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.
Funding source: Agence Nationale de la Recherche
Award Identifier / Grant number: ANR-17-CE40-0023
Funding statement: This work was initiated during visits of K. Langlois at IRMAR (Université de Rennes 1) and H. Mourtada at the Mathematisches Institut (Heinrich-Heine-Universität Düsseldorf), and some progresses were made while D. Bourqui and K. Langlois were visiting the Institut de Mathématiques de Jussieu-Paris Rive Gauche (Université Paris-Cité). We are grateful to these institutions for their hospitality and financial support. We have also benefited from the financial support of the ANR project LISA (ANR-17-CE40-0023). D. Bourqui and H. Mourtada were partially supported by the PICS project More Invariants in Arc Schemes.
Acknowledgements
We thank Tommaso de Fernex and Roi Docampo for useful discussions. We are grateful to Shihoko Ishii for her answers about the behavior of the Nash problem with respect to étale morphisms (see Remark 5.11). We owe many thanks to an anonymous referee for her/his very thorough reading of the paper and the numerous valuable comments and suggestions.
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Articles in the same Issue
- Frontmatter
- Geometric main conjectures in function fields
- Noncompact self-shrinkers for mean curvature flow with arbitrary genus
- K-stability of Casagrande–Druel varieties
- Geometry at infinity of the space of Kähler potentials and asymptotic properties of filtrations
- The Nash problem for torus actions of complexity one
- Min-max theory for capillary surfaces
- Floer homology and right-veering monodromy
- The fourth moment of the Hurwitz zeta function
Articles in the same Issue
- Frontmatter
- Geometric main conjectures in function fields
- Noncompact self-shrinkers for mean curvature flow with arbitrary genus
- K-stability of Casagrande–Druel varieties
- Geometry at infinity of the space of Kähler potentials and asymptotic properties of filtrations
- The Nash problem for torus actions of complexity one
- Min-max theory for capillary surfaces
- Floer homology and right-veering monodromy
- The fourth moment of the Hurwitz zeta function