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Deformations which preserve the non-immersive locus of a map-germ. (English) Zbl 0745.32019

Singular germs of maps \(f:\;(\mathbb{C}^ 2,0)\to f(\mathbb{C}^ 3,0)\) present three distinct kinds of degeneracy: non-immersiveness, non-transverse self-intersection, and triple self intersection. In this paper, the author introduces the notion of \(\Sigma\)-trivial unfoldings: that is, unfoldings in which the non-immersive locus of \(f\) is deformed trivially. The princial result is an infinitesimal criterion for versality within the category of \(\Sigma\)-trivial unfoldings, for germs of corank 1.
Theorem. Let \(f:\;(\mathbb{C}^ n,0)\to(\mathbb{C}^ p,0)\) be a finite map-germ with a singularity of type \(\Sigma^ 1\) at 0. Then a \(\Sigma^ 1\)- trivial \(d\)-parameter unfolding \(F\) of \(f\) is \(\Sigma^ 1\)-versal if and only if \(T{\mathcal A}_ ef+\mathbb{C}(\partial_ 1F,\dots,\partial_ dF)=\theta\Sigma^ 1(f)\);
In particular, the author obtains an infinitesimal criterion for \(\Sigma\)-stability.
In §3, he applies this to a variety of examples, and shows in particular that the germ parametrising the swallowtrail surface of catastrophe theory is \(\Sigma\)-stable; moreover, if \(\exp[\gamma]:\;\mathbb{R}\to\mathbb{R}^ 3\) is the map parametrising the tangent developable surface of real analytic space curve, then the germ of \(\exp[\gamma]\) at the point \((t,0)\in TR\) is \(\Sigma\)-stable if the curvature of \(\gamma\) at \(t\) is non-vanishing and the torsion is non-vanishing or vanishes to first order, or if there is a non-degenerate zero of curvature at \(t\).
[For special notations see the paper itself].

MSC:

32S30 Deformations of complex singularities; vanishing cycles
58C25 Differentiable maps on manifolds
58K99 Theory of singularities and catastrophe theory
58K35 Catastrophe theory
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