 -functor 205 219 234 238 240 243 282
 160—169
 -Process 386
-Adic completion 193 254
-adic topology 33 213 279
-Uple embedding 13 21 54 155 156 159 167 183 291 343 385
-Uple embedding is projectively normal 126 315
Abelian category 202—206
Abelian category with enough injectives 204 217
Abelian variety 105 140 422 437 447 451 see
Abhyankar, Shreeram 391
Abstract nonsingular curve 42—46 108 136
Abstract nonsingular curve is quasi-projective 45
Abstract variety 58 105
Acyclic resolution 205
Additive functor 203
Adjoint functors 68 110 124
Adjunction formula 361 368 374 382 387 390
Affine coordinate ring 4 8 17 20 23
Affine curve 4 7 8 47
Affine curve, any noncomplete curve is 297 298
Affine curve, locally free sheaves on 385
Affine formal scheme 195
Affine formal scheme, global section functor is exact 198 199
Affine line,  74
Affine line, is not proper 100
Affine line, with a point doubled 76 96 169
Affine morphism 128 222 252
Affine n-space,  1
Affine n-space, , automorphisms of 23
Affine plane,  75
Affine scheme 59 74 124
Affine scheme is quasi-compact 80
Affine scheme, cohomology of 114 213—216
Affine scheme, criterion for 81 215 216
Affine scheme, global section functor is exact 113
Affine scheme, sheaf  on 110—113
Affine variety 1—8 20 21 25
Algebraic equivalence of cycles 444
Algebraic equivalence of divisors 140 367 369
Algebraic family see "Family"
Algebraic integer 330—332 450 457
Algebraic set 2 4 5 9 47
Algebraic space 445
Algebraizable formal scheme 195
Algebraizable sheaf 281
Altman, Allen 268
Ample divisor see also "Ample invertible sheaf"
Ample divisor on a curve 156 307 308 372
Ample divisor on a surface 365 368 405
Ample divisor, Nakai — Moishezon criterion 356 365 382 405 434
Ample invertible sheaf 150 153—156 161 248
Ample invertible sheaf on  155
Ample invertible sheaf on a curve 307 308 365
Ample invertible sheaf on a quadric surface 156
Ample invertible sheaf, cohomological criterion of 229
Ample invertible sheaf, existence of  separated 169
Ample invertible sheaf, not very ample 156
Ample invertible sheaf, properties of 169 232
Ample invertible sheaf, scheme without an 169 171
Analytically isomorphic singularities 34 38 298 393—395
Annihilator 50 124
Arithmetic genus,  54 57 201 230 232 246 268
Arithmetic genus, is a birational invariant 230 409 413
Arithmetic genus, of a  253
Arithmetic genus, of a complete intersection 54 231
Arithmetic genus, of a curve 54 181 246 294 298
Arithmetic genus, of a curve on a surface 366 389 401
Arithmetic genus, of a product 54
Arithmetic genus, of a surface 188 247 362 371 409 421
Arithmetic genus, , constant in a family 263 265 289
Arithmetic genus, , invariant by monoidal transformation 388 394
Artin ring 266
Artin, Michael 417 445 449 452 453
Assigned base point of a linear system 395 399 400
Associated point (of a scheme) 257
Atiyah, Michael, F. 378 383
Automorphisms of 23
Automorphisms of  46
Automorphisms of 151 158 347
Automorphisms of a curve of genus  305 348 349 368
Automorphisms of a curve of genus 3 348
Automorphisms of an elliptic curve 318 321 336
Automorphisms of k(x) 46
Automorphisms of the configuration of 27 lines 405 408
Baily, W.L., Jr. 349
Banica, Constantin 439
Base extension 89 254 265
Base extension, behavior of cohomology 282 290 326 369
Base extension, behavior of differentials 175
Base extension, flat 255 287
Base extension, stable under 90
Base for topology 25 71
Base points of a linear system 158 307 395
Base points, scheme of 169
Base scheme 89
Base-point free 158 307 318
Baum, Paul 436
Berthelot, Pierre 452
Bertini, E. 316
Bertini’s theorem 179 183 187 188 245 250 281 425
Bertini’s theorem in characteristic 0 268 274
Bertini’s theorem with singularities 180
