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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.
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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
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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
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