of secant planes      146
Dimension of variety of tangent lines      191
Dimension of variety of tangent planes      189
Dimension pure      136
Directrix      86 92
Distinguishhed open sets      17 19
Dominant map      77
Double plane      33
Double point locus of a map      56
Dual projective space      6
Dual variety      196
Dual variety of plane conies      297
Dual variety, deficiency of      199
Dual variety, dimension of      197
Dual variety, dual variety of      198 208
Dual variety, tangent spaces to      208
Effective cycles      272
Elements (of a family)      41
Elimination theory      35
Embedding theorem      193
Equivalence, projective      4 22
Equivalence, projective, examples of      7 10 15 92 121 161 162 259 279 303—307
Euclidean coordinates      4
Exceptional divisor      81 82 254
Exceptional divisor, is projectivized tangent cone      255
Families      41
Families generically reduced      267
Families of conies      45 46 296 296
Families of hyperplane sections      43
Families of hyperplanes      42 46
Families of hypersurfaces      45
Families of lines      47
Families of quadrics      295
Families of quadrics, singular elements of pencils      301—303
Families of quadrics, subvarieties of singular quadrics      299
Families of rational normal curves      46
Families of twisted cubics      46
Families of varieties      41
Families reduced      267
Families sections of      43 46
Families tautological      278 279;
Fano varieties      70
Fano varieties of a Grassmannian      67—68 123
Fano varieties of a quadric hypersurface      288 289—294
Fano varieties of a quadric surface      26 67 71
Fano varieties of a Segre variety      27 67
Fano varieties of determinantal varieties      112
Fano varieties of hypersurfaces      152 210
Fano varieties tangent spaces to      209
Fiber products      30
Fibers of maps, closures of general      56 141
Fibers of maps, dimension of      138
Fine moduli space      279
Fine moduli space of plane cubics (does not exist)      279
Finite map      177 178
Finite sets      6
Finite sets, are not complete intersections      137
Finite sets, Hilbert functions of      163—164
Finite sets, impose independent conditions      56
Finite sets, resolutions of      170—171
Flag manifolds      95
Flag manifolds, dimension of      148
Flag manifolds, tangent spaces to      202
Flat map      43 267
Flex line      216
Free resolution      169
Free resolution of a complete intersection      172
Free resolution of four point      171
Free resolution of three points      170
Fulton — Hansen theorem      195
Function field      72 78 80 135 225
Fundamental class      226 239 259
Gauss map      188
Gauss map, graph of (Nash blow-up)      221
Gauss map, of a curve      189 211
Gauss map, of a hypersurface      188
Gauss map, of a tangential surface      213
General      53
General conic      54
General fibers of maps, closure of      56 141
General hyperplane section is irreducible      230
General hypersurfaces are smooth      218
General hypersurfaces, cut out complete intersections      218
General hypersurfaces, Fano variety of      152 210
General line      54 56
General linear group      114
General linear group, dimension of      138
General point      54
General polynomial as sum of powers      147
General position      7
General projection of twisted cubic      54
General projections intersect transversely      236
General set of points are not complete intersections      137
General set of points, impose independent conditions      56
General surface contains twisted cubics?      156
General twisted cubic      55
Generalized row/column of a matrix      102
Generate an ideal locally      50 51 165
GENERIC      54
Generic determinantal variety      98
Generically finite map      54 80
Generically reduced family      267
Generically reduced family, examples      273 276 277
Generically transverse intersection      227
Genus, arithmetic      166
Geometric invariant theory      117 124 280
Graph of a rational map      75
Graph of a regular map      29
Grassmannians      63
Grassmannians, affine coordinates on      65—66
Grassmannians, automorphisms of      122
Grassmannians, degree of some      245
Grassmannians, irreducibility, characterization of      139
Grassmannians, irreducibility, of dual variety      197
Grassmannians, irreducibility, of general hyperplane section      231
Grassmannians, irreducibility, of secant variety      144
Grassmannians, irreducibility, of tangent variety      190
Grassmannians, irreducibility, of universal hyperplane section      53
Grassmannians, irreducibility, of variety of tangent lines      191
Irreducible components      52
Irreducible decomposition      52
1 2 3
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