Revista Matemática Complutense

, Volume 26, Issue 1, pp 253–269 | Cite as

An infinitesimal condition to smooth ropes

  • Francisco Javier GallegoEmail author
  • Miguel González
  • Bangere P. Purnaprajna


In this article we give a condition, which holds in a very general setting, to smooth a rope, of any dimension, embedded in projective space. As a consequence of this we prove that canonically embedded carpets satisfying mild geometric conditions can be smoothed. Our condition for smoothing a rope \(\widetilde{Y}\) can be stated very transparently in terms of the cohomology class of a suitable first order infinitesimal deformation of a morphism ϕ associated to \(\widetilde{Y}\). In order to prove these results we find a sufficient condition, of independent interest, for a morphism ϕ from a smooth variety X to projective space, finite onto a smooth image, to be deformed to an embedding. Another application of this result on deformation of morphisms is the construction of smooth varieties in projective space with given invariants. We illustrate this by constructing canonically embedded surfaces with \(c_{1}^{2}=3p_{g}-7\) and deriving some interesting properties of their moduli spaces. The results of this article bear further evidence to the complexity of the moduli of surfaces of general type and its sharp contrast with the moduli of other objects such as curves or K3 surfaces.


Deformation of morphisms Embedding Smoothing Rope Canonical surface 

Mathematics Subject Classification (2000)

14B10 14D15 13D10 14D06 14A15 14J10 14J29 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Artin, M.: Algebraization of formal moduli I. In: Global Analysis: Papers in Honor of K. Kodaira. Princeton Math. Series, vol. 29, pp. 21–71 (1969) Google Scholar
  2. 2.
    Ashikaga, T., Konno, K.: Algebraic surfaces of general type with \(c\sp2\sb1=3p\sb g-7\). Tohoku Math. J. 42, 517–536 (1990) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Beauville, A., et al.: Géométrie des surfaces K3: modules et périodes. Astérisque 126 (1985) Google Scholar
  4. 4.
    Gallego, F.J., Purnaprajna, B.P.: Degenerations of K3 surfaces in projective space. Trans. Am. Math. Soc. 349, 2477–2492 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Gallego, F.J., González, M., Purnaprajna, B.P.: Deformation of finite morphisms and smoothing of ropes. Compos. Math. 144, 673–688 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Gallego, F.J., González, M., Purnaprajna, B.P.: K3 double structures on Enriques surfaces and their smoothings. J. Pure Appl. Algebra 212, 981–993 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Gallego, F.J., González, M., Purnaprajna, B.P.: Deformation of canonical morphisms and the moduli of surfaces of general type. Invent. Math. 182, 1–46 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Gallego, F.J., González, M., Purnaprajna, B.P.: On the deformations of canonical double covers of minimal rational surfaces. Preprint arXiv:1005.5399
  9. 9.
    Gallego, F.J., González, M., Purnaprajna, B.P.: Smoothings of a non Cohen–Macaulay double structures on curves (in preparation) Google Scholar
  10. 10.
    González, M.: Smoothing of ribbons over curves. J. Reine Angew. Math. 591, 201–235 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Grothendieck, A.: EGA III, Étude cohomologique des faisceaux cohérents (première partie). Publ. Math. IHES, vol. 11 (1961) Google Scholar
  12. 12.
    Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, vol. 52. Springer, New York (1977), zbMATHGoogle Scholar
  13. 13.
    Horikawa, E.: On deformations of holomorphic maps. I. J. Math. Soc. Jpn. 25, 372–396 (1973) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Hulek, K., Van de Ven, A.: The Horrocks-Mumford bundle and the Ferrand construction. Manuscr. Math. 50, 313–335 (1985) zbMATHCrossRefGoogle Scholar
  15. 15.
    Kollar, J., Mori, S.: Birational Geometry of Algebraic Varieties. Cambridge University Press, Cambridge (1998) zbMATHCrossRefGoogle Scholar
  16. 16.
    Matsumura, H.: Commutative Algebra. Benjamin/Cummings, Redwood City (1969) Google Scholar
  17. 17.
    Sernesi, E.: Deformations of Algebraic Schemes. Grundlehren der mathematischen Wissenschaften, vol. 334. Springer, Berlin (2006) zbMATHGoogle Scholar

Copyright information

© Revista Matemática Complutense 2011

Authors and Affiliations

  • Francisco Javier Gallego
    • 1
    Email author
  • Miguel González
    • 1
  • Bangere P. Purnaprajna
    • 2
  1. 1.Departamento de ÁlgebraUniversidad Complutense de MadridMadridSpain
  2. 2.Department of MathematicsUniversity of KansasLawrenceUSA

Personalised recommendations