Three-dimensional counter-examples to the Nash problem. (September 2013)

• Record Type: Journal Article
• Authors: de Fernex, Tommaso
• Is Part Of: Compositio mathematica. Volume 149:Number 9(2013)
• Page Start: 1519
• Page End: 1534
• Publication Date: 2013-09
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X13007252
• Languages: English
• ISSNs: 0010-437X

Abstract

<abstract abstract-type="normal"> <title>Abstract</title> <p>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 <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg8w9z0" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[${\geq }4$]]></tex-math></alternatives></inline-formula> 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.</p> </abstract>