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Ivan V. Arzhantsev

Publications and source records attributed to Ivan V. Arzhantsev.

17 recordsLinked to original sources
arXiv2012

Hassett-Tschinkel correspondence: Modality and projective hypersurfaces

B. Hassett and Yu. Tschinkel (1999) introduced a remarkable correspondence between generically transitive actions of a commutative unipotent algebraic group G and finite-dimensional local algebras. In this paper we develop Hassett-Tschinkel correspondence and calculate modality of generically transitive G-actions on projective spaces, classify actions of modality one, and characterize generically transitive G-actions on projective hypersurfaces of given degree. In particular, actions on degenerate projective quadrics are studied.

math.AG↗
arXiv2012

Torsors over Luna strata

Let G be a reductive group and X be a Luna stratum on the quotient space V//G of a rational G-module V. We consider torsors over X with both non-commutative and commutative structure groups. It allows us to compute the divisor class group and the Cox ring of a Luna stratum under mild assumptions. This techniques gives a simple cause why many Luna strata are singular along their boundary.

math.AG↗
arXiv2010

Homogeneous toric varieties

A description of transitive actions of a semisimple algebraic group G on toric varieties is obtained. Every toric variety admitting such an action lies between a product of punctured affine spaces and a product of projective spaces. The result is based on the Cox realization of a toric variety as a quotient space of an open subset of a vector space V by a quasitorus action and on investigation of the G-module structure of V.

math.AG↗
arXiv2009

Saturated subfields and invariants of finite groups

Every subfield $\kk(ϕ)$ of the field of rational functions $\kk(x_1,...,x_n)$ is contained in a unique maximal subfield of the form $\kk(ψ)$. The element $ψ$ is called generative for the element $ϕ$. A subfield of $\kk(x_1,...,x_n)$ is called saturated if it contains a generative element of each its element. We study the saturation property for subfields of invariants $\kk(x_1,...,x_n)^G$, where $G$ is a finite group of automorphisms of the field $\kk(x_1,...,x_n)$.

math.RA↗
arXiv2008

Cox rings, semigroups and automorphisms of affine algebraic varieties

We study the Cox realization of an affine variety, i.e., a canonical representation of a normal affine variety with finitely generated divisor class group as a quotient of a factorially graded affine variety by an action of the Neron-Severi quasitorus. The realization is described explicitly for the quotient space of a linear action of a finite group. A universal property of this realization is proved, and some results on the divisor theory of an abstract semigroup emerging in this context are given. We show that each automorphism of an affine variety can be lifted to an automorphism of the Cox ring normalizing the grading. It follows that the automorphism group of a non-degenerate affine toric variety of dimension $\geq 2$ has infinite dimension. We obtain a wild automorphism of the three-dimensional quadratic cone that rises to Anick's automorphism of the polynomial algebra in four variables.

math.AG↗
arXiv2008

Geometric Invariant Theory via Cox Rings

We consider actions of reductive groups on a varieties with finitely generated Cox ring, e.g., the classical case of a diagonal action on a product of projective spaces. Given such an action, we construct via combinatorial data in the Cox ring all maximal open subsets such that the quotient is quasiprojective or embeddable into a toric variety. As applications, we obtain an explicit description of the chamber structure of the linearized ample cone and several Gelfand-MacPherson type correspondences relating quotients of reductive groups to quotients of torus actions. Moreover, our approach provides information on the geometry of many of the resulting quotient spaces.

math.AG↗
arXiv2008

On factoriality of Cox rings

Generalized Cox's construction associates with an algebraic variety a remarkable invariant -- its total coordinate ring, or Cox ring. In this note we give a new proof of factoriality of the Cox ring when the divisor class group of the variety is finitely generated and free. The proof is based on a notion of graded factoriality. We show that if the divisor class group has torsion, then the Cox ring is again factorially graded, but factoriality may be lost.

math.AG↗
arXiv2008

Projective embeddings of homogeneous spaces with small boundary

We study open equivariant projective embeddings of homogeneous spaces such that the complement of the open orbit does not contain divisors. Criterions of existence of such an embedding are considered and finiteness of isomorphism classes of embeddings for a given homogeneous space is proved. Any embedding with small boundary is realized as a GIT-quotient associated with a linearization of the trivial line bundle on the space of the canonical embedding. The generalized Cox's construction and the theory of bunched rings allow us to describe basic geometric properties of embeddings with small boundary in combinatorial terms.

