Skip to main content
SpringerLink
Log in

Codimension 1 Subvarieties of \({\mathcal{M}_g }\) and Real Gonality of Real Curves

  • Published:
Czechoslovak Mathematical Journal Aims and scope Submit manuscript

Abstract

Let \({\mathcal{M}_g }\) be the moduli space of smooth complex projective curves of genus g. Here we prove that the subset of \({\mathcal{M}_g }\) formed by all curves for which some Brill-Noether locus has dimension larger than the expected one has codimension at least two in \({\mathcal{M}_g }\). As an application we show that if \({X \in \mathcal{M}_g }\) is defined over \(\mathbb{R}\) then there exists a low degree pencil \({u:X \to \mathbb{P}^1 }\) defined over \(\mathbb{R}.\)

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. E. Arbarello, M. Cornalba, P. Griffiths and J. Harris: Geometry of Algebraic Curves, I. Springer-Verlag, 1985.

  2. S. Chaudary: The Brill-Noether theorem for real algebraic curves. Ph.D. thesis. Duke University, 1995.

  3. M. Coppens: One-dimensional linear systems of type II on smooth curves. Ph.D. thesis. Utrecht, 1983.

    Google Scholar 

  4. M. Coppens and G. Martens: Linear series on a general k-gonal curve. Abh. Math. Sem. Univ. Hamburg 69 (1999), 347–371.

    MathSciNet  Google Scholar 

  5. S. Diaz: A bound on the dimension of complete subvarieties of Mg. Duke Math. J. 51 (1984), 405–408.

    Article  MATH  MathSciNet  Google Scholar 

  6. C. J. Earle: On moduli of closed Riemann surfaces with symmetries. Advances in the Theory of Riemann Surfaces. Annals of Math. Studies 66. Princeton, N. J., 1971, pp. 119–130.

    Google Scholar 

  7. B. Gross and J. Harris: Real algebraic curves. Ann. scient. Éc. Norm. Sup., 4e sÈrie 14 (1981), 157–182.

    MathSciNet  Google Scholar 

  8. J. Harris and D. Mumford: On the Kodaira dimension of the moduli space of curves. Invent. Math. 67 (1982), 23–86.

    Article  MathSciNet  Google Scholar 

  9. M. Homma: Separable gonality of a Gorenstein curve. Mat. Contemp. To appear.

  10. M. Seppälä: Teichm¸ller spaces of Klein surfaces. Ann. Acad. Sci. Fenn. Ser. A I. Math. Dissertationes 15 (1978), 1–37.

    Google Scholar 

  11. M. Seppälä: Real algebraic curves in the moduli spaces of complex curves. Compositio Math. 74 (1990), 259–283.

    MATH  MathSciNet  Google Scholar 

  12. C. S. Seshadri: FibrÈs vectoriels sur les courbes algÈbriques. AstÈrisque 96. Soc. Math. France, 1982.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. of Mathematics, University of Trento, 38050, Povo (TN), Italy

    E. Ballico

Authors
  1. E. Ballico
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ballico, E. Codimension 1 Subvarieties of \({\mathcal{M}_g }\) and Real Gonality of Real Curves. Czech Math J 53, 917–924 (2003). https://doi.org/10.1023/B:CMAJ.0000024530.02810.e9

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:CMAJ.0000024530.02810.e9

Search

Navigation