Bertini’s theorem, gives nonsingular curves 183 231 314 342 358 381 406
Betti numbers 451 456
Bezout’s Theorem 47 54 146 361 400 417
Bilinear form 364
Binomial coefficient 49 52
Birational equivalence 24 26—31 45 55 181 314 370 418
Birational invariant 56 181 190 387 409 421
Birational morphism 56 166 280 310 313
Birational transformation 409—420
Birational transformation of a ruled surface 416
Birational transformation, defined at a point 410
Birational transformation, factorization of 386 409 411—413 416
Birational transformation, fundamental point of 410
Birationally ruled surface 370 383 419 see
Birch, B.J. 317 334
Bitangent 305
Blowing down lines 398 416
Blowing up 28—31 163—171 356 see
Blowing up is birational 29 166
Blowing up is intrinsic 166
Blowing up is not flat 258
Blowing up, a curve on a surface 388—394
Blowing up, a nonsingular subvariety 186 188 394 443
Blowing up, canonical sheaf of 188
Blowing up, curve singularities 29 37 390
Blowing up, Picard group of 188
Blowing up, strict transform under see "Strict transform"
Blowing up, surfaces 56 395
Blowing up, to construct valuation rings 108 420
Blowing up, universal property of 164 412
Blowing up, vertex of cone 37 374 381
Bombieri, Enrico 356 421 451
Borel, Armand 149 239 432 435
Borelli, Mario 238
Branch divisor 306
Branch of a curve 30
Branch point 299 317
Branched covering 293
Brieskom, Egbert 420 421
Burnside, W. 349
Canonical curve 341 346 348 353 385
Canonical divisor on a curve 293 295 299 373
Canonical divisor on a hyperelliptic curve 343
Canonical divisor on a singular curve 298
Canonical divisor on a surface 361 373 387 421 424
Canonical embedding 293 340—349
Canonical linear system 340 341
| Canonical morphism 341 422
Canonical sheaf,  180 239 246
Canonical sheaf, of 182
Canonical sheaf, of a blowing-up 188
Canonical sheaf, of a complete intersection 188 315
Canonical sheaf, of a curve 294 295
Canonical sheaf, of a hypersurface 183 184
Canonical sheaf, of a nonsingular subvariety 182
Canonical sheaf, of a product 187
Cardinality of a variety 31
Cartan, Henri 172
Cartan’s theorem B 211 441
Cartesian product (of graded rings) 125
Cartier divisor 140—146 231 see
Cartier divisor on a singular curve 142 148 298
Cartier divisor, algebraic family of 261
Cartier divisor, associated Weil divisor of 141
Cartier divisor, effective 145 427
Cartier divisor, linear equivalence of 141
Cartier divisor, principal 141
Cartier divisor, sheaf  associated to 144
Cartier, Pierre 172
Cassels, J.W.S. 331 335
Castelnuovo, G. 351 368 422
Castelnuovo’s criterion for contracting a curve 356 410 414—417
Category Abelian 202—206
Category of abelian groups  61
Category of open sets,  61
Category of schemes over S,  78
Category of sheaves on X,  65
Category of varieties over k  78
Category, inverse limit in 192
Category, product in 22
Cech cocyle 232 240
Cech cohomology 201 211 218—225 255 262
Cech cohomology of a plane curve 224
Cech cohomology, computation of 219 220 225—227
Cech cohomology, computes  252 282
Cech cohomology, limit over coverings 223
Center of a valuation 106 108 137
Center of blowing up 163
Characteristic 2 312 317
Characteristic p 21 31 33 80 89 276 293 312 316 317 332—335 339 385 391 422 see
Characteristic p, funny curve in 305 385
Characteristic p, reduction mod p 89 334 340 451
Characteristic p, search for good cohomology 451
Characteristic zero 232 268 271—275 304 305 348 382 391 see
Chem class 363 425 429—431 433 435 437 450
Chem polynomial 429 435
Chem, S.S. 423
Chevalley, Claude 48 94 95 172 222 427 428
Chinese remainder theorem 138
Chow ring 426 428 429 433 437 454
Chow variety 349
Chow, Wei-Liang 105 324 441 443
Chow’s Lemma 107
