Nash  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.
János Kollár. Shihoko Shihoko Ishii. "The Nash problem on arc families of singularities." Duke Math. J. 120 (3) 601 - 620, 1 December 2003. https://doi.org/10.1215/S0012-7094-03-12034-7