The Nash problem of arcs and the rational double points \(D_{n}\)
type: Item
instance of: scholarly article
title: The Nash problem of arcs and the rational double points \(D_{n}\)
zbMATH Open document ID: 1168.14004
author: Camille Plénat
publication date: 2009-02-10T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/10377
review text: 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.
reviewed by: Shihoko Ishii
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 14J17
Mathematics Subject Classification ID: 14J10
zbMATH DE Number: 5505482
zbMATH Keywords: space of arcs
zbMATH Keywords: Nash problem
zbMATH Keywords: rational double points
MaRDI profile type: Publication
OpenAlex ID: W2328662802
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DOI: 10.5802/AIF.2413
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 5505482