## 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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