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Equinormalizability and topologically triviality of deformations of isolated curve singularities over smooth base spaces. (English) Zbl 1337.14004

Let \(f:X\to S\) be a morphism of complex spaces. A simultaneous normalization of \(f\) is a morphism \(n:\widetilde{X}\to X\) such that
(1)
\(n\) is finite,
(2)
\(\widetilde{f}:=f\circ n:\widetilde{X}\to S\) is normal, i.e. for each \(z\in \widetilde{X}, \widetilde{f}\) is flat at \(z\) and the fibre \(\widetilde{f}^{-1}(\widetilde{f}(z))\) is normal,
(3)
the induced map \(n_s:\widetilde{X}_s:=\widetilde{f}^{-1}(s)\to X_s\) is bimeromorphic for all \(s\in f(X)\).
\(f\) is called equinormalizable if the normalization \(n:\widetilde{X}\to X\) is a simultaneous normalization of \(f\). It is called equinormalizable at \(x\in X\) if the restriction of \(f\) to some neighbourhood of \(x\) is equinormalizable. It is known (B. Teissier) that a deformation of a reduced curve singularity over a normal base space is equinormalizable if and only if it is \(\delta\)-constant.
Let \(f:(X,x)\to (\mathbb{C}^n, 0)\) be a deformation of an isolated curve singularity \((X_0,x)\) with \((X,x)\) pure dimensional. Suppose that there exists a representative \(f:X\to S\) such that \(X\) is generically reduced over \(S\). It is proved that \(f\) being \(\delta\)-constant and the normalization \(\widetilde{X}\) of \(X\) being Cohen-Macaulay implies that \(f\) is equinormalizable. For one-parameter families of isolated curve singularities it is proved that their topologically triviality is equivalent to the admission of weak simultaneous resolutions.

MSC:

14B07 Deformations of singularities
14B12 Local deformation theory, Artin approximation, etc.
14B25 Local structure of morphisms in algebraic geometry: étale, flat, etc.
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