Equivariantly uniformly rational varieties

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 $\mathbb{G}_m$-uniform rationality of affine variety equipped with hyperbolic $\mathbb{G}_m$-action with a unique fixed point, formulated in term of their Altmann-Hausen presentation. We prove the $\mathbb{G}_m$-uniform rationality of Koras-Russel threefolds of the first kind and we also give example of non $\mathbb{G}_m$-uniformly rational but smooth rational $\mathbb{G}_m$-threefold associated to pairs of plane rational curves birationally non equivalent to a union of lines.
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Charlie Petitjean. Equivariantly uniformly rational varieties. The Michigan Mathematical Journal, Michigan Mathematical Journal, 2017, 66 (2), pp.245 - 268. ⟨10.1307/mmj/1496282443⟩. ⟨hal-01151427⟩

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