The Nash problem on arc families of singularities

2002/07/19 by Shihoko Ishii, Janós Kollár, Ishii, Shihoko +2 · 2 citations
Mathematics · #14J10 #14J17 #14M25 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AG #msc:14J10 #msc:14J17 #msc:14M25

paper · pdf · doi:10.48550/arxiv.math/0207171

17 pages

openalex publication_date 2002/07/19 · arxiv created 2003/02/03 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Nash proved that every irreducible component of the space of arcs through a singularity corresponds to an exceptional divisor that occurs on every resolution. He asked if the converse also holds: does every such exceptional divisor correspond to an arc family? We prove that the converse holds for toric singularities but fails in general.

Cited by

• On the Nash Problem over 3-Fold Terminal Singularities of Type cAx/2
• Arcs on rational double points in arbitrary characteristic

Related

• The Nash problem of arcs and the rational double point E₆
• Arc spaces of cA-type singularities
• The Local Nash problem on arc families of singularities
• Exceptional divisors which are not uniruled belong to the image of the Nash map
• A propos de la conjecture de Nash