## 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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