Terminal valuations and the Nash problem
Abstract
Let X be an algebraic variety of characteristic zero. Terminal valuations are defined in the sense of the minimal model program, as those valuations given by the exceptional divisors on a minimal model over X. We prove that every terminal valuation over X is in the image of the Nash map, and thus it corresponds to a maximal family of arcs through the singular locus of X. In dimension two, this result gives a new proof of the theorem of Fernández de Bobadilla and Pe Pereira stating that, for surfaces, the Nash map is a bijection.
 Publication:

Inventiones Mathematicae
 Pub Date:
 January 2016
 DOI:
 10.1007/s0022201505975
 arXiv:
 arXiv:1404.0762
 Bibcode:
 2016InMat.203..303D
 Keywords:

 Mathematics  Algebraic Geometry;
 Primary 14E18;
 Secondary 14E30;
 14J17
 EPrint:
 v2: 21 pages, minor changes and corrections following the referees' reports. To appear in Invent. Math