Related papers: Retract rational varieties are uniformly retract r…

Let k be a field of characteristic zero. We show that the norm variety associated to a prime $\ell$ and an ordered sequence of invertible elements of k is geometrically retract rational. This generalizes a recent result of…

A smooth, proper, retract rational variety over a field $k$ is known to be $\mathbb{A}^1$-connected. We improve on this result, in the case when $k$ is infinite, showing that such varieties are naively $\mathbb{A}^1$-connected.

Let K be any field and G be a finite group. We will prove that, if K is any field, p an odd prime number, and G is a non-abelian group of exponent p with |G|=p^3 or p^4 satisfying [K(\zeta_p):K] <= 2, then K(G) is rational over K. We will…

This paper studies absolute retracts in congruence modular varieties of universal algebras. It is shown that every absolute retract with finite dimensional congruence lattice is a product of subdirectly irreducible algebras. Further, every…

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.

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 prove that every nontrivial principal $G_m$-bundle over a complete uniformly rational variety is algebraically elliptic in the sense of Gromov.

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…

Nonsingular projective varieties which are both convex and rationally connected are considered. We ask whether such varieties must be algebraic homogeneous spaces G/P. In case X is a complete intersection, an affirmative answer is obtained…

A retract variety is defined as a class of algebras closed under isomorphisms, retracts and products. Let a principal retract variety be generated by one algebra and a set-principal retract variety be generated by some set of algebras. It…

Given a finite simplicial complex $\mathcal{K}$ in $\mathbb{R}^n$ and a real algebraic variety $Y,$ by a $\mathcal{K}$-regular map $|\mathcal{K}|\rightarrow Y$ we mean a continuous map whose restriction to every simplex in $\mathcal{K}$ is…

For any prime number $p$ and field $k$, we characterize the $p$-retract rationality of an algebraic $k$-torus in terms of its character lattice. We show that a $k$-torus is retract rational if and only if it is $p$-retract rational for…

We prove that every curve on a rationally connected variety is algebraically equivalent to a (non-effective) integral sum of rational curves.

We prove that every domain that fails to be algebraic admits the unit interval $[0, 1]$ as its Scott-continuous retract. As a result, every countable domain is algebraic.

A collection of varieties satisfies uniform potential density if each of them contains a dense subset of rational points after extending its ground field by a bounded degree. In this paper, we prove that uniform potential density holds for…

In the present paper, we prove the retract rationality of the classifying spaces $BG$ for several types of finite connected group schemes $G$ over algebraically closed fields of positive characteristic $p>0$. In particular, we prove the…

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…

We study unirational algebraic varieties and the fields of rational functions on them. We show that after adding a finite number of variables some of these fields admit an infinitely transitive model. The latter is an algebraic variety with…

We introduce the concept of the modularity of an abelian variety defined over the rational number field extending the modularity of an elliptic curve. We discuss the modularity of an abelian variety over the rational number field. We…

Let $X$ be a nonsingular rational variety. We prove that $X\times \mathbb{C}^2$ is uniformly rational. It follows that nonsingular stably rational varieties are stably uniformly rational.