Skip to main content
Log in

Reducible deformations and smoothing of primitive multiple curves

  • Published:
Manuscripta Mathematica Aims and scope Submit manuscript

Abstract

A primitive multiple curve is a Cohen–Macaulay irreducible projective curve Y that can be locally embedded in a smooth surface, and such that C = Y red is smooth. In this case, \({L=\mathcal{I}_{C}/{\mathcal{I}}_{C}^{2}}\) is a line bundle on C. This paper continues the study of deformations of Y to curves with smooth irreducible components, when the number of components is maximal (it is then the multiplicity n of Y). If a primitive double curve Y can be deformed to reduced curves with smooth components intersecting transversally, then \({h^{0}(L^{-1}){\neq}0}\). We prove that conversely, if L is the ideal sheaf of a divisor with no multiple points, then Y can be deformed to reduced curves with smooth components intersecting transversally. We give also some properties of reducible deformations in the case of multiplicity n > 2.

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

Access this article

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. Bănică, C., Forster, O.: Multiple Structures on Space Curves. In: Sundararaman, D. (Ed.) Proceedings of the Lefschetz Centennial Conference (10–14 Dec Mexico), Contemporary Mathematics 58, AMS, pp. 47–64 (1986)

  2. Bayer D., Eisenbud D.: Ribbons and their canonical embeddings. Trans. Am. Math. Soc. 347(3), 719–756 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  3. Drézet J.-M.: Faisceaux cohérents sur les courbes multiples. Collect. Math. 57(2), 121–171 (2006)

    MATH  MathSciNet  Google Scholar 

  4. Drézet J.-M.: Paramétrisation des courbes multiples primitives. Adv. Geom. 7, 559–612 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  5. Drézet J.-M.: Faisceaux sans torsion et faisceaux quasi localement libres sur les courbes multiples primitives. Mathematische Nachrichten 282(7), 919–952 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  6. Drézet J.-M.: Sur les conditions d’existence des faisceaux semi-stables sur les courbes multiples primitives. Pac. J. Math. 249(2), 291–319 (2011)

    Article  MATH  Google Scholar 

  7. Drézet J.-M.: Courbes multiples primitives et déformations de courbes lisses. Annales de la Faculté des Sciences de Toulouse 22(1), 133–154 (2013)

    Article  MATH  Google Scholar 

  8. Drézet J.-M.: Fragmented deformations of primitive multiple curves. Cent. Eur. J. Math. 11(12), 2106–2137 (2013)

    MATH  MathSciNet  Google Scholar 

  9. Eisenbud D., Green M.: Clifford indices of ribbons. Trans. Am. Math. Soc. 347(3), 757–765 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  10. Ferrand D.: Conducteur, descente et pincement. Bull. Soc. math. Fr. 131(4), 553–585 (2003)

    MATH  MathSciNet  Google Scholar 

  11. González M.: Smoothing of ribbons over curves. Journ. für die reine und angew. Math. 591, 201–235 (2006)

    MATH  Google Scholar 

  12. Grothendieck, A.: Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes. Inst. Hautes Études Sci. Publ. Math. No. 8 (1961)

  13. Grothendieck, A.: Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I. Inst. Hautes Études Sci. Publ. Math. No. 11 (1961)

  14. Hartshorne, R.: Deformation theory. Grad. Texts Math., Vol. 257, Springer (2010)

  15. Samuel P., Zariski O.: Commutative Algebra (vol. I, II). GTM 28,29. Springer, Berlin-Heidelberg-New York (1975)

    Google Scholar 

  16. Teixidor i Bigas M.: Moduli spaces of (semi-)stable vector bundles on tree-like curves. Math. Ann. 290, 341–348 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  17. Bigas M.T.: Moduli spaces of vector bundles on reducible curves. Am. J. Math. 117, 125–139 (1995)

    Article  MATH  Google Scholar 

  18. Bigas M.T.: Compactifications of moduli spaces of (semi)stable bundles on singular curves: two points of view. Dedicated to the memory of Fernando Serrano. Collect. Math. 49, 527–548 (1998)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jean-Marc Drézet.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Drézet, JM. Reducible deformations and smoothing of primitive multiple curves. manuscripta math. 148, 447–469 (2015). https://doi.org/10.1007/s00229-015-0755-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-015-0755-5

Mathematics Subject Classification

Navigation