The \(\mu\)-constant stratum is not smooth
In this work, we find several interesting examples of surface singularities in \({\mathbb{C}}^ 3\), for which the \(\mu\)-constant stratum \(S_{\mu}\), in the miniversal deformation is not smooth. We study surface singularities \((V,0)\subset ({\mathbb{C}}^ 3,0)\) that can be resolved by a quadratic transformation, called superisolated singularities. Let p:\({\mathcal B}\to T\) of a superisolated singularity with smooth base T be the \(\mu\)-constant deformation. Using the Perron's theorem and results of Neuman we can see that such a deformation \({\mathcal B}\) is equimultiple along \(\sigma\) (T) (section of p) and p has a strong simultaneous resolution, starting with the monoidal transformation with center \(\sigma\) (T). Also, we can see that if \(D\subset {\mathbb{P}}_ 2\) is the projectivized tangent cone of (V,0), then p induces a deformation \(\pi\) :\({\mathcal D}\to T\) of D, which is equisingular as a deformation of the projective plane curve \(D\subset {\mathbb{P}}_ 2\). In the second section, we study how to compute in the base of a miniversal deformation of (V,0), the stratum \(\mu\)-constant (resp. \(\mu^*\)-constant) which denoted by \(S_{\mu}\) (resp. \(S_{\mu^*})\). Next, we study in B the equimultiplicity stratum E and the deformation of D over E induced by the miniversal deformation, the stratum \(\Sigma_ D\subset E\) of the points where the corresponding plane curve has the same equisingularity type in its singularities as D. Following is the main result: If (V,0) is as above, then \(S_{\mu}=S_{\mu^*}=\Sigma_ D\).
scientific article; zbMATH DE number 4019449
scientific article