Skip to main content
SpringerLink
Log in

On integral points on degree four del Pezzo surfaces

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We report on our investigations concerning algebraic and transcendental Brauer–Manin obstructions to integral points on complements of a hyperplane section in degree four del Pezzo surfaces. We discuss two concepts of an obstruction at an archimedean place. Concrete examples are given of pairs of non-homogeneous quadratic polynomials in four variables representing (0, 0) over Q and over Z p for all primes p, but not over Z. By blow-up, these yield cubic polynomials in three variables all integral solutions of which satisfy a gcd condition.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Artin and D. Mumford, Some elementary examples of unirational varieties which are not rational, Proceedings of the London Mathematical Society 25 (1972), 75–95.

    Article  MATH  MathSciNet  Google Scholar 

  2. W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I. The user language, Journal of Symbolic Computation 24 (1997), 235–265.

    Article  MATH  MathSciNet  Google Scholar 

  3. E. Brussel, K. McKinnie and E. Tengan, Indecomposable and noncrossed product division algebras over function fields of smooth p-adic curves, Advances in Mathematics 226 (2011), 4316–4337.

    Article  MATH  MathSciNet  Google Scholar 

  4. J.-L. Colliot-Thélène and A. N. Skorobogatov, Descente galoisienne sur le groupe de Brauer, Journal für die reine und angewandte Mathematik 682 (2013), 141–165.

    MATH  MathSciNet  Google Scholar 

  5. J.-L. Colliot-Thélène and O. Wittenberg, Groupe de Brauer et points entiers de deux familles de surfaces cubiques affines, American Journal of Mathematics 134 (2012), 1303–1327.

    Article  MATH  MathSciNet  Google Scholar 

  6. J.-L. Colliot-Thélène and F. Xu, Brauer–Manin obstruction for integral points of homogeneous spaces and representation by integral quadratic forms. With an appendix by D. Wei and Xu, Compositio Mathematica 145 (2009), 309–363.

    Article  MATH  MathSciNet  Google Scholar 

  7. P. K. Corn, Del Pezzo surfaces and the Brauer–Manin obstruction, Ph.D. thesis, Harvard, 2005.

    MATH  Google Scholar 

  8. U. Derenthal and D. Wei, Strong approximation and descent, preprint, available at http://arxiv.org/abs/1311.3914.

  9. I. V. Dolgachev, Classical Algebraic Geometry: A Modern View, Cambridge University press, Cambridge, 2012.

    Book  MATH  Google Scholar 

  10. A.-S. Elsenhans and J. Jahnel, On cubic surfaces with a rational line, Archiv der Mathematik 98 (2012), 229–234.

    Article  MATH  MathSciNet  Google Scholar 

  11. P. Finsler, Über das Vorkommen definiter und semidefiniter Formen in Scharen quadratischer Formen, Commentarii Mathematici Helvetici 9 (1937), 188–192.

    Article  MATH  Google Scholar 

  12. A. Grothendieck, Le groupe de Brauer II: Théorie cohomologique, in Séminaire Bourbaki, Vol. 9, Société de Mathématique de France, Paris, 1995, pp. 287–307.

    Google Scholar 

  13. A. Grothendieck, Le groupe de Brauer III: Exemples et compléments, in Dix expos és sur la cohomologie des schémas, Advanced Studies in Pure Mathematics, Vol. 3, North-Holland, Amsterdam, 1968, 88–188.

    Google Scholar 

  14. Y. Harpaz, Geometry and arithmetic of certain log K3 surfaces, preprint, available at http:arxiv.org/abs/1511.01285.

  15. R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, Vol. 52, Springer, New York, 1977.

    Google Scholar 

  16. J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, Vol. 9, Springer, New York–Berlin, 1972.

    Google Scholar 

  17. J. Jahnel, Brauer Groups, Tamagawa Measures, and Rational Points on Algebraic Varieties, Mathematical Surveys and Monographs, Vol. 198, American Mathematical Society, Providence, RI, 2014.

    Google Scholar 

  18. A. Kresch and Yu. Tschinkel, Two examples of Brauer–Manin obstruction to integral points, Bulletin of the London Mathematical Society 40 (2008), 995–1001.

