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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
Article

Abstract

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.

Keywords

Deformation of morphisms Embedding Smoothing Rope Canonical surface 

Mathematics Subject Classification (2000)

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

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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

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