Quantitative Arithmetic of Projective Varieties

168 Pages·2009·1.99 MB·English

by  Timothy D. Browning (auth.)

Progress in Mathematics
Volume 277
Series Editors
H. Bass
J. Oesterlé
A. Weinstein
Timothy D. Browning
Quantitative Arithmetic
of Projective Varieties
Birkhäuser
Basel · Boston · Berlin
Author:
Timothy D. Browning
School of Mathematics
University of Bristol
Bristol BS8 1TW
United Kingdom
e-mail: [email protected]
2000 Mathematics Subject Classification: 11D45, 11G35, 11D72, 11E76, 11P55, 14G05,
14G10, 14G25
Library of Congress Control Number: 2009934090
Bibliographic information published by Die Deutsche Bibliothek.
Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie;
detailed bibliographic data is available in the Internet at http://dnb.ddb.de
ISBN 978-3-0346-0128-3 Birkhäuser Verlag AG, Basel · Boston · Berlin
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Ferran Sunyer i Balaguer (1912–1967) was a self-
taught Catalan mathematician who, in spite of a
seriousphysicaldisability,wasveryactiveinresearch
in classical mathematical analysis, an area in which
he acquired international recognition. His heirs cre-
ated the Fundacio´ Ferran Sunyer i Balaguer inside
theInstitutd’EstudisCatalanstohonorthememory
of Ferran Sunyer i Balaguer and to promote mathe-
matical research.
Each year, the Fundacio´ Ferran Sunyer i Balaguer
and the Institut d’Estudis Catalans award an in-
ternational researchprize for a mathematical mono-
graphofexpositorynature.Theprize-winningmono-
graphsarepublishedinthisseries.Detailsaboutthe
prizeandtheFundacio´FerranSunyeriBalaguercan
be found at
http://www.crm.es/FSBPrize/ffsb.htm
This book has been awarded the
Ferran Sunyer i Balaguer 2009 prize.
The members of the scientific commitee
of the 2009 prize were:
Hyman Bass
University of Michigan
Nu´ria Fagella
Universitat de Barcelona
Joan Verdera
Universitat Auto`noma de Barcelona
Alan Weinstein
University of California at Berkeley
Ferran Sunyer i Balaguer Prize winners since 1998:
1999 Patrick Dehornoy
Braids and Self-Distributivity, PM 192
2000 Juan-Pablo Ortega and Tudor Ratiu
Hamiltonian Singular Reduction, PM 222
2001 Martin Golubitsky and Ian Stewart
The Symmetry Perspective, PM 200
2002 Andr´e Unterberger
Automorphic Pseudodifferential Analysis
and Higher Level Weyl Calculi, PM 209
Alexander Lubotzky and Dan Segal
Subgroup Growth, PM 212
2003 Fuensanta Andreu-Vaillo, Vincent Caselles
and Jos´e M. Mazo´n
Parabolic Quasilinear Equations Minimizing
Linear Growth Functionals, PM 223
2004 Guy David
Singular Sets of Minimizers for the
Mumford-Shah Functional, PM 233
2005 Antonio Ambrosetti and Andrea Malchiodi
Perturbation Methods and Semilinear
Elliptic Problems on Rn, PM 240
Jos´e Seade
On the Topology of Isolated Singularities in
Analytic Spaces, PM 241
2006 Xiaonan Ma and George Marinescu
Holomorphic Morse Inequalities and
Bergman Kernels, PM 254
2007 Rosa Miro´-Roig
Determinantal Ideals, PM 264
2008 Luis Barreira
Dimension and Recurrence in Hyperbolic
Dynamics, PM 272
To my wife Sinead
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 (cid:2)5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.6 Universal torsors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Exercises for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
x Contents
3 The dimension growth conjecture 47
3.1 Linear spaces on hypersurfaces . . . . . . . . . . . . . . . . . . . . 49
3.2 Dimension growth for hypersurfaces . . . . . . . . . . . . . . . . . 52
3.3 Exponential sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
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
4 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 . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.4 Diagonal plane curves . . . . . . . . . . . . . . . . . . . . . . . . . 79
Exercises for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5 A del Pezzo surface of degree 6 83
1
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 D del Pezzo surface of degree 3 99
4
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 . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
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