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Deformation of finite morphisms and smoothing of ropes



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Gallego Rodrigo, Francisco Javier and González Andrés, Miguel and Purnaprajna, Bangere P. (2008) Deformation of finite morphisms and smoothing of ropes. Compositio Mathematica, 144 (3). pp. 673-688. ISSN 0010-437X


Official URL: http://journals.cambridge.org/action/displayJournal?jid=COM


In this paper we prove that most ropes of arbitrary multiplicity supported on smooth
curves can be smoothed. By a rope being smoothable we mean that the rope is the flat limit of a family of smooth, irreducible curves. To construct a smoothing, we connect, on the one hand, deformations of a finite morphism to projective space and, on the other hand, morphisms from a rope to projective space. We also prove a general result of independent interest, namely that finite covers onto smooth irreducible curves embedded in projective space can be deformed to a family of 1 : 1 maps. We apply our general theory to prove the smoothing of ropes of multiplicity 3 on P1. Even though this paper focuses on ropes of dimension 1, our method yields a general approach to deal with the smoothing of ropes of higher dimension.

Item Type:Article
Uncontrolled Keywords:Degenerations of curves, Multiple structures, Deformations of morphisms
Subjects:Sciences > Mathematics > Algebraic geometry
ID Code:12607
Deposited On:25 Apr 2011 20:51
Last Modified:06 Feb 2014 09:28

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