Arcs, valuations and the Nash map

By S. Ishii. 2004

52 citations recorded by the source corpus.

Abstract This paper gives a map from the set of families of arcs on a variety to the set of valuations on the rational function field of the variety. We characterize a family of arcs which corresponds to a divisorial valuation by this map. We can see that both the Nash map and a certain McKay correspondence are the restrictions of this map. This paper also gives the affirmative answer to the Nash problem for a non-normal variety in a certain category. As a corollary, we get the affirmative answer for a non-normal toric variety.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1dfa57533187ecbd98a6e091780c0adcdb79a4ce

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

1. Tommaso de Fernex
2. L. Ein
3. María Pe Pereira
4. J. Bobadilla
5. S. Ishii
6. M. Mustaţă
7. Karl Schwede
8. Song Sun

In Geometry and Topology

1. Tommaso de Fernex
2. S. Ishii
3. L. Ein
4. María Pe Pereira
5. J. Bobadilla
6. M. Mustaţă
7. H. Blum
8. Chi Li

In Mathematical Physics

1. Tommaso de Fernex
2. S. Ishii
3. L. Ein
4. María Pe Pereira
5. J. Bobadilla
6. M. Mustaţă
7. H. Blum
8. Martin Ulirsch

Citations in the corpus

• 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]).”
• Divisorial Valuations via Arcs (2007, methodology): “This link was first explored by Mustat¸ya in [M1, M2], and then further studied in [ELM, I2 , I3].”