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Three-dimensional counter-examples to the Nash problem

Part of: Birational geometry

Published online by Cambridge University Press:  13 August 2013

Tommaso de Fernex
Tommaso de Fernex*
Affiliation:
Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, UT 84112, USA email [email protected]
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Abstract

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The Nash problem asks about the existence of a correspondence between families of arcs through singularities of complex varieties and certain types of divisorial valuations. It has been positively settled in dimension 2 by Fernández de Bobadilla and Pe Pereira, and it was shown to have a negative answer in all dimensions ${\geq }4$ by Ishii and Kollár. In this note we discuss examples which show that the problem has a negative answer in dimension 3 as well. These examples also bring to light the different nature of the problem depending on whether it is formulated in the algebraic setting or in the analytic setting.

Keywords

arcsdivisorial valuationsresolution of singularities

MSC classification

Secondary: 14E05: Rational and birational maps 14E18: Arcs and motivic integration 14E15: Global theory and resolution of singularities
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 9 , September 2013 , pp. 1519 - 1534
Copyright
© The Author(s) 2013 

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