Around the uniform rationality I

TL;DR

This paper demonstrates the existence of rational smooth algebraic varieties that are not uniformly rational by analyzing numerical obstructions in compactifications of affine spaces.

Contribution

It introduces a new obstruction criterion and applies it to distinguish rational varieties that are not uniformly rational.

Findings

Existence of rational but not uniformly rational smooth algebraic varieties.

Numerical obstruction behaves differently in certain compactifications.

Provides a method to identify non-uniformly rational varieties.

Abstract

We prove that there exist rational but not uniformly rational smooth algebraic varieties. The proof is based on computing a certain numerical obstruction developed in the case of compactifications of affine spaces. We show that for some particular compactifications this obstruction behaves differently compared to the uniformly rational situation.