On the behavior of formal neighborhoods in the Nash sets associated with toric valuations: a comparison theorem

TL;DR

This paper explores the relationship between formal neighborhoods in Nash sets associated with toric valuations, revealing a product structure that links local and generic properties of arc schemes on toric varieties.

Contribution

It establishes a comparison theorem connecting the formal neighborhoods at rational arcs and generic points within Nash sets for toric valuations.

Findings

Formal neighborhoods at rational arcs and generic points are strongly connected.

Arc schemes of toric varieties locally resemble a product of a finite-dimensional singularity and an infinite-dimensional affine space.

Abstract

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.