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Universal topological stratification for the Pham example. (English) Zbl 0784.32029
The versal unfolding of a polynomial germ \(f(x,y)=x^ g+y^ 3\) is investigated. The authors describe the stratification of the moduli space of the corresponding bimodal complex plane curve singularity by the topological type of its versal unfolding. More precisely, let \(F(x,y,s,t)=f(x,y)+sx^ 7y+tx^ 6y\) be a semi-quasihomogeneous deformation of \(f(x,y)\) parametrized by the two moduli \((s,t)\). The main result of the paper under review states that the stratification of the \((s,t)\)-subspace such that the versal unfolding of \(f\) is topologically a product on strata is given by: the \(s\)-axis, the punctured lines for \(4t^ 3+27\neq 0\), and the complement. In large part the proof is based on knowledge of properties of the versality discriminant that has been explicitly determined with the aid of a computer.
It should be remarked that there is a series of works [see e.g. O. A. Laudal, B. Martin and G. Pfister, Singularities, Banach Cent. Publ. 20, 255-278 (1988; Zbl 0691.14018)] where closely related questions concerning the existence of analytical stratifications of the moduli space are discussed in full detail.

MSC:
32S60 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects)
32S30 Deformations of complex singularities; vanishing cycles
58C25 Differentiable maps on manifolds
58K99 Theory of singularities and catastrophe theory
Software:
MACSYMA
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