Skip to main content
SpringerLink
Log in

Characterizing gonality for two-component stable curves

  • Original Paper
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

It is a well-known result that a stable curve of compact type over \({\mathbb {C}}\) having two components is hyperelliptic if and only if both components are hyperelliptic and the point of intersection is a Weierstrass point for each of them. With the use of admissible covers, we generalize this characterization in two ways: for stable curves of higher gonality having two smooth components and one node; and for hyperelliptic and trigonal stable curves having two smooth non-rational components and any number of nodes.

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

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Abramovich, D., Vistoli, A.: Compactifying the space of stable maps. J. Am. Math. Soc. 15, 27–75 (2002)

    Article  MathSciNet  Google Scholar 

  2. Caporaso, L.: Gonality of algebraic curves and graphs, algebraic and complex geometry: in honour of Klaus Hulek’s 60th Birthday. Springer Proc. Mathe. Stat. 71, 77–109 (2014)

    Article  Google Scholar 

  3. Cavalieri, R., Markwig, H., Ranganathan, D.: Tropicalizing the space of admissible covers. Math. Ann. 364, 1275–1313 (2016)

    Article  MathSciNet  Google Scholar 

  4. Coelho, J., Sercio, F.: On the gonality of stable curves. Mathematische Nachrichten 292(11), 2338–2351 (2019)

    Article  MathSciNet  Google Scholar 

  5. Eisebud, D., Harris, J.: Limit linear series: basic theory. Invent. Math. 85, 337–371 (1986)

    Article  MathSciNet  Google Scholar 

  6. Esteves, E., Medeiros, N.: Limit canonical systems on curves with two components. Inventiones Mathematicae 149(2), 267–338 (2002)

    Article  MathSciNet  Google Scholar 

  7. Harris, J., Morrison, I.: Moduli of curves. Springer, New York (1998)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. Lara, D., Menezes Souza, J., Vidal Martins, R.: On gonality, scrolls, and canonical models of non-Gorenstein curves. Geom. Dedicata. 198, 1–23 (2019)

    Article  MathSciNet  Google Scholar 

  10. Osserman, B.: Limit linear series for curves not of compact type. Journal für die reine und angewandte Mathematik (Crelles Journal) 753, 57–88 (2019)

    Article  MathSciNet  Google Scholar 

  11. Rosa, R., Stöhr, K.-O.: Trigonal Gorenstein curves. J. Pure Appl. Algebra 174, 187–205 (2002)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Instituto de Matemática e Estatística - Rua Prof. Marcos Waldemar de Freitas Reis, Universidade Federal Fluminense (UFF), S/N, Gragoatá, Niterói, RJ, 24210-201, Brasil

    Juliana Coelho

  2. Departamento de Matemática, Rua José Lourenço Kelmer, Universidade Federal de Juiz de Fora (UFJF), s/n - Campus Universitário, Juiz de Fora, MG, 36036-900, Brasil

    Frederico Sercio

Authors
  1. Juliana Coelho
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Frederico Sercio
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Juliana Coelho.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Coelho, J., Sercio, F. Characterizing gonality for two-component stable curves. Geom Dedicata 214, 157–176 (2021). https://doi.org/10.1007/s10711-021-00609-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-021-00609-y

Keywords

Mathematics Subject Classification

Search

Navigation