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Liu Q., Erne R. — Algebraic Geometry and Arithmetic Curves
Liu Q., Erne R. — Algebraic Geometry and Arithmetic Curves



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Название: Algebraic Geometry and Arithmetic Curves

Авторы: Liu Q., Erne R.

Аннотация:

This book is a general introduction to the theory of schemes, followed by applications to arithmetic surfaces and to the theory of reduction of algebraic curves. The first part introduces basic objects such as schemes, morphisms, base change, local properties (normality, regularity, Zariski's Main Theorem). This is followed by the more global aspect: coherent sheaves and a finiteness theorem for their cohomology groups. Then follows a chapter on sheaves of differentials, dualizing sheaves, and grothendieck's duality theory. The first part ends with the theorem of Riemann-Roch and its application to the study of smooth projective curves over a field. Singular curves are treated through a detailed study of the Picard group. The second part starts with blowing-ups and desingularization (embedded or not) of fibered surfaces over a Dedekind ring that leads on to intersection theory on arithmetic surfaces. Castelnuovo's criterion is proved and also the existence of the minimal regular model. This leads to the study of reduction of algebraic curves. The case of elliptic curves is studied in detail. The book concludes with the fundamental theorem of stable reduction of Deligne-Mumford. The book is essentially self-contained, including the necessary material on commutative algebra. The prerequisites are therefore few, and the book should suit a graduate student. It contains many examples and nearly 600 exercises.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2002

