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
\(n\) is finite,
\(\widetilde{f}:=f\circ n:\widetilde{X}\to S\) 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,
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.


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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