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On the versal discriminant of \(J_{k,0}\) singularities. (English) Zbl 0848.32028

Let \(f : (\mathbb{C}^n, 0) \to (\mathbb{C},0)\) be the germ of an isolated hypersurface singularity. Let \(T\) be the \(\mu\)-const stratum in the base \(\Lambda\) of the miniversal deformation \(F : (\mathbb{C}^n \times \Lambda, 0) \to (\mathbb{C}, 0)\) of \(f\) (w.r.t. right equivalence). Given a projection \(\pi : \Lambda \to T\) the versality discriminant is the germ consisting of those \(\lambda \in \pi^{-1} (0)\) such that the deformation \(F_{\pi^{-1} (0)} : \mathbb{C}^n \times \pi^{-1} (0) \to \mathbb{C}\) is not versal for contact equivalence at \(\lambda\). For quasi-homogeneous singularities a natural choice of projection exist.
In this paper the author computes the versality discriminant for quasi-homogeneous singularities of type \(J_{k,0} : y^3 + \beta yx^{2k} + x^{3k}\). The fibres over the versality discriminant are polynomials equivalent to \(y^3 + eyd (x)^2 + d(x)^3\), \(y^3 + yb(x)\) or \(y^3 + c(x)\) with \(b,c,d\) polynomials of degree \(2k\), \(3k\) and \(k\) with no more than \(k - 1\) different roots.

MSC:

32S25 Complex surface and hypersurface singularities
32S15 Equisingularity (topological and analytic)
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