Chow’s Moving Lemma 427 434
circle 213 220
Class field theory 331
Class of a curve 305
Classical projective geometry 403 407
Classification of curve singularities 38 393—395
Classification of curves 56 341 345—347
Classification of curves in  349—355 354 409
Classification of curves of genus 2 304
Classification of elliptic curves 317 345
Classification of elliptic ruled surfaces 377
Classification of locally free sheaves of rank 2 on an elliptic curve 378
Classification of rational ruled surfaces 376 419
Classification of surfaces 56 421—423
Classification, problem 39 55—57 181 293 345
Clemens, C. Herbert 184 304
Clifford’s theorem 341 343 350 423
Closed immersion 85 92 see
Closed immersion is finite 124
Closed immersion is of finite type 93
Closed immersion is proper 102
Closed immersion is separated 99
Closed immersion, criterion for 151 152 158 307
Closed morphism 91 100
Closed points are dense 93
Closed subscheme see also "Closed immersion"
Closed subscheme of a Proj 92 119 125
Closed subscheme of an affine scheme 85 92 111 116
Closed subscheme of codimension one 145
Closed subscheme with nilpotents see "Nilpotent element"
Closed subscheme, associated Cartier divisor 145 149
Closed subscheme, associated cycle 425
Closed subscheme, completion along 86 190 194 279
Closed subscheme, criterion to be nonsingular 178
Closed subscheme, ideal sheaf of 115
Closed subscheme, image of a morphism 92
Closed subscheme, locally principal 145
Closed subscheme, reduced induced structure 86 92
Coarse moduli variety 347
Codimension 86 87
Coeffaceable functor 206 238 240 243
Cohen — Macaulay ring 184
Cohen — Macaulay scheme 185 239 243 276
Cohen, I.S. 34 187
Coherence of direct image sheaf 115
Coherence of direct image sheaf for a finite morphism 124
Coherence of direct image sheaf for a projective morphism 123 152 252 280
Coherent analytic sheaf 439 440 447
Coherent sheaf 111—115 see
Coherent sheaf, completion along a closed subscheme 194
Coherent sheaf, extending from an open set 126
Cohn-Vossen, S. 403
Cohomological dimension 224
Cohomology see also "Cech cohomology"
Cohomology and base change 282 290 326 369
Cohomology of 219
Cohomology of a circle 213 220
Cohomology of a complete intersection 231
Cohomology of a complex 203
Cohomology of an affine scheme 114 213—216
Cohomology of fibres 250 255 281 290
Cohomology of projective space 225—230
Cohomology of sheaves 206—211
Cohomology with supports 212 217
Cohomology, as a derived functor 207 211 439
Cohomology, Cech process 218—225
Cohomology, characterizes ample sheaf 154 229
Cohomology, class of a cycle 435 454
Cohomology, class of a divisor 367 418
Cohomology, class of a subvariety 249
Cohomology, commutes with  209
Cohomology, commutes with flat base extension 255 287
Cohomology, crystalline 452
Cohomology, etale 307 452 453
Cohomology, functor 207
Cohomology, gives numerical invariants 57 230 246 247
Cohomology, group 207 424
Cohomology, l-adic 435 449 452—457
Cohomology, p-adic 452
Cohomology, Picard group as 143 224 367 446
Cohomology, theories 211 449
Cokernel 63 65 109
Complete intersection 14 188 452
Complete intersection is connected 188 231
Complete intersection is every curve? 14
Complete intersection, a divisor on a quadric hypersurface is 147
Complete intersection, arithmetic genus of 54 231
Complete intersection, canonical sheaf of 188 315
Complete intersection, cohomology of 231
Complete intersection, curve 38 342 346 352 355
Complete intersection, elliptic quartic curve is 38
Complete intersection, geometric genus of 188
Complete intersection, hyperelliptic curve is not 348
Complete intersection, hyperplane section of 267
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