Local geometry of smooth curves passing through rational double points (Q1196014)

We study the local geometry of smooth curves on singular surfaces. This amounts to the following problem in local algebra: Classify pairs \((R,p)\) consisting of a complete local \(\mathbb{C}\)-algebra \(R\), together with a height one prime ideal \(p\) of \(R\) such that \(R/p\) is regular. It is assumed that \(R\) is the complete local ring of a singular point on a variety. Two pairs \((R,p)\) and \((R',p')\) are isomorphic if there exists an isomorphism of \(\mathbb{C}\)-algebras from \(R\) to \(R'\) which carries \(p\) to \(p'\). A complete solution is given in the special case where \(R\) corresponds to a rational double point.