Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

A stratification of \(B^4(2,K_C)\) over a general curve


Let C be a smooth curve of genus \(g\ge 10\) with general moduli. We show that the Brill–Noether locus \(B^4(2,K_C)\) contains irreducible subvarieties \({\mathcal {B}}_3\supset {\mathcal {B}}_4\supset \cdots \supset {\mathcal {B}}_n\), where each \({\mathcal {B}}_k\) has dimension \(3g-10-k\) and \({\mathcal {B}}_3\) is an irreducible component of the expected dimension the Brill–Noether number \(\rho =3g-13\).

This is a preview of subscription content, log in to check access.


  1. 1.

    Arbarello, E., Cornalba, M., Griffiths, P., Harris, J.: Geometry of Algebraic Curves Volume I. Series: Grundlehren der mathematischen Wissenschaften, vol. 267. Springer, New York (1985)

  2. 2.

    Castorena, A., Reyes-Ahumada, G.: Rank two bundles with canonical determinant and four sections. Rendiconti del Circolo Matematico di Palermo (1952 -). 64(2), 261–272 (2015)

  3. 3.

    Ciliberto, C., Flamini, F.: Extensions of line bundles and Brill-Noether loci of rank-two vector bundles on a general curve. Rev. Rumaine Math. Pures Appl. 60(3), 201–255 (2015)

  4. 4.

    Gieseker, D.: A lattice version of the KP equation. Acta Math. 168, 219–248 (1992)

  5. 5.

    Lange, H.: Higher secant varieties of curves and the theorem of Nagata on ruled surfaces. Manuscripta Math. 47, 263 (1984).

  6. 6.

    Lange, H., Narasimhan, M.S.: Maximal subbundles of rank two vector bundles on curves. Math. Ann. 266(1), 55–72 (1983)

  7. 7.

    Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, vol. 52. Springer, Heidelberg (1977)

  8. 8.

    Sernesi, E.: Deformations of Algebraic Schemes. Volume 334, Grundlehren der mathematischen Wissenschaften. A Series of Comprehensive Studies in Mathematics. Springer, Berlin (2006)

Download references

Author information

Correspondence to Abel Castorena.

Additional information

Publisher's Note

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

Abel Castorena is supported by PAPIIT IN100716, Universidad Nacional Autónoma de México).

Graciela Reyes-Ahumada is supported by a FORDECYT(CONACyT, México) fellowship.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Castorena, A., Reyes-Ahumada, G. A stratification of \(B^4(2,K_C)\) over a general curve. Bol. Soc. Mat. Mex. 26, 27–36 (2020).

Download citation


  • Vector bundles
  • Brill–Noether loci
  • Moduli of curves

Mathematics Subject Classification

  • Primary 14C20
  • Secondary 14H60
  • 14J26