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Perestroikas of vertex sets at umbilic points. (English) Zbl 1177.53013

Consider a smooth surface \(M\) and a generic umbilic point p\( \in M\). Intersecting \(M\) with planes parallel to the tangent plane at p yields a one-parameter set of curves. The collection of vertices of these curves is called the vertex set through p. The authors study the evolution of the vertex set under generic \(n\)-parameter deformations of \(M\).
One of the main results states that, in a certain sense, the study of the discriminants of \(n\)-parameter deformations reduces to that of \(2\)-parameter transformations. The vertex set of the unperturbed surface consists of three smooth transverse branches through p. After a small perturbation the vertex set consist of three branches, two of them transverse in a point close to p and disjoint to the third.
The evolution of the vertex set under one-parameter deformations of \(M\) is studied as well. All results are illustrated by instructive figures.

MSC:

53A05 Surfaces in Euclidean and related spaces
53A04 Curves in Euclidean and related spaces
53A55 Differential invariants (local theory), geometric objects
58K60 Deformation of singularities
58K65 Topological invariants on manifolds
14B05 Singularities in algebraic geometry
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References:

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