    Article  MATH  MathSciNet  Google Scholar 

  19. B. E. Kunyavskiĭ, A. N. Skorobogatov and M. A. Tsfasman, Del Pezzo surfaces of degree four, Mémoires de la Société Mathématique de France 37 (1989), 1–113.

    Article  MATH  Google Scholar 

  20. Yu. I. Manin, Cubic Forms, Algebra, Geometry, Arithmetic, North-Holland, Amsterdam–London and American Elsevier, New York, 1974.

    MATH  Google Scholar 

  21. J. S. Milne, Étale Cohomology, Princeton Mathematical Series, Vol. 33, Princeton University Press, Princeton, NJ, 1980.

    Google Scholar 

  22. M. V. Nori, Zariski’s conjecture and related problems, Annales Scientifiques de lÉcole Normale Supérieure 16 (1983), 305–344.

    Article  MATH  MathSciNet  Google Scholar 

  23. R. S. Pierce, Associative Algebras, Graduate Texts in Mathematics, Vol. 88, Springer, New York–Berlin, 1982.

    Google Scholar 

  24. V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, Pure and Applied Mathematics, Vol. 139, Academic Press, Boston, 1994.

    Google Scholar 

  25. A. Grothendieck, Cohomologie l-adique et Fonctions L. Séminaire de Géométrie Algébrique du Bois Marie 1965–1966 (SGA 5), Lecture Notes in Mathematics, Vol. 589, Springer, Berlin–Heidelberg–New York, 1977.

    Google Scholar 

  26. R. Silhol, Classification birationnelle des surfaces rationnelles réelles, in Real Analytic and Algebraic Geometry (Trento 1988), Lecture Notes in Mathematics, Vol. 1420, Springer, Berlin, 1990, pp. 308–324.

    Chapter  Google Scholar 

  27. A. N. Skorobogatov, Beyond the Manin obstruction, Inventiones Mathematicae 135 (1999), 399–424.

    Article  MATH  MathSciNet  Google Scholar 

  28. A. N. Skorobogatov, Torsors and Rational Points, Cambridge Tracts in Mathematics, Vol. 144, Cambridge University Press, Cambridge, 2001.

    Google Scholar 

  29. P. Swinnerton-Dyer, The Brauer group of cubic surfaces, Mathematical Proceedings of the Cambridge Philosophical Society 113 (1993), 449–460.

    Article  MATH  MathSciNet  Google Scholar 

  30. C. Tu, W. Wang, B. Mourrain and J. Wang, Signature sequence of intersection curve of two quadrics for exact morphological classification, preprint, available at http://arxiv.org/abs/cs/0701121.

  31. A. Várilly-Alvarado and B. Viray, Arithmetic of del Pezzo surfaces of degree 4 and vertical Brauer groups, Advances in Mathematics 255 (2014), 153–181.

    Article  MATH  MathSciNet  Google Scholar 

  32. O. Wittenberg, Intersections de deux quadriques et pinceaux de courbes de genre 1, Lecture Notes in Mathematics, Vo. 1901, Springer, Berlin, 2007.

    Google Scholar 

  33. F. Xu, Strong approximation for certain quadric fibrations with compact fibers, Advances in Mathematics 281 (2015), 279–295.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Département Mathematik, Universität Siegen, Walter-Flex-Str. 3, D-57068, Siegen, Germany

    Jörg Jahnel

  2. Mathematical Institute, Universiteit Utrecht, Hans Freudenthalgebouw, Budapestlaan 6, 3584 CD, Utrecht, Netherlands

    Damaris Schindler

Authors
  1. Jörg Jahnel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Damaris Schindler
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jörg Jahnel.

Additional information

All computations are with magma [BCP].

The second author was supported by the NSF under agreement No. DMS-1128155 and by an NWO grant 016.Veni.173.016.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jahnel, J., Schindler, D. On integral points on degree four del Pezzo surfaces. Isr. J. Math. 222, 21–62 (2017). https://doi.org/10.1007/s11856-017-1581-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-017-1581-0

Search

Navigation