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Families of smooth curves on surface singularities and wedges. (English) Zbl 0894.14017

This paper concerns the study of families of smooth curves lying on a surface singularity \((S,O)\). Let \(\pi:X\to S\) be a minimal desingularization of \((S,O)\). The authors prove that for any irreducible component \(E\) of \(\pi^{-1}(0)\) such that \(\text{ord}_E (mO_X)=1\) (where \(\text{ord}_E\) is the divisorial valuation and \(m\) is the maximal ideal corresponding to \(O\) in \(S)\) the family of smooth curves \({\mathcal L}_E\) is nonempty. In that case a smooth curve in \({\mathcal L}_E\) corresponds to a point in \(E\). The authors conclude the existence of smooth curves lying on sandwich singularities. They also study wedge morphisms, that are morphisms \(h:\text{Spec} (k[[u,v]]) \to(S,O)\) such that the image of \(h\) is Zariski dense in some analytically irreducible component of \((S,O)\).

MSC:

14J17 Singularities of surfaces or higher-dimensional varieties
14H50 Plane and space curves
14B05 Singularities in algebraic geometry
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
32S45 Modifications; resolution of singularities (complex-analytic aspects)
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