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Simple singularities of curves V.I.Arnold
 

Summary: Simple singularities of curves
V.I.Arnold 
Steklov Mathematics Institute, Moscow
& CEREMADE, Universite Paris-Dauphine
Dedicated to L.D.Faddeev
1 Introduction
A singularity of a curve below means a germ of a holomorphic mapping of the complex line
into the complex space at a singular point (where the derivative of the mapping vanishes)
considered up to the biholomorphic mappings of the image space.
The singularity is called simple if all neighbouring singularities belong to a nite set of
equivalence classes (have no moduli). The simple singularities of curves in the plane have been
classi ed by J.W.Bruce and T.Ga ney [1], that of curves in three-space { by C.Gibson and
C.Hobbs [7].
The singularity is called stably simple if it is simple and remains simple when the ambient
space is embedded into a larger space.
Two curves, obtained one from the other by such an embedding, are called stably equivalent.
We classify below the simple curve singularities in spaces of any dimension up to stable
equivalence.
We nd, that almost all those singularities, whose Tailor series starts from a term of degree
2 or 3 or has the form

  

Source: Arnold, Vladimir Igorevich - Steklov Mathematical Institute

 

Collections: Mathematics