math.AG↗
arXiv2007

Closed and Irreducible Polynomials in Several Variables

New and old results on closed polynomials, i.e., such polynomials f in K[x_1,...,x_n] that the subalgebra K[f] is integrally closed in K[x_1,...,x_n], are collected. Using some properties of closed polynomials we prove the following factorization theorem: Let f be an element of K[x_1,...,x_n], where K is an algebraically closed field. Then for all but finite number of a's the polynomial f+a can be decomposed into a product of irreducible polynomials of the same degree (not depending on a) such that their defferences are constants. An algorithm for finding of a generative polynomial of a given polynomial f, which is a closed polynomial h with f=F(h) for some F(t) in K[t], is given. Some types of saturated subalgebras A in K[x_1,...,x_n] are considered, i.e., such that for any f in A a generative polynomial of f is contained in A.

math.AC↗
arXiv2006

On the multiplication map of a multigraded algebra

Given a multigraded algebra $A$, it is a natural question whether or not for two homogeneous components $A_u$ and $A_v$, the product $A_{nu}A_{nv}$ is the whole component $A_{nu+nv}$ for $n$ big enough. We give combinatorial and geometric answers to this question.

math.AC↗
arXiv2005

On embeddings of homogeneous spaces with small boundary

We study equivariant embeddings with small boundary of a given homogeneous space $G/H$, where $G$ is a connected, linear algebraic group with trivial Picard group and only trivial characters, and $H \subset G$ is an extension of a connected Grosshans subgroup by a torus. Under certain maximality conditions, like completeness, we obtain finiteness of the number of isomorphism classes of such embeddings, and we provide a combinatorial description the embbeddings and their morphisms. The latter allows a systematic treatment of examples and basic statements on the geometry of the equivariant embeddings of a given homogeneous space $G/H$.

math.AG↗
arXiv2005

Invariant Ideals and Matsushima's Criterion

Let G be a reductive algebraic group and H a closed subgroup of G. Explicit constructions of G-invariant ideals in the algebra K[G/H] are given. This allows to obtain an elementary proof of Matsushima's criterion: a homogeneous space G/H is an affine variety if and only if H is reductive.

math.AG↗
arXiv2005

Affine embeddings of homogeneous spaces

Let G be a reductive algebraic group and H a closed subgroup of G. An affine embedding of the homogeneous space G/H is an affine G-variety with an open G-orbit isomorphic to G/H. We start with some basic properties of affine embeddings and consider the cases, where the theory is well-developed: toric varieties, normal SL(2)-embeddings, S-varieties, and algebraic monoids. We discuss connections between the theory of affine embeddings and Hilbert's 14th Problem via a theorem of Grosshans. We characterize affine homogeneous spaces G/H such that any affine embedding of G/H contains a finite number of G-orbits. The maximal value of modality over all affine embeddings of a given affine homogeneous space is computed and the group of equivariant automorphisms of an embedding is studied. As applications of the theory, we describe invariant algebras on homogeneous spaces of a compact Lie group and G-algebras with finitely generated invariant subalgebras.

math.AG↗
arXiv2004

On the canonical embeddings of certain homogeneous spaces

We study equivariant affine embeddings of homogeneous spaces and their equivariant automorphisms. An example of a quasiaffine, but not affine, homogeneous space with finitely many equivariant automorphisms is presented. We prove the solvability of any connected group of equivariant automorphisms for an affine embedding with a fixed point and finitely many orbits. This is applied to studying the orbital decomposition for algebraic monoids and canonical embeddings of quasiaffine homogeneous spaces, i.e., those affine embeddings associated with the coordinate algebras of homogeneous spaces, provided the latter algebras are finitely generated. We pay special attention to the canonical embeddings of quotient spaces of reductive groups modulo the unipotent radicals of parabolic subgroups. For these varieties, we describe the orbital decomposition, compute the modality of the group action, and find out which of them are smooth. We also describe minimal ambient modules for these canonical embeddings provided that the acting group is simply connected.

math.AG↗
arXiv2004

Classification of affine homogeneous spaces of complexity one

The complexity of an action of a reductive algebraic group G on an algebraic variety X is the codimension of a generic B-orbit in X, where B is a Borel subgroup of G. We classify affine homogeneous spaces G/H of complexity one. These results are the natural continuation of the classification of spherical affine homogeneous spaces, i.e., spaces of complexity zero.

math.AG↗
arXiv2004

On affinely closed homogeneous spaces

Affinely closed homogeneous spaces G/H, i.e., affine homogeneous spaces that admit only the trivial affine embedding, are characterized for any affine algebraic group G. As a corollary, a description of affine G-algebras with finitely generated invariant subalgebras is obtained.

math.AG↗