Количество страниц: 463

Добавлена в каталог: 21.05.2007

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Dimension, of a ring      69
Dimension, of a topological space      68
Dimension, of a toric group      312
Dimension, of a unipotent group      306
Dimension, of fibers      137 154—156
Dimension, relative      240
Dimension, virtual relative      240
Discrete valuation ring      64 106
Discrete valuation ring, extension with prescribed residue field      517
Discrete valuation ring, linear equivalence      268
Discrete valuation ring, numerically effective      423
Discrete valuation ring, of poles      269
Discrete valuation ring, vertical      350
Discrete valuation ring, Weil      268
Discrete valuation ring, with normal crossings      see “Normal crossings”
Discrete valuation ring, with strictly normal crossings      378
Discriminant      446
div(f), principal Cartier divisor associated to a rational function      256
div(s), Cartier divisor associated to a rational section of an invertible sheaf      266
Div(X), group of Cartier divisors      256
Divisor, horizontal      349
Domination, a ring dominating a subring      106
Domination, a scheme dominating another one      330
Dual graph of a curve      473 511
Duality      243—247 281
Duality, for global complete intersections      250
Dualizing sheaf      243 389
Dualizing sheaf, and birational morphisms      403
Dualizing sheaf, degree      282
Dualizing sheaf, of a semi-stable curve      509
Dualizing sheaf, of a stable curve      510—511
Dualizing sheaf, of the projective space      242
Dualizing sheaf, on the canonical model      440
Dynkin diagrams      476
Eisenstein polynomial      116
Elimination of indeterminacy      112 396
Elliptic curve      217 (see also “Hyperelliptic curve”)
Elliptic curve, differential forms      218
Elliptic curve, dualizing sheaf on the minimal regular model      423 447
Elliptic curve, filtration on the rational points      498
Elliptic curve, genus      280
Elliptic curve, group of components      497
Elliptic curve, has a hyperelliptic involution      288
Elliptic curve, integral equation      442
Elliptic curve, is an Abelian variety      491
Elliptic curve, minimal discriminant      447
Elliptic curve, minimal regular model      447
Elliptic curve, minimal Weierstrass model      447 449
Elliptic curve, Neron model      494
Elliptic curve, potential semi-stable reduction      501
Elliptic curve, reduction of the minimal regular model      485
Elliptic curve, Weierstrass equation      442
Elliptic curve, Weierstrass model      442 445
Elliptic equation      296
Embedded point      254
End vertex      471
Equidimensional      138 154 234 335
Etale, homomorphism      139
Etale, morphism      139—141 153 176 225
Etale, quasi-section      227
Etale, standard      153
Etale, topology      223
Euler — Poincare characteristic      205 278
Exact sequence      4
Excellent ring      342—344
Excellent scheme      342—347 361—362
Exceptional divisor      412 416—417
Exceptional locus      272 397 411
Extension of morphisms from a normal scheme to a proper scheme      119
Extension of scalars      3
Extension, purely inseparable      89—90 105 214
Extension, separable      89 215 226 342
Exterior algebra      236
f*D, inverse image of a Cartier divisor      262
Fermat curve      550
Fiber of a morphism      83
Fibered product      78
Fibered product, universal property      78
Fibered surface, minimal      418
Fibered surface, normal      347
Fibered surface, over a Dedekind scheme      347
Fibered surface, regular      347
Fibered surface, relatively minimal      418
Fibered surface, the set of regular points of a      343
Field, finitely generated      77
Filtration      16 20 498
Flat homomorphism      6
Flatness      6—15 137 144 144—146 176 207 235 346 347
Flatness, is a local property      10
Flatness, of a quotient algebra      14 139
Flatness, over a Dedekind domain      11
Flatness, over a Dedekind scheme      137
Formal completion      18—24 344
Formal completion, and action of a group      146
Formal completion, does not change the dimension      132
Formal completion, of a local ring of a semi-stable curve      512
Formal completion, of a regular local ring is regular      132
Formal completion, of a semi-local ring      146
Formal completion, of a topological group      17
Formal fiber      341 469—482
Formal functions (theorem on)      200
Formal group      482
Formal power series      18
Frac(A), total ring of fractions      255
Frobenius      94—99 301 394
Frobenius, absolute      94
Frobenius, and differentials      226
Frobenius, and flatness      145
Frobenius, and semi-stable curves      532
Frobenius, is purely inseparable      207
Frobenius, relative      94
Function field      66 77 277
Function, meromorphic      255
Function, rational      66
Fundamental divisor      436—438 450
g(X), genus of a smooth projective curve      279
Generic fiber      83 347
Genus      282
Genus, arithmetic      279 393
Genus, geometric      279
Geometric multiplicity      392 409
Geometrically connected      90—98 105 155 201 208 350—351 496
Geometrically integral      90—98 144 369
Geometrically irreducible      90—98 421
Geometrically reduced      90—98 105 131 209 369
Geometrically regular      342
Germ      see “Section”
Glueing, morphisms      40
Glueing, schemes      49
Glueing, sheaves      40
Going-down Theorem      545
Good reduction      462—464 479 502 545
Good reduction, potential      465 505 548 551
Grading      20
Graph      471 (see also “Dual graph”)
Graph, branch in a graph      476
Graph, of a birational map      112
Graph, of a morphism      110
Graph, of a rational map      111
Grothendieck's vanishing theorem      188
