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© 2007

Intersections de deux quadriques et pinceaux de courbes de genre 1

Intersections of two quadrics and pencils of curves of genus 1

Book

Part of the Lecture Notes in Mathematics book series (LNM, volume 1901)

Table of contents

About this book

Introduction

Cet ouvrage est consacré à l'arithmétique des surfaces fibrées en courbes de genre 1 au-dessus de la droite projective, et à l'arithmétique des intersections de deux quadriques dans l'espace projectif. Swinnerton-Dyer introduisit en 1993 une technique permettant d'étudier les points rationnels des pinceaux de courbes de genre 1. La première moitié de l'ouvrage reprend et développe cette technique ainsi que ses généralisations ultérieures. La seconde moitié, qui repose sur la première, porte sur les surfaces de del Pezzo de degré 4 et sur les intersections de deux quadriques de dimension supérieure; les résultats annoncés dans [C. R. Math. Acad. Sci. Paris 342 (2006), no. 4, 223--227] y sont démontrés.

This research monograph focuses on the arithmetic, over number fields, of
surfaces fibred into curves of genus 1 over the projective line, and of
intersections of two quadrics in projective space.  The first half contains a
complete account of the technique initiated by Swinnerton-Dyer in 1993 for
studying rational points on pencils of curves of genus 1, while incorporating
and generalising most of its subsequent refinements.  The second half, which
builds upon the first, is devoted to quartic del Pezzo surfaces and
higher-dimensional intersections of two quadrics.  It culminates in the proof
of the results announced in [C. R. Math. Acad. Sci. Paris 342 (2006), no. 4,
223--227].

Keywords

Brauer group Brauer-Manin obstruction Hasse principle MSC(2000):11G35, 14J20, 11D09, 14G05 Pinceau de courbes elliptiques del Pezzo surface groupe de Brauer obstruction de Brauer-Manin pencil of elliptic curves principe de Hasse surface de del Pezzo

Authors and affiliations

  1. 1.Department of MathematicsMS 136 Rice University77251-1892HoustonUSA

Bibliographic information

  • Book Title Intersections de deux quadriques et pinceaux de courbes de genre 1
  • Book Subtitle Intersections of two quadrics and pencils of curves of genus 1
  • Authors Olivier Wittenberg
  • Series Title Lecture Notes in Mathematics
  • DOI https://doi.org/10.1007/3-540-69137-5
  • Copyright Information Springer-Verlag Berlin Heidelberg 2007
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Softcover ISBN 978-3-540-69137-2
  • eBook ISBN 978-3-540-69141-9
  • Series ISSN 0075-8434
  • Series E-ISSN 1617-9692
  • Edition Number 1
  • Number of Pages VIII, 218
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Number Theory
    Algebraic Geometry
  • Buy this book on publisher's site
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Reviews

From the reviews:

"The monograph under review grew out of the author’s Ph.D thesis. … The monograph is written in extremely careful manner. … There are two introductions: the English one makes emphasis on the general context, whereas the French one contains more technical descriptions and references. To sum up, the monograph can be recommended to everyone interested in the state of the art of methods and results concerning arithmetic of rational varieties." (Boris Kunyavskii, Zentralblatt MATH, Vol. 1122 (24), 2007)