Algebraic geometry and arithmetic curves:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Weitere Verfasser: | |
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Oxford
Oxford University Press
2008
|
| Ausgabe: | 1. publ. in paperback, reprint. |
| Schriftenreihe: | Oxford graduate texts in mathematics
6 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Hier auch später erschienene, unveränderte Nachdrucke |
| Beschreibung: | XV, 577 Seiten Diagramme |
| ISBN: | 0199202494 9780199202492 0198502842 9780198502845 |
Internformat
MARC
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| 100 | 1 | |a Liu, Qing |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Algebraic geometry and arithmetic curves |c Qing Liu (CNRS Laboratoire de Théorie des Nombres et d'Algorithmique Arithmétique, Université Bordeaux 1) ; translated by Reinie Erné (Institut de Recherche Mathématique de Rennes, Université Rennes 1) |
| 250 | |a 1. publ. in paperback, reprint. | ||
| 264 | 1 | |a Oxford |b Oxford University Press |c 2008 | |
| 300 | |a XV, 577 Seiten |b Diagramme | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Oxford graduate texts in mathematics |v 6 | |
| 490 | 0 | |a Oxford mathematics | |
| 500 | |a Hier auch später erschienene, unveränderte Nachdrucke | ||
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| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Erné, Reinie |4 trl | |
| 830 | 0 | |a Oxford graduate texts in mathematics |v 6 |w (DE-604)BV011416591 |9 6 | |
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Datensatz im Suchindex
| _version_ | 1804138515232980992 |
|---|---|
| adam_text | Contents
Some topics in commutative algebra
1
1.1
Tensor products
1
1.1.1
Tensor product of modules
1
1.1.2
Right-exactness of the tensor product
4
1.1.3
Tensor product of algebras
5
1.2
Flatness
6
1.2.1
Left-exactness: flatness
6
1.2.2
Local nature of flatness
9
1.2.3
Faithful flatness
12
1.3
Formal completion
15
1.3.1
Inverse limits and completions
15
1.3.2
The Artin-Rees lemma and applications
20
1.3.3
The case of Noetherian local rings
22
General properties of schemes
26
2.1
Spectrum of a ring
26
2.1.1
Zariski topology
26
2.1.2
Algebraic sets
29
2.2
Ringed topological spaces
33
2.2.1
Sheaves
33
2.2.2
Ringed topological spaces
37
2.3
Schemes
41
2.3.1
Definition of schemes and examples
42
2.3.2
Morphisms of schemes
45
2.3.3
Projeetive schemes
50
2.3.4
Noetherian schemes, algebraic varieties
55
2.4
Reduced schemes and integral schemes
59
2.4.1
Reduced schemes
59
xij Contents
2.4.2
Irreducible components
61
2.4.3
Integral schemes
64
2.5
Dimension
67
2.5.1
Dimension of schemes
68
2.5.2
The case of Noetherian schemes
70
2.5.3
Dimension of algebraic varieties
73
3
Morphisms and base change
78
3.1
The technique of base change
78
3.1.1
Fibered product
78
3.1.2
Base change
81
3.2
Applications to algebraic varieties
87
3.2.1
Morphisms of finite type
87
3.2.2
Algebraic varieties and extension of the base field
89
3.2.3
Points with values in an extension of the base field
92
3.2.4
Frobenius
94
3.3
Some global properties of morphisms
99
3.3.1
Separated morphisms
99
3.3.2
Proper morphisms
103
3.3.3
Projective
morphisms
107
4
Some local properties
115
4.1
Normal schemes
115
4.1.1
Normal schemes and extensions of regular functions
115
4.1.2
Normalization
119
4.2
Regular schemes
126
4.2.1
Tangent space to a scheme
126
4.2.2
Regular schemes and the Jacobian criterion
128
4.3
Flat morphisms and smooth morphisms
135
4.3.1
Flat morphisms
136
4.3.2
Etale
morphisms
139
4.3.3
Smooth morphisms
141
4.4
Zariski s Main Theorem and applications
