Skip to main content
SpringerLink
Log in

Stable degenerations of symmetric squares of curves

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

The stable (in the sense of the relative minimal model program) degenerations of symmetric squares of smooth curves of genus g>2 are computed. This information is used to prove that the component of the moduli space of stable surfaces parameterizing such surfaces is isomorphic to the moduli space of stable curves of genus g.

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. Abramovich, D., Vistoli, A.: Complete moduli for fibered surfaces. In: ``Recent progress in intersection theory (Bologna, 1997)'', Trends Math., 1–31. Birkhäuser Boston, Boston, MA, 2000

  2. Barth, W.P., Hulek, K., Peters, C.A.M., Van de Ven, A.: Compact complex surfaces, vol. 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics. Springer-Verlag, Berlin, second edition 2004

  3. Ciliberto, C., Kouvidakis, A.: On the symmetric product of a curve with general moduli. Geom. Dedicata 78 (3), 327–343 (1999)

    Article  MathSciNet  Google Scholar 

  4. Ciliberto, C., Sernesi, E.: On the symmetric products of a curve. Arch. Math. (Basel) 61 (3), 285–290 (1993)

    MathSciNet  Google Scholar 

  5. Fantechi, B.: Deformations of symmetric products of curves. In: ``Classification of algebraic varieties (L'Aquila, 1992)'', vol. 162 of Contemp. Math., 135–141. Amer. Math. Soc., Providence, RI, 1994

  6. Harris, J., Morrison, I.: Moduli of curves, vol. 187 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1998

  7. Kani, E.: Elliptic curves on abelian surfaces. Manuscripta Math. 84 (2), 199–223 (1994)

    MathSciNet  Google Scholar 

  8. Karu, K.: Minimal models and boundedness of stable varieties. J. Algebraic Geom. 9 (1), 93–109 (2000)

    MathSciNet  Google Scholar 

  9. Kollár, J., Mori, S.: Birational geometry of algebraic varieties, vol. 134 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1998

  10. Kollár, J.: Projectivity of complete moduli. J. Differential Geom. 32 (1), 235–268 (1990)

    MathSciNet  Google Scholar 

  11. Kollár, J., Shepherd-Barron, N.I.: Threefolds and deformations of surface singularities. Invent. Math. 91 (2), 299–338 (1988)

    Article  Google Scholar 

  12. La Nave, G.: Explicit stable models of elliptic surfaces with sections. Preprint, arXiv:math.AG/0205035

  13. van Opstall, M.A.: Moduli of products of curves. Arch. Math. (Basel) 84 (2), 148–154 (2005)

    MathSciNet  Google Scholar 

  14. van Opstall, M.A.: Stable degenerations of surfaces isogenous to products of curves. To appear in Proc. AMS.

Download references

Author information

Authors and Affiliations

  1. University of Utah, USA

    Michael A. van Opstall

Authors
  1. Michael A. van Opstall
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael A. van Opstall.

Rights and permissions

Reprints and permissions

About this article

Cite this article

van Opstall, M. Stable degenerations of symmetric squares of curves. manuscripta math. 119, 115–127 (2006). https://doi.org/10.1007/s00229-005-0609-7

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-005-0609-7

Keywords

Search

Navigation