The Nash problem for torus actions of complexity one

TL;DR

This paper solves the equivariant generalized Nash problem for non-rational normal varieties with torus actions of complexity one, providing a combinatorial description and addressing classical Nash problem questions.

Contribution

It offers an explicit combinatorial description of the Nash order for these varieties and demonstrates that all essential valuations are Nash valuations.

Findings

Every essential valuation is a Nash valuation.

Constructed examples of Nash valuations that are neither minimal nor terminal.

Provided a negative answer to a question by de Fernex and Docampo.

Abstract

We solve the equivariant generalized Nash problem for any non-rational normal variety with torus action of complexity one. Namely, we give an explicit combinatorial description of the Nash order on the set of equivariant divisorial valuations on any such variety. Using this description, we positively solve the classical Nash problem in this setting, showing that every essential valuation is a Nash valuation. We also describe terminal valuations and use our results to answer negatively a question of de Fernex and Docampo by constructing examples of Nash valuations which are neither minimal nor terminal, thus illustrating a striking new feature of the class of singularities under consideration.