149
5
Coherent sheaves and
Čech cohomology
157
5.1
Coherent sheaves on a scheme
157
5.1.1
Sheaves of modules
157
5.1.2
Quasi-coherent sheaves on an
affine
scheme
159
5.1.3
Coherent sheaves
161
5.1.4
Quasi-coherent sheaves on
a projective
scheme
164
Contents xiii
5.2
Čech
cohomology 178
5.2.1 Differential
modules and
cohomology
with values
in a sheaf
178
5.2.2
Cech
cohomology on a separated scheme
185
5.2.3
Higher direct image and flat base change
188
5.3
Cohomology of
projective
schemes
195
5.3.1
Direct image theorem
195
5.3.2
Connectedness principle
198
5.3.3
Cohomology of the fibers
201
6
Sheaves of differentials
210
6.1 Kahler
differentials
210
6.1.1
Modules of relative differential forms
210
6.1.2
Sheaves of relative differentials (of degree
1) 215
6.2
Differential study of smooth morphisms
220
6.2.1
Smoothness criteria
220
6.2.2
Local structure and lifting of sections
223
6.3
Local complete intersection
227
6.3.1
Regular immersions
228
6.3.2
Local complete intersections
232
6.4
Duality theory
236
6.4.1
Determinant
236
6.4.2
Canonical sheaf
238
6.4.3
Grothendieck duality
243
7
Divisors and applications to curves
252
7.1
Cartier
divisors
252
7.1.1
Meromorphic functions
252
7.1.2
Cartier
divisors
256
7.1.3
Inverse image of
Cartier
divisors
260
7.2
Weil divisors
267
7.2.1
Cycles of codimension
1 267
7.2.2
Van
der Waerden s
purity theorem
272
7.3
Riemann-Roch theorem
275
7.3.1
Degree of a divisor
275
7.3.2
Riemann-Roch for
projective
curves
278
■„т
Contents
7.4
Algebraic curves
284
7.4.1
Classification of curves of small genus
284
7.4.2
Hurwitz formula
289
7.4.3
Hyperelliptic curves
292
7.4.4
Group schemes and
Picard
varieties
297
7.5
Singular curves, structure of Pic°(X)
303
8 Birational
geometry of surfaces
317
8.1
Blowing-ups
317
8.1.1
Definition and elementary properties
318
8.1.2
Universal property of blowing-up
323
8.1.3
Blowing-ups and
birational
morphisms
326
8.1.4
Normalization of curves by blowing-up points
330
8.2
Excellent schemes
332
8.2.1
Universally catenary schemes and the
dimension formula
332
8.2.2
Cohen-Macaulay rings
335
8.2.3
Excellent schemes
341
8.3
Fibered surfaces
347
8.3.1
Properties of the fibers
347
8.3.2
Valuations and
birational
classes of fibered surfaces
353
8.3.3
Contraction
356
8.3.4
Desingularization
361
9
Regular surfaces
375
9.1
Intersection theory on a regular surface
376
9.1.1
Local intersection
376
9.1.2
Intersection on a fibered surface
381
9.1.3
Intersection with a horizontal divisor,
adjunction formula
388
9.2
Intersection and morphisms
394
9.2.1
Factorization theorem
394
9.2.2
Projection formula
397
9.2.3 Birational
morphisms and
Picard
groups
401
9.2.4
Embedded resolutions
404
9.3
Minimal surfaces
411
9.3.1
Exceptional divisors and Castebuovo s criterion
412
9.3.2
Relatively minimal surfaces
418
9.3.3
Existence of the
minimal
regular model
421
9.3.4
Minimal desingularization and minimal
embedded resolution
424
Contents xv
9.4 Applications
to contraction; canonical model
429
9.4.1
Artin s contractability criterion
430
9.4.2
Determination of the tangent spaces
434
9.4.3
Canonical models
438
9.4.4
Weierstrass
models and regular models of
elliptic curves
442
10
Reduction of algebraic curves
454
10.1
Models and reductions
454
10.1.1
Models of algebraic curves
455
10.1.2
Reduction
462
10.1.3
Reduction map
467
10.1.4
Graphs
471
10.2
Reduction of elliptic curves
483
10.2.1
Reduction of the minimal regular model
484
10.2.2
Néron
models of elliptic curves
489
10.2.3
Potential semi-stable reduction
498
10.3
Stable reduction of algebraic curves
505
10.3.1
