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.

v2: 21 pages, minor changes and corrections following the referees' reports. To appear in Invent. Math

References in corpus (3)

• Bijectiveness of the Nash Map for Quasi-Ordinary Hypersurface Singularities
• Exceptional divisors which are not uniruled belong to the image of the Nash map
• Familles d'espaces de m-jets et d'espaces d'arcs

Cited by in corpus (10)

• Normalized Berkovich spaces and surface singularities
• Differentials on the arc space
• Familles d'espaces de m-jets et d'espaces d'arcs
• Kawamata log terminal singularities of full rank
• The arc space of the Grassmannian
• On Lipschitz Normally Embedded singularities
• Nash multiplicity sequences and Hironaka's order function
• On the embedded Nash problem
• Contact loci and Hironaka's order
• On the Nash problem for terminal threefolds of type $cA/r$