On the behavior of formal neighborhoods in the Nash sets associated with toric valuations: a comparison theorem
Bourqui, David1; Morán Cañón, Mario2; Sebag, Julien1
1Institut de recherche mathématique de Rennes, UMR 6625 du CNRS, Université de Rennes, Campus de Beaulieu, 35042 Rennes cedex (France)
2Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, and, Instituto de Ciencias Matemáticas, ICMAT, CSIC-UAM-UC3M-UCM, Campus de Cantoblanco, 28049 Madrid (Spain)
Annales de l'Institut Fourier, Online first, 33 p.

We show that there exists a strong connection between the generic formal neighborhood at a rational arc lying in the Nash set associated with a toric divisorial valuation on a toric variety and the formal neighborhood at the generic point of the same Nash set. This may be interpreted as the fact that, analytically along such a Nash set, the arc scheme of a toric variety is a product of a finite dimensional singularity and an infinite dimensional affine space.

Nous montrons qu’il existe un lien étroit entre le voisinage formel générique d’un arc rationnel situé sur l’ensemble de Nash associé à une valuation divisorielle torique et le voisinage formel du point générique de cet ensemble de Nash. Cela peut être interprété comme le fait que, analytiquement le long de cet ensemble de Nash, le schéma des arcs d’une variété torique est le produit d’une singularité de dimension finie et d’un espace affine de dimension infinie.

Received:
Revised:
Accepted:
Online First:
DOI: 10.5802/aif.3757
Classification: 14B20, 14E18, 14M25
Keywords: arc schemes, formal neighborhoods, toric varieties
Mots-clés : schéma des arcs, voisinages formels, variétés toriques

Bourqui, David  1 ; Morán Cañón, Mario  2 ; Sebag, Julien  1

1 Institut de recherche mathématique de Rennes, UMR 6625 du CNRS, Université de Rennes, Campus de Beaulieu, 35042 Rennes cedex (France)
2 Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, and, Instituto de Ciencias Matemáticas, ICMAT, CSIC-UAM-UC3M-UCM, Campus de Cantoblanco, 28049 Madrid (Spain) 
Bourqui, David; Morán Cañón, Mario; Sebag, Julien. On the behavior of formal neighborhoods in the Nash sets associated with toric valuations: a comparison theorem. Annales de l'Institut Fourier, Online first, 33 p.
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     author = {Bourqui, David and Mor\'an Ca\~n\'on, Mario and Sebag, Julien},
     title = {On the behavior of formal neighborhoods in the {Nash} sets associated with toric valuations: a comparison theorem},
     journal = {Annales de l'Institut Fourier},
     year = {2026},
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