Three-dimensional counter-examples to the Nash problem. (September 2013) - de Fernex, Tommaso
Abstract 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 <alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg8w9z0" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[≥ 4]]> </alternatives> 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.