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The Brauer-Manin Obstruction and III[2]

Published online by Cambridge University Press:  01 February 2010

M.J. Bright ,
N. Bruin ,
E.V Flynn  and
A. Logan
M.J. Bright
Affiliation:
Department of Mathematics, University Walk, Bristol BS8 1TWUnited KingdomM.Bright@bristol.ac.uk
N. Bruin
Affiliation:
Department of Mathematics, Simon Fraser University, BurnabyCanada V5A 1S6nbruin@cecm.sfu.cahttp://www.cecm.sfu.ca/~nbruin
E.V Flynn
Affiliation:
Mathematical Institute, University of Oxford, 24-29 St. Giles, Oxford 0X1 3LB, United Kingdomflynn@maths.ox.ac.ukhttp://www.maths.ox.ac.uk/~flynn
A. Logan
Affiliation:
Department of Pure Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada N2L 3G1, a51ogan@math.uuaterloo.CA

Abstract

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We discuss the Brauer-Manin obstruction on del Pezzo surfaces of degree 4. We outline a detailed algorithm for computing the obstruction and provide associated programs in MAGMA. This is illustrated with the computation of an example with an irreducible cubic factor in the singular locus of the defining pencil of quadrics (in contrast to previous examples, which had at worst quadratic irreducible factors). We exploit the relationship with the Tate-Shafarevich group to give new types of examples of III [2], for families of curves of genus 2 of the form y2 = f(x), where f(x) is a quintic containing an irreducible cubic factor.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 10 , 2007 , pp. 354 - 377
Copyright
Copyright © London Mathematical Society 2007

References

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Supplementary material: File

JCM 10 Bright et al Appendix A Part 1

Bright et al Appendix A Part 1

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JCM 10 Bright et al Appendix A Part 2

Bright et al Appendix A Part 2

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JCM 10 Bright et al Appendix A Part 3

Bright et al Appendix A Part 3

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