# Uniformly rational varieties with torus action

## TL;DR

This paper proves that smooth, rational affine varieties with a torus action and low-dimensional quotients are uniformly rational, meaning they locally resemble affine space, which advances understanding of their geometric structure.

## Contribution

It establishes that smooth, rational affine varieties with torus actions and quotients of dimension 0 or 1 are uniformly rational, a new class of varieties with this property.

## Key findings

- Such varieties are uniformly rational.
- The result applies to varieties with low-dimensional algebraic quotients.
- Provides new insights into the structure of torus actions on rational varieties.

## Abstract

A smooth variety is called uniformly rational if every point admits a Zariski open neighborhood isomorphic to a Zariski open subset of the affine space. In this note we show that every smooth and rational affine variety endowed with an algebraic torus action such that the algebraic quotient has dimension 0 or 1 is uniformly rational.