Related papers: Uniformly rational varieties with torus action

We investigate basic properties of uniformly rational varieties, i.e. those smooth varieties for which every point has a Zariski open neighborhood isomorphic to an open subset of A^n. It is an open question of Gromov whether all smooth…

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…

In this paper we provide a characterization of smooth algebraic varieties endowed with a faithful algebraic torus action in terms of a combinatorial description given by Altmann and Hausen. Our main result is that such a variety X is smooth…

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…

In this paper, we classify smooth, contractible affine varieties equipped with faithful torus actions of complexity two, having a unique fixed point and a two-dimensional algebraic quotient isomorphic to a toric blow-up of a toric surface.…

A real algebraic variety W of dimension m is said to be uniformly rational if each of its points has a Zariski open neighborhood which is biregularly isomorphic to a Zariski open subset of R^m. Let l be any nonnegative integer. We prove…

Given an action of an algebraic torus on a normal affine variety, we describe all open subsets admitting a complete orbit space.

In this paper, we prove that: For any given finitely many distinct points $P_1,...,P_r$ and a closed subvariety $S$ of codimension $\geq 2$ in a complete toric variety over a uncountable (characteristic 0) algebraically closed field, there…

This note proves the existence of universal rational parametrizations. The description involves homogeneous coordinates on a toric variety coming from a lattice polytope. We first describe how smooth toric varieties lead to universal…

Algebraic varieties which are locally isomorphic to open subsets of affine space will be called {\em plain}. Plain varieties are smooth and rational. The converse is true for curves and surfaces, and unknown in general. It is shown that…

It was shown in [S. Kaliman, M. Zaidenberg, Gromov ellipticity of cones over projective manifolds, Math. Res. Lett. (to appear), arXiv:2303.02036 (2023)] that the affine cones over flag manifolds and rational smooth projective surfaces are…

The main result of this paper is that every (separated) toric variety which has a semigroup structure compatible with multiplication on the underlying torus is necessarily affine. In the course of proving this statement, we also give a…

Let X=spec A be a normal affine variety over an algebraically closed field k of characteristic 0 endowed with an effective action of a torus T of dimension n. Let also D be a homogeneous locally nilpotent derivation on the normal affine…

We consider rational varieties with a torus action of complexity one and extend the combinatorial approach via the Cox ring developed for the complete case in earlier work to the non-complete, e.g. affine, case. This includes in particular…

It is shown that any compact semistable quotient (in the sense of Heinzner and Snow) of a normal algebraic variety by a complex reductive Lie group $G$ is a good quotient. This reduces the investigation and classification of such…

We show how to construct certain homogeneous deformations for rational normal varieties with codimension one torus action. This can then be used to construct homogeneous deformations of any toric variety in arbitrary degree. For locally…

Let X be a normal affine T-variety of complexity at most one over a perfect field k, where T stands for the split algebraic torus. Our main result is a classification of additive group actions on X that are normalized by the T-action. This…

Algebraic actions of unipotent groups $U$ actions on affine $k-$varieties $X$ ($k$ an algebraically closed field of characteristic 0) for which the algebraic quotient $X//U$ has small dimension are considered$.$ In case $X$ is factorial,…

In this work, it is shown that a simply-connected, rationally-elliptic torus orbifold is equivariantly rationally homotopy equivalent to the quotient of a product of spheres by an almost-free, linear torus action, where this torus has rank…

We give a characterization of all complete smooth toric varieties whose rational homotopy is of elliptic type. All such toric varieties of complex dimension not more than three are explicitly described.