Skip to main content
SpringerLink
Log in

Quelques résultats sur les déformations équivariantes des courbes stables

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

Let G be a finite group, let g≥2 and g ′ ≥ 0 be integers. We introduce the algebraic stack classifying the stable curves of genus g endowed with an action of G faithful in each geometric fiber and such that the quotient of each fiber is a semi-stable curve of genus g′. We study the completion of the local rings of this algebraic stack. They are closely related to universal equivariant deformation rings R C,G of stable curves endowed with a faithful action of G. A useful tool for this purpose is a local-global principle generalizing the one obtained by Bertin and Mézard in [BM00]. We then use the results we already proved in [Mau03b] and [Mau03a] to describe some properties of the space (purity, dimension).

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. Bertin, J., Mézard, A.: Déformations formelles des revêtements sauvagement ramifiés de courbes algébriques. Invent. Math. 141 (1), 195–238 (2000)

    Article  MathSciNet  Google Scholar 

  2. Bertin, J., Maugeais, S.: Déformation équivariante des courbes semi-stables. Ann. Inst. Fourier 55 (6), 1905–1941 (2005)

    MathSciNet  Google Scholar 

  3. Cornelissen, G., Kato, F.: Equivariant deformation of Mumford curves and of ordinary curves in positive characteristic. Duke Math. J. 116 (3), 431–470 (2003)

    Article  MathSciNet  Google Scholar 

  4. Cornelissen, G., Mézard, A.: Relèvements des revêtements de courbes faiblement remifiés. arXiv:math.AG/0412189, preprint

  5. de Jong, A.J.: Families of curves and alterations. Ann. Inst. Fourier (Grenoble) 47 (2), 599–621 (1997)

    Google Scholar 

  6. Deligne, P., Mumford, D.: The irreducibility of the space of curves of given genus. Inst. Hautes Études Sci. Publ. Math. 36, 75–109 (1969)

    MATH  MathSciNet  Google Scholar 

  7. Ekedahl, T.: Boundary behaviour of Hurwitz schemes, The moduli space of curves (Texel Island, 1994), Progr. Math. vol. 129, Birkhäuser Boston, Boston, MA, 1995, pp. 173–198

  8. Grothendieck, A.: Sur quelques points d'algèbre homologique. Tôhoku Math. J. 9 (2), 119–221 (1957)

    MathSciNet  Google Scholar 

  9. Harbater, D.: Moduli of p-covers of curves. Comm. Algebra 8 (12), 1095–1122 (1980)

    MathSciNet  Google Scholar 

  10. Liu, Q.: Algebraic geometry and arithmetic curves. Oxford University Press, 2002, Oxford Graduate Texts in Mathematics, No. 6

  11. Maugeais, S.: Déformations équivariantes des courbes stables i, études cohomologiques, 2003, math.AG/0310137, preprint

  12. Maugeais, S.: Relèvement des revêtements p-cycliques des courbes rationnelles semi-stables. Math. Ann. 327 (2), 365–393 (2003)

    Article  MathSciNet  Google Scholar 

  13. Maugeais, S.: Théorie des déformations équivariantes des morphismes localement d'intersections complètes, 2003, math.AG/0310136, preprint

  14. Tufféry, S.: Déformations de courbes avec action de groupe. Forum Math. 5 (3), 243–259 (1993)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. SFB 478 - Geometrische Strukturen in der Mathematik, Hittorfstr. 27, 48149, Münster, Germany

    Sylvain Maugeais

Authors
  1. Sylvain Maugeais
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sylvain Maugeais.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Maugeais, S. Quelques résultats sur les déformations équivariantes des courbes stables. manuscripta math. 120, 53–82 (2006). https://doi.org/10.1007/s00229-006-0633-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-006-0633-2

Keywords

Search

Navigation