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

Abstract

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 \(\rho =3g-13\).

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References

  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). https://doi.org/10.1007/BF01174597

  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)

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Correspondence to Abel Castorena.

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

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

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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). https://doi.org/10.1007/s40590-018-0227-5

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Keywords

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

Mathematics Subject Classification

  • Primary 14C20
  • Secondary 14H60
  • 14J26