Families of arcs on rational surface singularities. (English) Zbl 0867.14012

The problem treated in this paper was posed by J. Nash, who proposed to study the space \({\mathcal H}\) of arcs in a germ of a singular variety \((V,P)\); he showed that the arc families in \({\mathcal H}\) correspond to irreducible components of the exceptional locus of any given desingularization of \((V,P)\) and raised the question whether every component which (modulo birational equivalence) appears in every desingularization (such components are called essential) is associated to some family in \({\mathcal H}\).
The paper gives an affirmative answer in the case when \(V\) is a surface and the singularity is rational. In this case the essential components \(\{E_\alpha\}_{\alpha\in \Delta}\) are represented by the irreducible components of a minimal desingularization \(S\to V\). – The idea is to define, for each \(E_\alpha\), a \(\mathbb{Q}\)-Cartier divisor \(D_\alpha\) in \(S\) (\(D_\alpha\) is the exceptional part of the total transform of the projection of any germ of curve in \(S\) which is transversal to \(E_\alpha\) and does not meet any \(E_\gamma\), \(\gamma\neq \alpha\)); then a closed subset \({\mathcal H}_\alpha\) of \({\mathcal H}\) is defined by imposing valuative conditions determined by \(D_\alpha\), and those \({\mathcal H}_\alpha\) give the answer to the Nash problem.


14J17 Singularities of surfaces or higher-dimensional varieties
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
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