Stable curves
505
10.3.2
Stable reduction
511
10.3.3
Some sufficient conditions for the existence of
the stable model
521
10.4
Deligne-Mumford theorem
532
10.4.1
Simplifications on the base scheme
533
10.4.2
Proof of Artin-Winters
537
10.4.3
Examples of computations of the potential
stable reduction
543
Bibliography
557
Index
562
|
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| author | Liu, Qing |
| author2 | Erné, Reinie |
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| id | DE-604.BV035242196 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T21:29:25Z |
| institution | BVB |
| isbn | 0199202494 9780199202492 0198502842 9780198502845 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017047965 |
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| physical | XV, 577 Seiten Diagramme |
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| series2 | Oxford graduate texts in mathematics Oxford mathematics |
| spelling | Liu, Qing Verfasser aut Algebraic geometry and arithmetic curves Qing Liu (CNRS Laboratoire de Théorie des Nombres et d'Algorithmique Arithmétique, Université Bordeaux 1) ; translated by Reinie Erné (Institut de Recherche Mathématique de Rennes, Université Rennes 1) 1. publ. in paperback, reprint. Oxford Oxford University Press 2008 XV, 577 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Oxford graduate texts in mathematics 6 Oxford mathematics Hier auch später erschienene, unveränderte Nachdrucke Algebraische Geometrie (DE-588)4001161-6 gnd rswk-swf Algebraische Kurve (DE-588)4001165-3 gnd rswk-swf Algebraische Geometrie (DE-588)4001161-6 s Algebraische Kurve (DE-588)4001165-3 s DE-604 Erné, Reinie trl Oxford graduate texts in mathematics 6 (DE-604)BV011416591 6 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017047965&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Liu, Qing Algebraic geometry and arithmetic curves Oxford graduate texts in mathematics Algebraische Geometrie (DE-588)4001161-6 gnd Algebraische Kurve (DE-588)4001165-3 gnd |
| subject_GND | (DE-588)4001161-6 (DE-588)4001165-3 |
| title | Algebraic geometry and arithmetic curves |
| title_auth | Algebraic geometry and arithmetic curves |
| title_exact_search | Algebraic geometry and arithmetic curves |
| title_full | Algebraic geometry and arithmetic curves Qing Liu (CNRS Laboratoire de Théorie des Nombres et d'Algorithmique Arithmétique, Université Bordeaux 1) ; translated by Reinie Erné (Institut de Recherche Mathématique de Rennes, Université Rennes 1) |
| title_fullStr | Algebraic geometry and arithmetic curves Qing Liu (CNRS Laboratoire de Théorie des Nombres et d'Algorithmique Arithmétique, Université Bordeaux 1) ; translated by Reinie Erné (Institut de Recherche Mathématique de Rennes, Université Rennes 1) |
| title_full_unstemmed | Algebraic geometry and arithmetic curves Qing Liu (CNRS Laboratoire de Théorie des Nombres et d'Algorithmique Arithmétique, Université Bordeaux 1) ; translated by Reinie Erné (Institut de Recherche Mathématique de Rennes, Université Rennes 1) |
| title_short | Algebraic geometry and arithmetic curves |
| title_sort | algebraic geometry and arithmetic curves |
| topic | Algebraische Geometrie (DE-588)4001161-6 gnd Algebraische Kurve (DE-588)4001165-3 gnd |
| topic_facet | Algebraische Geometrie Algebraische Kurve |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017047965&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV011416591 |
| work_keys_str_mv | AT liuqing algebraicgeometryandarithmeticcurves AT ernereinie algebraicgeometryandarithmeticcurves |