## 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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