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Equianalytic and equisingular families of curves on surfaces. (English) Zbl 0874.14021
The authors consider flat families of embedded reduced curves on a smooth surface \(S\) such that for each member \(C\) of the family the number of singular points of \(C\) and for each singular point \(x\in C\) the analytic (resp. the equisingular) type of \((C,x)\) is fixed, then they obtain a locally closed subscheme \(H_S^{ea}\) (resp. \(H_S^{es}\)) of the Hilbert scheme \(H_S\) of \(S\). They study local properties of \(H_S^{ea}\) (resp. \(H_S^{es})\) and give conditions on the smoothness of these subschemes. They improve previous results for curves in \(\mathbb{P}^2\) and give answer to some existing problem of curves of given degree having a fixed number of singularities of given analytic type.

14J17 Singularities of surfaces or higher-dimensional varieties
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14H20 Singularities of curves, local rings
32S15 Equisingularity (topological and analytic)
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