The Nash problem on arc families of singularities

By S. Ishii, J. Kollár. 2002

125 citations recorded by the source corpus.

Nash [21] proved that every irreducible component of the space of arcs through a singularity corresponds to an exceptional divisor that appears 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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/23f794715241a00ada0d1382858c7d081cccd314

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In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Quasi-ordinary Power Series and Their Zeta Functions (2003, result): “,T) but in general the other way around is not true. Consider for instance h= x3 1 +x 3 2 +x 3 3 +x 3 4 +x 6 5 and the diﬀerential form ω= dx.This example has been recently studied by Ishii-Kollar in [23] to disprove a J. Nash conjecture. Then hhas non-degenerated principal part and its Newton polyh”
Motivic integration on smooth rigid varieties and invariants of degenerations (2001, background): “Koll ár [17] gave an example where these two sets are not in bijection.”
Nash problem for surfaces (2011, background): “On the other hand bijectivity of the Nash mapping has been shown for many classes of surfaces (see [4],[5],[7],[8],[9],[10],[14],[16],[18],[19],[22], [20],[23],[24]).”
Arc spaces, motivic integration and stringy invariants (2004, background): “† Ishii and Koll¶ar [ IK ] found counter examples in dimensions at least 4 to the Nash problem, which relates irreducible components of the space of arcs through a singularity to exceptional components of a resolution.”