Related papers: Finite separating sets and quasi-affine quotients

We study Hilbert's fourteenth problem from a geometric point of view. Nagata's celebrated counterexample demonstrates that for an arbitrary group action on a variety the ring of invariant functions need not be isomorphic to the ring of…

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…

This paper studies separating invariants of finite groups acting on affine varieties through automorphisms. Several results, proved by Serre, Dufresne, Kac-Watanabe and Gordeev, and Jeffries and Dufresne exist that relate properties of the…

This paper gives the first explicit example of a finite separating set in an invariant ring which is not finitely generated, namely, for Daigle and Freudenburg's 5-dimensional counterexample to Hilbert's Fourteenth Problem.

If V is a representation of a linear algebraic group G, a set S of G-invariant regular functions on V is called separating if the following holds: If two elements v,v' from V can be separated by an invariant function, then there is an f…

We say that an algebraic group $G$ over a field is anti-affine if every regular function on $G$ is constant. We obtain a classification of these groups, with applications to the structure of algebraic groups in positive characteristics, and…

Hilbert's 14th problem studies the finite generation property of the intersection of an integral algebra of finite type with a subfield of the field of fractions of the algebra. It has a negative answer due to the counterexample of Nagata.…

We formulate a notion of "geometric reductivity" in an abstract categorical setting which we refer to as adequacy. The main theorem states that the adequacy condition implies that the ring of invariants is finitely generated. This result…

A quandle will be called quasi-affine, if it embeds into an affine quandle. Our main result is a characterization of quasi-affine quandles, by group-theoretic properties of their displacement group, by a universal algebraic condition coming…

This is a survey on the finite basis problem for varieties of algebraic systems. Our exposition is in two directions: (i) We give numerous examples of varieties which are not finitely based. (ii) We give examples of important varieties with…

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…

In the case of finite groups, a separating algebra is a subalgebra of the ring of invariants which separates the orbits. Although separating algebras are often better behaved than the ring of invariants, we show that many of the criteria…

We interpret a counterexample to Hilbert's 14th problem by S. Kuroda geometrically in two ways: As ring of regular functions on a smooth rational quasiprojective variety over any field K of characteristic 0, and, in the special case where K…

Given an action of an affine algebraic group with only trivial characters on a factorial variety, we ask for categorical quotients. We characterize existence in the category of algebraic varieties. Moreover, allowing constructible sets as…

Let G be an affine algebraic group acting on an affine variety X. We present an algorithm for computing generators of the invariant ring K[X]^G in the case where G is reductive. Furthermore, we address the case where G is connected and…

We prove that a variety of algebras whose finitely generated members are free must be definitionally equivalent to the variety of sets, the variety of pointed sets, a variety of vector spaces over a division ring, or a variety of affine…

The paper deals with a construction of a separating system of rational invariants for finite dimensional generic algebras. In the process of dealing an approach to a rough classification of finite dimensional algebras is offered by…

The study of separating invariants is a recent trend in invariant theory. For a finite group acting linearly on a vector space, a separating set is a set of invariants whose elements separate the orbits of G. In some ways, separating sets…

A separating algebra is, roughly speaking, a subalgebra of the ring of invariants whose elements distinguish between any two orbits that can be distinguished using invariants. In this paper, we introduce a geometric notion of separating…

We introduce a new family of invariants of real algebraic sets defined in terms of the topology of their complexifications and compute some of these invariants for spheres. This allows us to completely classify topological isomorphism…