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Sergey Gaifullin

Publications and source records attributed to Sergey Gaifullin.

At least 19 recordsLinked to original sources
arXiv2026

Non-cancellative varieties, maximal tori, and the Makar-Limanov invariant

In this paper, we construct a wide class of new counterexamples to the generalized Zariski cancellation problem. The cylinders over these counterexamples have infinitely many non-conjugate maximal tori in their regular automorphism groups. We provide an example of a variety with maximal tori of different dimensions in its automorphism group. Additionally, we prove that a cylinder over a non-rigid trinomial variety without a line factor is generically flexible, and hence, has a trivial Makar-Limanov invariant.

math.AG↗
arXiv2025

Flexibility criterion for affine horospherical varieties

In this paper we obtain a criterion of flexibility for an affine complexity-zero horospherical variety. This result generalizes previously known results on flexibility of normal horospherical varieties, horospherical varieties with an action of a semisimple group, and non-normal toric varieties.

math.AG↗
arXiv2025

The isotropy group of a derivation on a Danielewski-type algebra

Given an algebraically closed field $k$ of characteristic zero, we consider in this paper $k$-algebras of the form $$A_{c,q}=k[x,y,z]/\big(c(x)z-q(x,y)\big),$$ where $c(x)\in k[x]$ is a polynomial of degree at least two and $q(x,y)\in k[x,y]$ is a quasi-monic polynomial of degree at least two with respect to $y$. We give a complete description of the $k$-automorphism group of $A_{c,q}$ as an abstract group. Moreover, for every non-locally nilpotent $k$-derivation $\delta$ of $A_{c,q}$ we prove that the isotropy group of $\delta$ is a linear algebraic group of dimension at most three.

math.RA↗
arXiv2024

On finite-dimensional homogeneous Lie algebras of derivations of polynomial rings

For a finite set of homogeneous locally nilpotent derivations of the algebra of polynomials in several variables, a finite dimensionality criterion for the Lie algebra generated by these derivations is known. Also the structure of the corresponding finite-dimensional Lie algebras is described in previous works. In this paper, we obtain a finite dimensionality criterion for a Lie algebra generated by a finite set of homogeneous derivations, each of which is not locally nilpotent.

math.RA↗
arXiv2023

Isolated torus invariants and automorphism groups of rigid varieties

Perepechko and Zaidenberg conjectured that the neutral component of the automorphism group of a rigid affine variety is a torus. We prove this conjecture for toric varieties and varieties with a torus action of complexity one. We also obtain a criterion for an $m$-suspension over a rigid variety to be rigid (for every rigid variety and every regular function). Additionally, we study the automorphism group of $m$-suspensions satisfying this criterion.

math.AG↗
arXiv2023

Modified Derksen invariant

Modified Derksen invariant HD*(X) of an affine variety X is a subalgebra in K[X] generated by kernels of all locally nilpotent derivations of K[X] with slices. If there is a locally nilpotent derivation of K[X] with a slice then X is a product of Y and a line, where Y is an affine variety. We prove that there are three possibilities: A) HD*(X) = K[X]; B) HD*(X) is a proper infintely generated subalgebra; C) HD*(X) = \BK[Y]. We give examples for each case, and also provide sufficient conditions for the variety Y so that the variety X belongs to one of the type.

math.AG↗
arXiv2023

On locally nilpotent derivations of polynomial algebra in three variables

In this paper we investigate locally nilpotent derivations on the polynomial algebra in three variables over a field of characteristic zero. We introduce an iterating construction giving all locally nilpotent derivations of rank $2$. This construction allows to get examples of non-triangularizable locally nilpotent derivations of rank $2$. We also show that the well-known example of a locally nilpotent derivation of rank $3$, given by Freudenburg, is a member of a large family of new examples of rank $3$ locally nilpotent derivations. Our approach is based on considering all locally nilpotent derivations commuting with a given one. We obtain a characterization of locally nilpotent derivations with a given rank in terms of sets of commuting locally nilpotent derivations.

