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On innermost circles of the sets of singular values for generic deformations of isolated singularities. (English) Zbl 1387.57044

A real analytic map \(f:\mathbb R^4\to\mathbb R^2\) is said to be excellent if the only singularities are isolated cusp points and folds. The image of the singular set is a collection of (possibly cusped) curves. A component of the singular set is said to be innermost if there exists a disk in the plane which contains the component and does not intersect any other component. Finally, a cusp in a closed simple curve is called outward if it points towards the unbounded component of the complement of the curve and inwards otherwise.
The authors prove that there exists a real analytic map \(f:\mathbb R^4\to\mathbb R^2\) with isolated singularity which admits a deformation which is excellent and such that the image of the singular set of the deformation has an innermost component with \(k\) outward cusps and no inward cusp if and only if \(k\neq 1\).

MSC:

57R45 Singularities of differentiable mappings in differential topology
14B05 Singularities in algebraic geometry
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