Advertisement

Manuscripta Mathematica

, Volume 142, Issue 3–4, pp 525–544 | Cite as

On the Hilbert scheme of the moduli space of vector bundles over an algebraic curve

  • L. Brambila-PazEmail author
  • O. Mata-Gutiérrez
Article

Abstract

Let M(n, ξ) be the moduli space of stable vector bundles of rank n ≥ 3 and fixed determinant ξ over a complex smooth projective algebraic curve X of genus g ≥ 4. We use the gonality of the curve and r-Hecke morphisms to describe a smooth open set of an irreducible component of the Hilbert scheme of M(n, ξ), and to compute its dimension. We prove similar results for the scheme of morphisms \({M or_P (\mathbb{G}, M(n, \xi))}\) and the moduli space of stable bundles over \({X \times \mathbb{G}}\), where \({\mathbb{G}}\) is the Grassmannian \({\mathbb{G}(n - r, \mathbb{C}^n)}\). Moreover, we give sufficient conditions for \({M or_{2ns}(\mathbb{P}^1, M(n, \xi))}\) to be non-empty, when s ≥ 1.

Mathematics Subject Classification (2000)

14H60 14J60 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Balaji V., Brambila-Paz L., Newstead P.E.: Stability of the Poincaré bundle. Math. Nachr. 188, 5–15 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bertram, A.: Stable maps and Gromov-Witten invariants. In: Arbarello E., Ellingsrud G., Göttsche L. (eds.) Intersection theory and moduli, ICTP Lecture Notes, XIX, 1–40. Abdus Salam International Center for Theoretical Physics, Trieste (2004)Google Scholar
  3. 3.
    Bertram A., Daskalopoulos G., Wentworth R.: Gromov invariants for holomorphic maps from Riemann surfaces to Grassmannians. J. Am. Math. Soc. 9(2), 529–571 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Castravet A.M.: Rational families of vector bundles on curves. Int. J. Math 15(1), 13–45 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Choe I., Choy J., Kiem Y.H.: Cohomology of the moduli space of Hecke cycles. Topology 44(3), 585–608 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Desale, U.V., Ramanan, S.: Classification of vector bundles of rank 2 on hyperelliptic curves. Invent. Math. 38(2), 161–185 (1976/1977)Google Scholar
  7. 7.
    Grothendieck, A.: Techniques de construction et théorèmes d’existence en géométrie algébrique IV: Les Schémas de Hilbert. Séminaire Bourbaki. 1960/61, Exp.221, Astérisque hors série 6, Soc. Math. Fr. (1997)Google Scholar
  8. 8.
    Hwang J.M.: Tangent vectors to Hecke curves on the moduli space of rank 2 bundles over an algebraic curve. Duke Math. J. 101(1), 179–187 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hwang, J.M.: Hecke curves on the moduli space of vector bundles over an algebraic curve. In: Algebraic Geometry in East Asia (Kyoto, 2001), pp. 155–164. World Scientific (2002)Google Scholar
  10. 10.
    Hwang, J.M., Ramanan, S.: Hecke curves and Hitchin discriminant. Ann. Sci. Éc. Norm. Sup. (4). 37(5), 801–817 (2004)Google Scholar
  11. 11.
    Kiem Y.H.: Hecke correspondence, stable maps, and Kirwan desingularization. Duke Math. J. 136(3), 585–618 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kilaru S.: Rational curves on moduli spaces of vector bundles. Proc. Indian Acad. Sci. Math. Sci. 108, 217–226 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kollár, J.: Rational curves on algebraic varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Springer, Berlin (1996)Google Scholar
  14. 14.
    Li J., Tian G.: The quantum cohomology of homogeneous varieties. J. Algebr. Geom. 6(2), 269–305 (1997)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Maruyama, M.: Elementary transformations in the theory of algebraic vector bundles. In: Algebraic Geometry (La Rábida, 1981). Lecture Notes in Math. vol. 961, pp. 241–266, Springer, Berlin (1982)Google Scholar
  16. 16.
    Mata-Gutiérrez, O.: On (k, l)-stable vector bundles over algebraic curves. arXiv:1202.1632.Google Scholar
  17. 17.
    Mok N., Sun X.T.: Remarks on lines and minimal rational curves. Sci. China Ser. A 52((4), 617–630 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Munoz V.: Quantum cohomology of the moduli space of stable bundles over a Riemann surface. Duke Math. J. 98(3), 525–540 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Narasimhan M.S., Ramanan S.: Deformations of the moduli of vector bundles. Ann. Math. 101, 391–417 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Narasimhan, M.S., Ramanan, S.: Geometry of Hecke cycles-1. In: Ramanathan K., Ramanujam C.P. (eds.) A tribute. Tata Institute of Fundamental Research Studies in Mathematics, vol. 8, pp. 291–345. Springer, Berlin (1978)Google Scholar
  21. 21.
    Newstead P.E.: A non-existence theorem for families of stable bundles. J. London Math. Soc. 6, 259–266 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Pandharipande, R., Thomas, R.P.: 13/2 ways of counting curves. arXiv:1111.1552Google Scholar
  23. 23.
    Ramanan S.: The moduli spaces of vector bundles over an algebraic curve. Math. Ann. 200, 69–84 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Sun X.: Minimal rational curves on moduli spaces of stable bundles. Math. Ann. 331, 925–937 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Tyurin A.N.: The geometry of moduli of vector bundles. Uspekhi Mat. Nauk 29, 59–88 (1974)zbMATHGoogle Scholar
  26. 26.
    Witten, E.: The Verlinde algebra and the cohomology of the Grassmannian. In: Yau S.T. (ed.) Geometry, topology and physics. Conference Proceedings and Lectures Notes in Geometry and Topology IV, pp. 357–422. International Press, Cambridge (1995)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Centro de Investigación en MatemáticasGuanajuatoMexico

Personalised recommendations