The Nash problem on arc families of singularities
  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. 
Shihoko Ishii (Tokyo Institute of Technology) and J\'anos Koll\'ar   (Princeton University)