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Ivan Beldiev

Publications and source records attributed to Ivan Beldiev.

8 recordsLinked to original sources
arXiv2026

The Lie algebra of polynomial vector fields on the affine space with constant divergence is $1.5$-generated

A Lie algebra is said to be $1.5$-generated if every nonzero element can be completed to a two-element generating set. We prove that the Lie algebra of polynomial vector fields with constant divergence on the affine space is $1.5$-generated. We also show that the Lie algebra of all polynomial vector fields on the complex affine space is generated by two completely integrable elements.

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arXiv2025

Projective Hypersurfaces of High Degree Admitting an Induced Additive Action

We study induced additive actions on projective hypersurfaces, i.e. effective regular actions of the algebraic group $\mathbb G_a^m$ with an open orbit that can be extended to a regular action on the ambient projective space. It is known that the degree of a hypersurface $X\subseteq\mathbb{P}^n$ admitting an induced additive action cannot be greater than $n$ and there is a unique such hypersurface of degree $n$. We give a complete classification of hypersurfaces $X\subseteq \mathbb{P}^n$ admitting an induced additive action of degrees from $n-1$ to $n-3$.

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arXiv2024

On Picard Number and Dimension of Algebraic Homogeneous Spaces

An algebraic variety $X$ is called a homogeneous space if there exists a transitive regular action of an algebraic group on $X$. We prove inequalities between the dimension of a homogeneous space of a linear algebraic group and its Picard number.

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arXiv2024

On Normality of Projective Hypersurfaces with an Additive Action

We study projective hypersurfaces $X$ admitting an induced additive action, i.e., an effective action ${\mathbb G_a^m\times X\to X}$ of the vector group $\mathbb G_a^m$ with an open orbit that can be extended to an action on the ambient projective space. A criterion for normality of such a hypersurface $X$ is given. Also, we prove that for any projective hypersurface $Z$ there exists a hypersurface $X$ with an induced additive action such that the complement to the open $\mathbb G_a^m$-orbit in $X$ is a projective cone over $Z$. We introduce a construction that produces non-degenerate hypersurfaces with induced additive action from Young diagrams and study the properties of the hypersurfaces obtained in this way.

math.AG↗
arXiv2023

Gorenstein Algebras and Uniqueness of Additive Actions

We study induced additive actions on projective hypersurfaces, i.e. regular actions of the algebraic group $\mathbb G_a^m$ with an open orbit that can be extended to a regular action on the ambient projective space. We prove that if a projective hypersurface admits an induced additive action, then it is unique if and only if the hypersurface is non-degenerate. We also show that for any $n\geq 2$, there exists a non-degenerate hypersurface in $\mathbb P^n$ of each degree $d$ from $2$ to $n$.

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arXiv2022

The Group of Isometries of $K_0(\mathbb P_n)$

We study the group of isometries of the Grothendieck group $K_0(\mathbb P_n)$ equipped with the standard Euler form $\chi$ defined by $\chi(E, F) = \sum_{\nu}(-1)^\nu\dim Ext^\nu(E, F)$. We prove several properties of this group, in particular, we show that it is essentially a free abelian group of rank $[\frac{n+1}{2}]$. Also, we compute explicitly its generators for $n\leqslant 6$.

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