arXiv subjects

Kirill Shakhmatov

Publications and source records attributed to Kirill Shakhmatov.

8 recordsLinked to original sources

arXiv2026

Automorphisms of toric varieties and Gale duality

We classify complete toric threefolds $X$ such that the automorphism group $\text{Aut}(X)$ acts on $X$ with an open orbit whose complement does not contain a divisor. The latter condition means that for any ray $ρ$ of the fan $Σ_X$ there is a Demazure root of $Σ_X$ associated with $ρ$. We also find among these varieties those $X$ for which the group $\text{Aut}(X)$ is transitive on the smooth locus $X^{\text{reg}}$. The classifications are based on Gale-dual interpretations of these properties.

math.AG↗

arXiv2026

Cylinders and the zero locus of the plinth ideal

Given a $\mathbb{G}_\mathrm{a}$-action on an affine variety $X$, we show that the complement of the union of all principal invariant cylinders in $X$ is equal to the zero locus of the plinth ideal of the corresponding locally nilpotent derivation.

math.AG↗

arXiv2026

On flexibility of affine factorial varieties

We give a criterion of factoriality of a suspension. This allows to construct many examples of flexible affine factorial varieties. In particular, we find a homogeneous affine factorial 3-fold that is not a homogeneous space of an algebraic group.

math.AG↗

arXiv2025

Flexibility of affine cones over a smooth complete intersection of two quadrics

We prove flexibility of two families of affine varieties: the complement in $\mathbb{P}^n$ of a projective quadric of rank at least three and affine cones over a smooth complete intersection of two quadrics in $\mathbb{P}^{n + 2}$, $n \ge 3$.

math.AG↗

arXiv2024

Radiant toric varieties and unipotent group actions

We consider complete toric varieties $X$ such that a maximal unipotent subgroup $U$ of the automorphism group $\text{Aut}(X)$ acts on $X$ with an open orbit. It turns out that such varieties can be characterized by several remarkable properties. We study the set of Demazure roots of the corresponding complete fan, describe the structure of a maximal unipotent subgroup $U$ in $\text{Aut}(X)$, and find all regular subgroups in $U$ that act on $X$ with an open orbit.

math.AG↗

arXiv2022

Homogeneous algebraic varieties and transitivity degree

Let $X$ be an algebraic variety such that the group $\text{Aut}(X)$ acts on $X$ transitively. We define the transitivity degree of $X$ as a maximal number $m$ such that the action of $\text{Aut}(X)$ on $X$ is $m$-transitive. If the action of $\text{Aut}(X)$ is $m$-transitive for all $m$, the transitivity degree is infinite. We compute the transitivity degree for all quasi-affine toric varieties and for many homogeneous spaces of algebraic groups. Also we discuss a conjecture and open questions related to this invariant.

math.AG↗

arXiv2022

Some finiteness results on triangular automorphisms

In this note we prove that every finite collection of connected algebraic subgroups of the group of triangular automorphisms of the affine space generates a connected solvable algebraic subgroup.

math.AG↗

arXiv2020

Smooth non-projective equivariant completions of affine spaces

In this paper we construct an equivariant embedding of the affine space $\mathbb{A}^n$ with the translation group action into a complete non-projective algebraic variety $X$ for all $n \geq 3$. The theory of toric varieties is used as the main tool for this construction. In the case of $n = 3$ we describe the orbit structure on the variety $X$.

math.AG↗