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Decompositions of hypersurface singularities of type \(J_{k,0}\). (English) Zbl 0819.32013

The author studies singularities of type \(J_{k,0}\). These are multimodal singularities \(f\) with leading part \[ y^ 3+\lambda y^ 2x^ k+x^{3k}. \] Let \(W\) be in the zero set of \(f\). \(W\) has an isolated critical point. When one deforms \(W\) then this singular point may decompose into several less complicated critical points. The author investigates how these decompositions of critical points depends on the modulus \(\lambda\). It turns out that the exceptional values of \(\lambda\) are 0 and \(d\), where \(2d^ 3 + 27 = 0\).
Moreover these values are universal, i.e. every decomposition of any hypersurface occurs for the leading part with \(\lambda = 0\) or \(d\).
The paper has interesting consequences for the theory of Legendrian singularities, i.e. the singularities of wave fronts. For \(k = 2\) the situation was well known; the decomposition does not depend on the modulus.

MSC:

32S30 Deformations of complex singularities; vanishing cycles
32S25 Complex surface and hypersurface singularities
14B07 Deformations of singularities
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