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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
Normalization, and flatness      136
Normalization, by blowing-up      341
Normalization, in an extension      120 278
Normalization, of a semi-stable curve      507—508 529
Nullstellensatz      30—31
Number field      58
Number field, ring of integers      58 69 112 139 268 454
Open subscheme      44
Open subscheme, of an affine Dedekind scheme      123
Ordinary double point      310 315 483 506 526
Ordinary double point, split      508 514
Ordinary multiple point      310 316 521 531
Ordinary multiple point, universal property      316
p-adic integers      18
Parameters (system of)      129
Path      471
Pic(X), Picard group      173
Picard group      192 261 408 538
Picard group, and group of Cartier divisors      257 261 264
Picard group, of a conic      418
Picard group, of a factorial domain      273
Picard group, of an affine line      346
Picard group, restriction to an open subset      401
Picard group, torsion points      299 540
Point, general      64
Point, generic      63 65
Point, of codimension 1      119 267
Point, regular      128
Point, singular      128
Point, with value in an algebraic extension      92
Poles      see “Divisor of poles”
Presheaf      33
Presheaf, injective morphism      35
Presheaf, morphism      35
Presheaf, surjective morphism      35
Principal closed subset      27 74
Principal open subset      27 51
Proj $\mathcal{B}$, scheme associated to a homogeneous sheaf of algebras      321
Proj B, scheme associated to a graded algebra      52
Proj B, set of homogeneous prime ideals of a graded algebra      50
Projection formula      190 398
Projection, and base change      81
Projection, from a center      87 246
Projection, to a component      78
Projective space      50
Projective variety      56
Property local, on the base      84
Pure (dimension)      68
Quasi-coherent sheaf      158
Quasi-compact (topological space)      33
Quasi-section      227
Quotient, of a ringed topological space by a group      41
Quotient, of a scheme by a group      see scheme (quotient)
Quotient, of an algebraic variety by an algebraic group      59
Radical ideal      63
Radical of an ideal      27
Ramification, index      265 289
Ramification, locus      290 347
Ramification, point      290
Rational functions      see “Function field”
Rational map      111 348 367 494
Rational map, domain of definition      111
Rational point      148
Rational points      53 388 468 498
Rational points, and Galois extension      93
Rational points, are closed      76
Rational points, of a projective scheme      54
Rational points, of a subscheme      49
Rational points, of an affine scheme      49
Rational section of a sheaf      266
Reduction      462
Reduction map      467—469 481 497
Reduction, additive      485
Reduction, bad reduction      462 466
Reduction, non-split multiplicative      485
Reduction, of rational points      468
Reduction, potential multiplicative      499 505
Reduction, potential semi-stable      499
Reduction, semi-Abelian      533 542—543 551
Reduction, split multiplicative      485
Reg(X), set of regular points of X      131
Regular element      see “Regular sequence”
Regular function      44 60 61 66
Regular function, extension of      118
Regular immersion      228
Regular ring      128—135 337 346
Regular scheme      128—135 140—144
Regular scheme, and branch locus      347
Regular scheme, and divisors      271
Regular scheme, and local complete intersection      232
Regular scheme, birational morphism to a      272
Regular scheme, blowing-up      325
Regular scheme, embedding in a smooth scheme      226
Regular scheme, is Cohen — Macaulay      337
Regular scheme, is universally catenary      338
Regular scheme, not smooth      142
Regular sequence      228 335
Regular sequence, and flat base change      338
Relative differential forms      210 (see also “Differential forms” “Dualizing
Relative differential forms, module of      210—215 219
Relative differential forms, sheaf of      216—235 529
Relative differential forms, universal property      210
Residue field      11 37 46
Resolution of singularities      see “Desingularization”
Restriction map      33
Restriction, of a section      34
Restriction, of a sheaf      34
Riemann hypothesis      394
Riemann — Roch theorem      281—282
Riemann's theorem      279
Ring, Artinian      70
Ring, Cohen — Macaulay      337—345
Ring, factorial      130
Ring, graded      20 23 135
Ring, local      9
Ring, reduced      59
Ring, semi-local      146 344
Ringed topological space      37
Ringed topological space, morphism      37
S-scheme      47
S-valuation      354
Scheme      44
Scheme, base      47
Scheme, Cohen — Macaulay      338—346
Scheme, connected      66 97 110—111 200 208 266 331
Scheme, finite      76
Scheme, integral      65
Scheme, irreducible      62 (see also “Geometrically irreducible”)
Scheme, local      66
Scheme, locally noetherian      55
