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Anton Trushin

Publications and source records attributed to Anton Trushin.

6 recordsLinked to original sources
arXiv2026

Contracted divisors and Degree-Two Maps

We consider polynomial maps of affine space over an algebraically closed field of characteristic zero. We prove that every irreducible component of the zero locus of the Jacobian determinant corresponds to either a contracted divisor or a branching divisor. We further consider polynomial maps of degree two without contracted divisors and show that the Jacobian determinant is irreducible, anti-invariant under the Galois involution, and coincides with the defining equation of the unique branching divisor.

math.AG↗
arXiv2024

Automorphisms of rigid hypersurfaces with separable variables

Consider a polynomial F such that each variable appears in exactly one monomial. The hypersurface defined by the polynomial F is called a hypersurface with separable variables. A variety is called rigid if there are no nontrivial actions of the additive group of the ground field on it. If a variety is rigid, then it is known that in the automorphism group there exists a unique maximal torus. We describe the automorphism group of a rigid hypersurface with separable variables, in particular we show that it is a finite extension of the maximal torus.

math.AG↗
arXiv2023

On the automorphism group of a toral variety

Let $\mathbb{K}$ be an algebraically closed field of characteristic zero. An affine algebraic variety $X$ over $\mathbb{K}$ is toral if it is isomorphic to a closed subvariety of a torus $(\mathbb{K}^*)^d$. We study the group $\mathrm{Aut}(X)$ of regular automorpshims of a toral variety $X$. We prove that if $T$ is a maximal torus in $\mathrm{Aut}(X)$, then $X$ is a direct product $Y\times T$, where $Y$ is a toral variety with a trivial maximal torus in the automorphism group. We show that knowing $\mathrm{Aut}(Y)$, one can compute $\mathrm{Aut}(X)$. In the case when the rank of the group $\mathbb{K}[Y]^*/\mathbb{K}^*$ is $\dim Y + 1$, the group $\mathrm{Aut}(Y)$ can be described explicitly.

math.AG↗
arXiv2021

Gradings allowing wild automorphisms

In 2004 Shestakov and Umirbaev proved that the Nagata automorphism of the polynomial algebra in three variables is wild. We fix a Z-grading on this algebra and consider graded-wild automorphisms, i.e. such automorphisms that can not be decomposed onto elementary automorphisms respecting the grading. We describe all gradings allowing graded-wild automorphisms.

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↗