## 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.

### MSC:

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

### Keywords:

rational singularity; singular variety; desingularization
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### References:

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