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Deformation of finite morphisms and smoothing of ropes

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

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Abstract

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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