Related papers: Limit points of one-parameter subgroups for additi…
An induced additive action on a projective variety $X \subseteq \mathbb{P}^n$ is a regular action of the group $\mathbb{G}_a^m$ on $X$ with an open orbit, which can be extended to a regular action on the ambient projective space…
Let $\mathbb{K}$ be an algebraically closed field of characteristic zero and $\mathbb{G}_a$ be the additive group of $\mathbb{K}$. We say that an irreducible algebraic variety $X$ of dimension $n$ over the field $\mathbb{K}$ admits an…
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…
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…
An additive action on an algebraic variety is an effective action of the vector group with an open orbit. We describe projective surfaces with du Val singularities that admit an additive action with a finite number of orbits. In particular,…
By an additive action on an algebraic variety $X$ we mean a regular effective action $\mathbb{G}_a^n\times X\to X$ with an open orbit of the commutative unipotent group $\mathbb{G}_a^n$. In this paper, we give a classification of additive…
By an additive action on a hypersurface H in the projective space P^{n+1} we mean an effective action of a commutative unipotent group on P^{n+1} which leaves H invariant and acts on H with an open orbit. Brendan Hassett and Yuri Tschinkel…
For a projective variety $X$ in $\mathbb{P}^{m}$ of dimension $n$, an additive action on $X$ is an effective action of $\mathbb{G}_{a}^{n}$ on $\mathbb{P}^{m}$ such that $X$ is $\mathbb{G}_{a}^{n}$-invariant and the induced action on $X$…
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…
By an additive structure on a hypersurface S in projective space we mean an effective action of commutative unipotent group on projective space which leaves S invariant and acts on S with an open orbit. It is known that these structures…
By an additive action on an algebraic variety $X$ we mean a regular effective action $\mathbb{G}_a^n\times X\to X$ with an open orbit of the commutative unipotent group $\mathbb{G}_a^n$. In this paper, we give a uniqueness criterion for…
We consider endomorphism actions of arbitrary discrete semigroups on a connected metrizable topological group G. We give necessary and sufficient conditions for expansiveness of such actions when G is a Lie group or a compact…
We show that an effective action of the one-dimensional torus $\mathbb{G}_m$ on a normal affine algebraic variety $X$ can be extended to an effective action of a semi-direct product $\mathbb{G}_m\rightthreetimes\mathbb{G}_a$ with the same…
Let G be an affine algebraic group and let X be an affine algebraic variety. An action $G\times X \to X$ is called observable if for any G-invariant, proper, closed subset Y of X there is a nonzero invariant $f\in K[X]^G$ such that f(Y) =0.…
In this paper we survey some recent results on actions of finite groups on topological manifolds. Given an action of a finite group $G$ on a manifold $X$, these results provide information on the restriction of the action to a subgroup of…
By an additive action on an algebraic variety $X$ of dimension $n$ we mean a regular action $\mathbb{G}_a^n \times X \to X$ with an open orbit of the commutative unipotent group $\mathbb{G}_a^n$. We prove that if a complete toric variety…
An additive action on an irreducible algebraic variety $X$ is an effective action $\mathbb{G}_a^n\times X\to X$ with an open orbit of the vector group $\mathbb{G}_a^n$. Any two additive actions on $X$ are conjugate by a birational…
Let $X$ be an irreducible affine algebraic variety that is spherical with respect to an action of a connected reductive group $G$. In this paper we provide sufficient conditions, formulated in terms of weight combinatorics, for the…
Given a connected reductive algebraic group $G$ and a Borel subgroup $B \subseteq G$, we study $B$-normalized one-parameter additive group actions on affine spherical $G$-varieties. We establish basic properties of such actions and their…
In this paper we study the orbit closure problem for a reductive group $G\subseteq GL(X)$ acting on a finite dimensional vector space $V$ over $\C$. We assume that the center of $GL(X)$ lies within $G$ and acts on $V$ through a fixed…