The Nash problem of arcs and the rational double points \(D_{n}\). (English) Zbl 1168.14004

The Nash problem is asking whether the number of irreducible families of arcs passing through the singular locus is the same as the number of the essential divisors of resolutions of singularities. This problem is negatively answered for varieties of dimension greater than or equal to 4. But for 2- and 3-dimensional cases, the problem is still open. This paper answers the problem affirmatively if the singularity is 2-dimensional rational and of type \(D_n\). The problem is reduced to show that there is no inclusion relation between the irreducible families of arcs corresponding to the essential divisors. The author first obtain an inclusion relation among the irreducible families using “usual order” of valuations and then check the other inclusion relations case by case.


14B05 Singularities in algebraic geometry
14J17 Singularities of surfaces or higher-dimensional varieties
14J10 Families, moduli, classification: algebraic theory
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