Scheme, Noetherian      55
Scheme, non-separated      101
Scheme, of connected components      496—497 (see also “Group of components”)
Scheme, of dimension 0      76
Scheme, projective      53 83
Scheme, proper      103
Scheme, pure      see “Equidimensional”
Scheme, quasi-compact      57—60 67 160
Scheme, quasi-projective      109 113 124 171 188
Scheme, quotient      59 113 124 146 526 531
Scheme, reduced      59
Scheme, separated      100 110
Scheme, singular      128
Scheme, smooth      see “Smooth morphism”
Scheme, unibranch      223
Scheme-theoretic closure      58
Section, germ of a      35
Section, intersection with the special fiber      388
Section, lifting      224
Section, of a fibered product      81
Section, of a presheaf      34
Section, of a scheme      49
Section, of a separated morphism      110
Section, with values in an extension      81
Segre embedding      108 177
Self-intersection      383
Semi-stable curve      510
Semi-stable curve, dualizing sheaf      509
Semi-stable curve, graph      511
Semi-stable curve, local structure      514 531
Semi-stable curve, quotient by a group      526
Semi-stable model      515 522
Semi-stable reduction      515 518 533
Semi-stable reduction, in characteristic 0      536
Separated topological space      27
Serre duality      236
Serre's criterion for affine schemes      187 193
Serre's criterion for normality      339
Sheaf      34
Sheaf, associated to a presheaf      36 39
Sheaf, conormal      229
Sheaf, constant      39
Sheaf, direct image      37 163
Sheaf, direct limit      192
Sheaf, flasque      190 264
Sheaf, flat      189
Sheaf, free      173
Sheaf, generated by global sections      158 169—171 177—178 285 359 440
Sheaf, higher direct image      189
Sheaf, image      36
Sheaf, inverse image      37 163
Sheaf, invertible      165 173
Sheaf, locally free      173
Sheaf, normal      229
Sheaf, of algebras      34 175
Sheaf, of graded algebras      321
Sheaf, of homogeneous algebras      321
Sheaf, of ideals      38
Sheaf, of meromorphic functions      255 263—266
Sheaf, of modules      157
Sheaf, of relative differential forms      see “Relative differential forms”
Sheaf, of rings      34
Sheaf, pull-back      163
Sheaf, quasi-coherent      158
Sheaf, quotient      36
Sheaf, skyscraper      40
Sheaf, stalk      35
Sheaf, structure      37
Sheaf, surjective morphism      35
Sheaf, torsion-free      174
Sheaf, twists      166
Sing(X), set of singular points of X      131
Smooth locus      224 227 494
Smooth morphism      142—149
Smooth morphism, and differentials      222 227 230 247
Smooth morphism, and lifting of rational points      224
Smooth morphism, and normality      339
Smooth morphism, and quasi-sections      227
Smooth morphism, and regularity      142
Smooth morphism, embedding      226 407
Smooth morphism, local structure      224 226
Smooth morphism, smooth fibers      352 (see also “Good reduction”)
sp(X), underlying topological space of a scheme X      81
Spec A, spectrum of A      26
Spec$ \mathcal{A}$, spectrum of a quasi-coherent $\mathcal{O} _{X}$-algebra      175
Spec$ \varphi$, morphism of schemes associated to a ring homomorphism $\varphi$      28
Special fiber      see “Closed fiber”
Specialization      63—64 467
Spectral sequence      196 522
Spectrum      26
Stable curve      506 510
Stable curve, automorphisms      520 529
Stable curve, dualizing sheaf      510—511
Stable model      515
Stable model, and canonical model      518
Stable reduction      515 523 533
Stable reduction, and base change      520 533
Stable reduction, and covering      544
Stable reduction, and purely inseparable base extension      552
Stable reduction, and semi-Abelian reduction      543
Stable reduction, Deligne-Mumford's theorem      533
Stable reduction, extension realizing the stable reduction      551—552
Stable reduction, of a Fermat curve      548
Stable reduction, potential      543 544 547
Stein factorization      208
Strict transform      324 331 402 411
Strictly normal crossings      378
Subgraph      471 483
Subscheme      see “Closed subscheme” “Open
Subsheaf      34
Subvariety      56
Supp $\mathcal{F}$, support of a sheaf $\mathcal{F}$      40
Supp D, support of a Cartier divisor      260
Supp M, support of a module      336
Supp Z, support of a cycle      267
Support      173
Support, of a Cartier divisor      260 274
Support, of a cycle      267
Support, of a sheaf      40 56
surface      75 317
Surface, any proper regular algebraic surface is projective      413
Surface, rational      427
Surface, ruled      427
System of parameters      129
t(X), toric rank of a curve      314
u(X), unipotent rank of curve      314
V(f), principal closed subset associated to f      27 74
V(I), closed subset defined by an ideal I      26
X(K), set of points of X with values in a field K      92
X(S), set of sections of an S-scheme X      49
Z(I), set of common zeros of the polynomials contained in I      31
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