The Nash problem and its solution: a survey

Camille Plenat (1) , Mark Spivakovsky (2)

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités
2 IMT - Institut de Mathématiques de Toulouse UMR5219

The goal of this survey is to give a historical overview of the Nash Problem of arcs in arbitrary dimension, as well as its solution. This problem was stated by J. Nash around 1963 and has been an important subject of research in singularity theory. In dimension two the problem has been solved affirmatively by J. Fernández de Bobadilla and M. Pe Pereira in 2011. In 2002 S. Ishii and J. Kollár gave a counterexample in dimension four and higher, and in May 2012 T. de Fernex settled (negatively) the last remaining case - that of dimension three. After some history, we give an outline of the solution of the Nash problem for surfaces by Fernández de Bobadilla and Pe Pereira. We end this survey with the latest series of counterexamples, as well as the Revised Nash problem, both due to J. Johnson and J. Kollár.

Mots clés: Nash problem; Arc space

Domaines: Géométrie algébrique [math.AG]; Algèbre commutative [math.AC]

Camille Plenat, Mark Spivakovsky. The Nash problem and its solution: a survey. Journal of Singularities, 2015, Proc. of Geometry and Topology of Singular Spaces, CIRM, 2012, 13, pp.229-244. ⟨hal-00966629⟩