The Nash problem on arc families of singularities

TL;DR

This paper investigates Nash's problem relating arc families and exceptional divisors in singularities, proving the correspondence for toric cases and showing counterexamples in general.

Contribution

It establishes the validity of Nash's correspondence for toric singularities and demonstrates its failure in broader contexts.

Findings

Nash's correspondence holds for toric singularities.

Counterexamples show the correspondence fails in general.

Provides insight into the structure of arc spaces and singularities.

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.