Related papers: Cox rings, semigroups and automorphisms of affine …

Let $G$ be an affine algebraic group over an algebraically closed field $k$ of characteristic zero. In this paper, we consider finite $G$-equivariant morphisms $F:X\to Y$ of irreducible affine $G$-varieties. First we determine under which…

We study a toric degeneration of the Cox ring of the moduli of principal $SL_m(\mathbb{C})$ bundles on the projective line, with quasi parabolic data given by the the stabilizer of the highest weight vector in $\mathbb{C}^m$ and its dual…

Given a positive definite even lattice and a commutative ring, there is a standard construction of a lattice vertex algebra over the commutative ring, and it admits a natural grading by non-negative integers. We describe the groups of…

It is shown that any compact semistable quotient (in the sense of Heinzner and Snow) of a normal algebraic variety by a complex reductive Lie group $G$ is a good quotient. This reduces the investigation and classification of such…

Consider a smooth connected algebraic group $G$ acting on a normal projective variety $X$ with an open dense orbit. We show that Aut($X$) is a linear algebraic group if so is $G$; for an arbitrary $G$, the group of components of Aut($X$) is…

In this paper we consider some classical varieties of linear algebras over the field which has characteristic 0. For every considered variety we take a category of the finite generated free algebras of this variety. And for every this…

We study over a number field, the iterates of automorphisms of the affine space. More precisely, we are interested in the periodic and non-periodic points; for the former the questions are similar to the ones about torsion points on abelian…

Let $R$ be the coordinate ring of an affine toric variety. We show that the endomorphism ring $End_R(\mathbb A),$ where $\mathbb A$ is the (finite) direct sum of all (isomorphism classes of) conic $R$-modules, has finite global dimension.…

We construct affine charts of a smooth projective toric variety which contain its nonnegative points, and which admit a closed embedding into the total coordinate space of Cox's quotient construction. We show that such positive charts arise…

For a variety with a finitely generated total coordinate ring, we describe basic geometric properties in terms of certain combinatorial structures living in its divisor class group. For example, we describe the singularities, we calculate…

Given an affine algebraic variety X of dimension at least 2, we let SAut (X) denote the special automorphism group of X i.e., the subgroup of the full automorphism group Aut (X) generated by all one-parameter unipotent subgroups. We show…

We compute an explicit algebraic deformation quantization for an affine Poisson variety described by an ideal in a polynomial ring, and inheriting its Poisson structure from the ambient space.

For an affine algebraic variety $X$, we study the subgroup $\mathrm{Aut}_{\text{alg}}(X)$ of the group of regular automorphisms $\mathrm{Aut}(X)$ of $X$ generated by all the connected algebraic subgroups. We prove that…

The first part of this paper is a refinement of Winkelmann's work on invariant rings and quotients of algebraic groups actions on affine varieties, where we take a more geometric point of view. We show that the (algebraic) quotient…

We study affine toric varieties with an action of group $SL_n$ with a dense orbit. A characterisation in terms of $SL_n \times Q$-modules is given where $Q$ is a quasitorus. This characterisation is more explicitly expanded in case $n=3$.…

The object of study is the group of units O^\ast(X) in the coordinate ring of a normal affine variety X over an algebraically closed field k. Methods of Galois cohomology are applied to those varieties that can be presented as a finite…

K. Altmann and J. Hausen have shown that affine T-varieties can be described in terms of p-divisors. Given a p-divisor describing a T-variety X, we show how to construct new p-divisors describing X with respect to actions by larger tori.…

We explore orbits, rational invariant functions, and quotients of the natural actions of connected, not necessarily finite dimensional subgroups of the automorphism groups of irreducible algebraic varieties. The applications of the results…

We show that any compact quantum group having the same fusion rules as the ones of $SO(3)$ is the quantum automorphism group of a pair $(A, \varphi)$, where $A$ is a finite dimensional $C^*$-algebra endowed with a homogeneous faithful…

We prove that the canonical ring of a canonical variety in the sense of de Fernex and Hacon is finitely generated. We prove that canonical varieties are klt if and only if R(-K_X) is finitely generated. We introduce a notion of nefness for…