math.AC↗
arXiv2023

Rigid Trinomial Varieties

An algebraic variety $X$ is called rigid if there is no non-trivial action on $X$ of the additive group of the base field. A trinomial variety is an affine variety that is given by a set of equations consisting of polynomials with three monomials; see Definition 1. In this paper, we complete the classification of rigid trinomial varieties.

math.AG↗
arXiv2022

Modified Makar-Limanov and Derksen invariants

We investigate modified Makar-Limanov and Derksen invariants of an affine algebraic variety. The modified Makar-Limanov invariant is the intersection of kernels of all locally nilpotent derivations with slices and the modified Derksen invariant is the subalgebra generated by these kernels. We prove that modified Makar-Limanov invariant coincide with Makar-Limanov invariant if there exists a locally nilpotent derivation with a slice. Also we construct an example of a variety admitting a locally nilpotent derivation with a slice such that modified Derksen invariant does not coincide with Derksen invariant.

math.AG↗
arXiv2022

Automorphism group orbits on horospherical varieties and divisor class group

In 2013 Bazhov proved a criterium for two points on a complete toric variety to lie in the same orbit of the neutral component of automorphism group. This criterium is in terms of divisor class group. Arzhantsev-Bazhov (2013) obtained a similar criterium for affine toric varieties. We prove a necessary condition similar this criteria to the cases of affine and projective horospherical varieties.

math.AG↗
arXiv2022

Orbits of automorphism group of trinomial hypersurfaces

Trinomial hypersurfaces form a natural class of affine algebraic varieties closely connected with varieties admitting a torus action of complexity one. We investigate orbits of the automorphism group on these hypersurfaces. We prove that each nonrigid trinomial variety has finite number of orbits. We investigate singular orbits and this gives us description of all orbits for some classes of flexible trinomial hypersurfaces. Also we obtain a description of orbits for a class of hypersurfaces having a unique variable with power one in the equation.

math.AG↗
arXiv2021

On orbits of automorphism groups on horospherical varieties

In this paper we describe orbits of automorphism group on a horospherical variety in terms of degrees of homogeneous with respect to natural grading locally nilpotent derivations. In case of (may be non-normal) toric varieties a description of orbits of automorphism group in terms of corresponding weight monoid is obtained.

math.AG↗
arXiv2020

Automorphisms of nonnormal toric varieties

In this paper we prove criteria for a nonnormal toric variety to be flexible, to be rigid and to be almost rigid. For rigid and almost rigid toric varieties we describe the automorphism group explicitly.

math.AG↗
arXiv2020

Commutative actions on smooth projective quadrics

By a commutative action on a smooth quadric $Q_n$ in $P^{n+1}$ we mean an effective action of a commutative connected algebraic group on $Q_n$ with an open orbit. We show that for $n \geq 3$ all commutative actions on $Q_n$ are additive actions described by Sharoiko in 2009. So there is a unique commutative action on $Q_n$ up to equivalence. For $n = 2$ there are three commutative actions on $Q_2$ up to equivalence, for $n = 1$ there are two commutative actions on $Q_1$ up to equivalence.

math.AG↗
arXiv2019

Flexibility of normal affine horospherical varieties

We investigate flexibility of affine varieties with an action of a linear algebraic group. Flexibility of a smooth affine variety with only constant invertible functions and a locally transitive action of a reductive group is proved. Also we show that a normal affine complexity-zero horospherical variety is flexible.

math.AG↗
arXiv2019

On rigidity of trinomial hypersurfaces and factorial trinomial varieties

Trinomial varieties are affine varieties given by some special system of equations consisting of polynomials with three terms. Such varieties are total coordinate spaces of normal rational varieties with torus action of complexity one. For an affine variety X we consider the subgroup SAut(X) of the automorphism group generated by all algebraic subgroups isomorphic to the additive group of the ground field. An affine variety X is rigid if SAut(X) is trivial. In opposite an affine variety is flexible if SAut(X) acts transitively on the regular locus. Arzhantsev proved a criterium for a factorial trinomial hypersurface to be rigid. We give two generalizations of Arzhantsev's result: a criterium for an arbitrary trinomial hypersurface to be rigid and a criterium for a factorial trinomial variety to be rigid. Also a sufficient condition for a trinomial hypersurface to be flexible is obtained.

math.AG↗