Equivariantly uniformly rational varieties

TL;DR

This paper introduces equivariant versions of uniform rationality for algebraic varieties with group actions, providing criteria and examples, including Gm-uniform rationality of certain affine varieties and Koras-Russel threefolds.

Contribution

It develops a new framework for G-equivariant uniform rationality, establishes a criterion for Gm-uniform rationality, and provides concrete examples and counterexamples.

Findings

Established a criterion for Gm-uniform rationality using Altmann-Hausen presentation.

Proved Gm-uniform rationality of Koras-Russel threefolds of the first kind.

Presented examples of Gm-threefolds that are not Gm-uniformly rational.

Abstract

We introduce equivariant versions of uniform rationality: given an algebraic group G, a G-variety is called G-uniformly rational (resp. G-linearly uniformly rational) if every point has a G-invariant open neighborhood equivariantly isomorphic to a G-invariant open subset of the affine space endowed with a G-action (resp. linear G-action). We establish a criterion for Gm-uniform rationality of affine variety equipped with hyperbolic Gm-action with a unique fixed point, formulated in term of their Altmann-Hausen presentation. We prove the Gm-uniform rationality of Koras-Russel threefolds of the first kind and we also give example of non Gm-uniformly rational but smooth rational Gm-threefold associated to pairs of plane rational curves birationally non equivalent to a union of lines.