Families of arcs on rational surface singularities
type: Item
instance of: scholarly article
title: Families of arcs on rational surface singularities
zbMATH Open document ID: 0867.14012
DOI: 10.1007/BF02567826
published in: Manuscripta Mathematica
publication date: 1996-06-18T00:00:00Z
full work available at URL: https://eudml.org/doc/156131
review text: 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.
reviewed by: Alessandro Gimigliano
Mathematics Subject Classification ID: 14J17
Mathematics Subject Classification ID: 14E15
zbMATH DE Number: 871771
zbMATH Keywords: rational singularity
zbMATH Keywords: singular variety
zbMATH Keywords: desingularization
author: Ana J. Reguera-López
MaRDI profile type: Publication
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OpenAlex ID: W2053531918
description: scientific article; zbMATH DE number 871771