Skip to main content
SpringerLink
Log in

Exemples de comptages de courbes sur les surfaces

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Résumé

Soit \(X\) une surface dont l’anneau de Cox a une seule relation, laquelle vérifie en outre une certaine propriété de linéarité. Nous montrons que les conjectures de Manin géométriques valent pour certains degrés du cône effectif dual de \(X\) (notamment pour ces degrés l’espace de modules de morphismes a la dimension attendue). Le résultat s’applique à une classe de surfaces de del Pezzo généralisées qui a été intensément étudiée dans le cadre de la conjecture de Manin arithmétique.

Abstract

Some examples of curves countings on surfaces Let \(X\) be a surface whose Cox ring has a single relation satisfying moreover a kind of linearity property. We show that the geometric Manin’s conjectures hold for some degrees lying in the dual of the effective cone of \(X\) (in particular, for those degrees the moduli space of morphisms has the expected dimension). The result applies to a class of generalized del Pezzo surfaces which has been intensively studied in the context of the arithmetic Manin’s conjecture.

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

References

  1. Arzhantsev, I.V., Derenthal, U., Hausen, J., Laface, A.: Cox rings. http://www.mathematik.uni-tuebingen.de/~hausen/CoxRings/download.php?name=coxrings.pdf (2010)

  2. Berchtold, F., Hausen, J.: Cox rings and combinatorics. Trans. Am. Math. Soc. 359(3):1205–1252 (2007)

    Google Scholar 

  3. Bourqui, D.: Comptage de courbes sur le plan projectif éclaté en trois points alignés. Ann. Inst. Fourier (Grenoble) 59(5), 1847–1895 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bourqui, D.: Produit eulérien motivique et courbes rationnelles sur les variétés toriques. Compos. Math. 145(6), 1360–1400 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bourqui, D.: La conjecture de Manin géométrique pour une famille de quadriques intrinsèques. Manuscr. Math. 135(1–2), 1–41 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bourqui, D.: Moduli spaces of curves and Cox rings. Michigan Math. J. 61, 593–613 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. de la Bretèche, R., Browning, T.D.: On Manin’s conjecture for singular del Pezzo surfaces of degree 4. I. Michigan Math. J. 55(1), 51–80 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. de la Bretèche, R., Browning, T.D., Derenthal, U.: On Manin’s conjecture for a certain singular cubic surface. Ann. Sci. École Norm. Sup. (4) 40(1):1–50 (2007)

    Google Scholar 

  9. Browning, T.D., Derenthal, U.: Manin’s conjecture for a cubic surface with \(D_5\) singularity. Int. Math. Res. Not. IMRN 14, 2620–2647 (2009)

    MathSciNet  Google Scholar 

  10. Browning, T.D., Derenthal, U.: Manin’s conjecture for a quartic del Pezzo surface with \(A_4\) singularity. Ann. Inst. Fourier (Grenoble) 59(3), 1231–1265 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Debarre, O.: Higher-dimensional algebraic geometry. Universitext. Springer, New York (2001)

    Book  Google Scholar 

  12. Derenthal, U.: Counting integral points on universal torsors. Int. Math. Res. Not. IMRN 14, 2648–2699 (2009)

    MathSciNet  Google Scholar 

  13. Derenthal, U.: Singular Del Pezzo surfaces whose universal torsors are hypersurfaces. arXiv:math/0604194v2 (2012)

  14. Derenthal, U., Tschinkel, Y.: Universal torsors over del Pezzo surfaces and rational points. In Equidistribution in number theory, an introduction, volume 237 of NATO Sci. Ser. II Math. Phys. Chem., pp. 169–196. Springer, Dordrecht (2007)

  15. Franz, M.: Convex—a Maple package for convex geometry—version 1.1.3 (2009)

  16. Hassett, B.: Equations of universal torsors and Cox rings. In Mathematisches Institut, Georg-August-Universität Göttingen: Seminars Summer Term 2004, pp 135–143. Universitätsdrucke Göttingen, Göttingen (2004)

  17. Hassett, B.: Rational surfaces over nonclosed fields. In Arithmetic geometry, volume 8 of Clay Math. Proc., pp. 155–209. Am. Math. Soc., Providence (2009)

  18. Le Boudec, P.: Manin’s conjecture for a cubic surface with \(2A_2+A_1\) singularity type. arXiv:1105.3495 (2011)

  19. Le Boudec, P.: Manin’s conjecture for two quartic del Pezzo surfaces with \(3A_1\) and \(A_1+A_2\) singularity types. Acta Arith. 151(2), 109–163 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Loughran, D.: Manin’s conjecture for a singular sextic del Pezzo surface. J. Théor. Nombres Bordeaux 22(3), 675–701 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Peyre, E.: Hauteurs et mesures de Tamagawa sur les variétés de Fano. Duke Math. J. 79(1), 101–218 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  22. Peyre, E.: Points de hauteur bornée sur les variétés de drapeaux en caractéristique finie. Acta Arith. 152(2), 185–216 (2012)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042 , Rennes Cedex, France

    David Bourqui

Authors
  1. David Bourqui
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David Bourqui.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bourqui, D. Exemples de comptages de courbes sur les surfaces. Math. Ann. 357, 1291–1327 (2013). https://doi.org/10.1007/s00208-013-0933-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-013-0933-2

Mathematics Subject Classification

Search

Navigation