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
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.
|Uncontrolled Keywords:||Degenerations of curves, Multiple structures, Deformations of morphisms|
|Subjects:||Sciences > Mathematics > Algebraic geometry|
|Deposited On:||25 Apr 2011 22:51|
|Last Modified:||29 Aug 2011 09:24|
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