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Three-dimensional counter-examples to the Nash problem

Published online by Cambridge University Press:  13 August 2013

Tommaso de Fernex*
Affiliation:
Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, UT 84112, USA email defernex@math.utah.edu
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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.

Type
Research Article
Copyright
© The Author(s) 2013 

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