Quantitative arithmetic of projective varieties:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Basel [u.a.]
Birkhäuser
2009
|
| Series: | Progress in mathematics
277 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XI, 160 S. Ill. 235 mm x 155 mm |
| ISBN: | 9783034601283 9783034601290 |
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MARC
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Record in the Search Index
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| adam_text | Titel: Quantitative arithmetic of projective varieties
Autor: Browning, Timothy D.
Jahr: 2009
Contents
Preface xiii
1 Introduction 1
1.1 A naive heuristic............................ 2
1.2 The basic counting function...................... 5
1.3 Influence of analytic number theory ................. 8
1.3.1 Paucity results......................... 9
1.3.2 Waring s problem........................ 10
1.3.3 Vinogradov s mean value theorem .............. 11
1.3.4 Small solutions......................... 12
1.3.5 Divisor problems........................ 14
Exercises for Chapter 1 ........................... 15
2 The Manin conjectures 17
2.1 Divisors on varieties.......................... 17
2.1.1 The Picard group........................ 18
2.1.2 The canonical divisor ..................... 19
2.1.3 The intersection form ..................... 19
2.1.4 Cubic surfaces ......................... 19
2.2 The conjectures............................. 20
2.3 Degree 3................................. 24
2.3.1 Non-singular surfaces ..................... 24
2.3.2 Singular surfaces........................ 27
2.4 Degree 4................................. 33
2.4.1 Non-singular surfaces ..................... 33
2.4.2 Singular surfaces........................ 37
2.5 Degree ^ 5 ............................... 41
2.6 Universal torsors............................ 43
Exercises for Chapter 2 ........................... 46
Contents
i The dimension growth conjecture
3.1 Linear spaces on hypersurfaces.................... 49
3.2 Dimension growth for hypersurfaces ................. 52
3.3 Exponential sums............................
3.3.1 Singular X........................... 57
3.3.2 Non-singular X......................... 58
3.4 Covering with linear spaces...................... 59
Exercises for Chapter 3 ........................... 62
1 Uniform bounds for curves and surfaces 63
4.1 The determinant method ....................... 70
4.2 The geometry of numbers....................... 72
4.3 General plane curves.......................... 7°
4.4 Diagonal plane curves......................... 79
Exercises for Chapter 4 ........................... 82
5 Ai del Pezzo surface of degree 6 83
5.1 Passage to the universal torsor.................... 84
5.2 The asymptotic formula........................ 88
5.3 Perron s formula............................ 91
Exercises for Chapter 5 ........................... 97
6 D4 del Pezzo surface of degree 3 99
6.1 Passage to the universal torsor.................... 100
6.2 A crude upper bound ......................... 103
6.3 A better upper bound......................... 105
Exercises for Chapter 6 ........................... Ill
7 Siegel s lemma and non-singular surfaces 113
7.1 Dual variety............................... 115
7.2 Non-singular del Pezzo surfaces of degree 3............. 116
7.3 Non-singular del Pezzo surfaces of degree 4............. 118
Exercises for Chapter 7 ........................... 122
8 The Hardy-Littlewood circle method 123
8.1 Major arcs and minor arcs....................... 125
8.2 Quartic hypersurfaces......................... 128
8.2.1 The minor arcs......................... 129
8.2.2 The major arcs......................... 134
8.2.3 Improving Birch s argument.................. 136
8.3 Diagonal cubic surfaces ........................ 137
8.3.1 The lines on a diagonal cubic surface............. 138
8.3.2 Cubic characters and Jacobi sums.............. 140
8.3.3 The heuristic................. 142
Contents xi
Exercises for Chapter 8 ...........................149
Bibliography 151
Index 159
|
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| indexdate | 2024-07-09T22:10:57Z |
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| isbn | 9783034601283 9783034601290 |
| language | English |
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| publishDate | 2009 |
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| spelling | Browning, Timothy D. Verfasser aut Quantitative arithmetic of projective varieties Timothy D. Browning Basel [u.a.] Birkhäuser 2009 XI, 160 S. Ill. 235 mm x 155 mm txt rdacontent n rdamedia nc rdacarrier Progress in mathematics 277 Algebraic varieties Arithmetical algebraic geometry Diophantine equations Geometry, Algebraic Number theory Projektive Varietät (DE-588)4327070-0 gnd rswk-swf Analytische Zahlentheorie (DE-588)4001870-2 gnd rswk-swf Diophantische Geometrie (DE-588)4150021-0 gnd rswk-swf Projektive Varietät (DE-588)4327070-0 s Diophantische Geometrie (DE-588)4150021-0 s Analytische Zahlentheorie (DE-588)4001870-2 s DE-604 Progress in mathematics 277 (DE-604)BV000004120 277 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018966301&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Browning, Timothy D. Quantitative arithmetic of projective varieties Progress in mathematics Algebraic varieties Arithmetical algebraic geometry Diophantine equations Geometry, Algebraic Number theory Projektive Varietät (DE-588)4327070-0 gnd Analytische Zahlentheorie (DE-588)4001870-2 gnd Diophantische Geometrie (DE-588)4150021-0 gnd |
| subject_GND | (DE-588)4327070-0 (DE-588)4001870-2 (DE-588)4150021-0 |
| title | Quantitative arithmetic of projective varieties |
| title_auth | Quantitative arithmetic of projective varieties |
| title_exact_search | Quantitative arithmetic of projective varieties |
| title_full | Quantitative arithmetic of projective varieties Timothy D. Browning |
| title_fullStr | Quantitative arithmetic of projective varieties Timothy D. Browning |
| title_full_unstemmed | Quantitative arithmetic of projective varieties Timothy D. Browning |
| title_short | Quantitative arithmetic of projective varieties |
| title_sort | quantitative arithmetic of projective varieties |
| topic | Algebraic varieties Arithmetical algebraic geometry Diophantine equations Geometry, Algebraic Number theory Projektive Varietät (DE-588)4327070-0 gnd Analytische Zahlentheorie (DE-588)4001870-2 gnd Diophantische Geometrie (DE-588)4150021-0 gnd |
| topic_facet | Algebraic varieties Arithmetical algebraic geometry Diophantine equations Geometry, Algebraic Number theory Projektive Varietät Analytische Zahlentheorie Diophantische Geometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018966301&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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