Group of components      496 538 541 554
Group scheme      297—299 302
Group scheme, additive      298
Group scheme, commutative      298
Group scheme, multiplicative      298
Group scheme, toric      312
Group scheme, unipotent      306 316
Group, divisible      538
Group, uniquely $\mathfrak{l}$-divisible      538
Height      69—72 76 332 337
Henselian discrete valuation ring      360 410 467—469
Henselian ring      360
Henselization      360
Higher direct image      189 204 208
Hilbert polynomial      284
Hodge index theorem      383
Homogeneous algebra      53
Homogeneous coordinates      53
Homogeneous element      20
Homogeneous ideal      50
Homomorphism, finite      29
Homomorphism, integral      29 33 69
Homomorphism, local      37
Horizontal divisor      349
ht(I), height of an ideal I      69
Hurwitz formula      289
Hyperelliptic curve      287 292
Hyperelliptic curve, bad reduction      466 480
Hyperelliptic curve, equation      296
Hyperelliptic curve, good reduction      464 480
Hyperelliptic curve, involution      294
I-adic topology      16
Idempotent      66
Identity component      496 504
Immersion      96 200
Immersion, closed      38 45 113
Immersion, open      38 45
Index of a curve      393
Inertia group      147 526 532
Integral closure      120 330 340 345
Integral equation      29
Intersection number      376 383
Inverse limit      16
Inverse limit, universal property      16
Inverse system      16
Irreducible component      62—63 97
Irreducible component, geometric      97
Isomorphism and base change      194
j(E), modular invariant of an elliptic curve      500
Jacobian conjecture      134
Jacobian criterion      130 134 147
Jacobian criterion, for projective varieties      134
Jacobian matrix      130
Jacobian variety      299 504 542
Jordan — Holder theorem      258
K(X), field of rational functions      66
k(x), residue field at a point x      37
Kahler differentials      see “Relative differential forms”
Klein curve      479 551
Krull's principal ideal theorem      71
Krull's theorem on the intersection of submodules      21
l(D), dimension of L(D)      279
L(D), global sections of ${\mathcal{O} _{X}} (D)$      279
l.c.i.      see “Local complete intersection”
Leibniz rule      210
Length, of a module      258—259 262—265 274
Length, of a path      472
Leray acyclicity theorem      183
Lifting, infinitesimal      225
Lifting, of a closed point      468
Lifting, of rational points      224
Lifting, of the formal structure      235
Local complete intersection      232 338 339 438 451 510
Local ring      9
localization      9
Luroth's theorem      292
Meromorphic functions      255
Mittag — Leffler condition      24
Model      456
Model, minimal regular      418 422 455
Model, morphism      418 455
Model, of $\mathbb{P} _{K}^{1}$      482 (see also “Conic”)
Model, of a surface      418
Model, of algebraic curves      455
Model, regular      455
Model, relatively minimal      418
Model, smooth      455
Model, with normal crossings      455 456 503 534 552
Modular curve      551
Modular invariant      500
Module, Artinian      259
Module, Cohen-Macaulay      337 346
Module, differential      178
Module, faithful      122
Module, faithfully flat      13
Module, flat      6 11 193
Module, graded      20 164
Module, of relative differential forms      see “Relative differential forms”
Module, simple      213 258 264
Module, torsion-free      8
Monoidal transformation      395
Mor(X, Y), set of morphisms from X to Y      47
Morphism, closed      103
Morphism, constant      137
Morphism, diagonal      100 216
Morphism, dominant      67 86 98 278
Morphism, faithfully flat      145
Morphism, finite      112—113 155 176 194 246 301
Morphism, finite and birational      362
Morphism, flat      136—139 145
Morphism, generically etale      227
Morphism, generically separable      227 249
Morphism, integral      112 120 545
Morphism, into a projective scheme      57 169
Morphism, into an affine scheme      48 57
Morphism, l.c.i.      see “Local complete intersection”
Morphism, of finite type      87 96 96—110
Morphism, of projective schemes      53
Morphism, of schemes      45
Morphism, open      58 145
Morphism, projection      78
Morphism, projective      83 108
Morphism, proper      103—107 110—112 194
Morphism, proper birational      150 371 373 408
Morphism, purely inseparable      200 207 207—208 291 419 532
Morphism, quasi-compact      58 67 96 163 188
Morphism, quasi-finite      97 140 152 155
Morphism, quasi-projective      109 112 152 155
Morphism, quotient      41 146
Morphism, ramified      290 294
Morphism, separable      289
Morphism, separated      100—102
Morphism, structural      47
Morphism, tamely ramified      290 300
Morphism, universally closed      103
Moving Lemma      379 380 391 429
Multiplicative subset      9
Multiplicity, intersection multiplicity      376
Multiplicity, of a Cartier divisor      260
Multiplicity, of a cycle      267
Multiplicity, of a hypersurface at a point      402
Multiplicity, of an irreducible component      305 352 368 392
Nagata ring      340—341 343 371—372
Nagata scheme      341 356 362
Nakayama's lemma      9
Neron model      489 494 504
Neron model, universal property      490
Neron — Ogg — Shafarevich's criterion      502
Nilpotent      28 32
Nilradical      31 60
Node in a graph      475
Noether's normalization lemma      29 155
Norm      77 272—274
Normal crossings      378 391 474
Normal crossings, strictly      378
Normal law      493
Normal ring      115—123 130 133 273 344
Normal scheme      115—126 130—131 152 201 269 339 545
Normalization      119—126 303 343 373 534
Normalization, and arithmetic genus      304
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