Affine Algebraic Variety Research Articles

On rigidity of factorial trinomial hypersurfaces

An affine algebraic variety [Formula: see text] is rigid if the algebra of regular functions [Formula: see text] admits no nonzero locally nilpotent derivation. We prove that a factorial trinomial hypersurface is rigid if and only if every exponent in the trinomial is at least [Formula: see text].

Algebraic embeddings of smooth almost complex structures

The goal of this work is to prove an embedding theorem for compact almost complex manifolds into complex algebraic varieties. It is shown that every almost complex structure can be realized by the transverse structure to an algebraic distribution on an affine algebraic variety, namely an algebraic subbundle of the tangent bundle. In fact, there even exist universal embedding spaces for this problem, and their dimensions grow quadratically with respect to the dimension of the almost complex manifold to embed. We give precise variation formulas for the induced almost complex structures and study the related versality conditions. At the end, we discuss the original question raised by F. Bogomolov: can one embed every compact complex manifold as a \mathcal C^\infty smooth subvariety that is transverse to an algebraic foliation on a complex projective algebraic variety?

Rationally integrable vector fields and rational additive group actions

We characterize rational actions of the additive group on algebraic varieties defined over a field of characteristic zero in terms of a suitable integrability property of their associated velocity vector fields. This extends the classical correspondence between regular actions of the additive group on affine algebraic varieties and the so-called locally nilpotent derivations of their coordinate rings. Our results lead in particular to a complete characterization of regular additive group actions on semi-affine varieties in terms of their associated vector fields. Among other applications, we review properties of the rational counterpart of the Makar–Limanov invariant for affine varieties and describe the structure of rational homogeneous additive group actions on toric varieties.

Local topological algebraicity of analytic function germs

Mostowski showed that every (real or complex) germ of an analytic set is homeomorphic to the germ of an algebraic set. In this paper we show that every (real or complex) analytic function germ, defined on a possibly singular analytic space, is topologically equivalent to a polynomial function germ defined on an affine algebraic variety.

Some results of algebraic geometry over Henselian rank one valued fields

We develop geometry of affine algebraic varieties in K^{n} over Henselian rank one valued fields K of equicharacteristic zero. Several results are provided including: the projection K^{n} times {mathbb {P}}^{m}(K) rightarrow K^{n} and blowups of the K-rational points of smooth K-varieties are definably closed maps; a descent property for blowups; curve selection for definable sets; a general version of the Łojasiewicz inequality for continuous definable functions on subsets locally closed in the K-topology; and extending continuous hereditarily rational functions, established for the real and p-adic varieties in our joint paper with J. Kollár. The descent property enables application of resolution of singularities and transformation to a normal crossing by blowing up in much the same way as over the locally compact ground field. Our approach relies on quantifier elimination due to Pas and a concept of fiber shrinking for definable sets, which is a relaxed version of curve selection. The last three sections are devoted to the theory of regulous functions and sets over such valued fields. Regulous geometry over the real ground field {mathbb {R}} was developed by Fichou–Huisman–Mangolte–Monnier. The main results here are regulous versions of Nullstellensatz and Cartan’s theorems A and B.

A Note on the Canonical Divisor of Quasi-homogeneous Affine Algebraic Varieties

We give a necessary and sufficient condition for the canonical divisor to vanish on a quasi-homogeneous affine algebraic variety.

ON ALGEBRAIC VOLUME DENSITY PROPERTY

A smooth affine algebraic variety X equipped with an algebraic volume form ω has the algebraic volume density property (AVDP) if the Lie algebra generated by complete algebraic vector fields of ω-divergence zero coincides with the space of all algebraic vector fields of ω-divergence zero. We develop an effective criterion of verifying whether a given X has AVDP. As an application of this method we establish AVDP for any homogeneous space X = G/R that admits a G-invariant algebraic volume form where G is a linear algebraic group and R is a closed reductive subgroup of G.

The relative Brauer group and generalized cyclic crossed products for a ramified covering

Let T/A be an integral extension of noetherian integrally closed integral domains whose quotient field extension is a finite cyclic Galois extension. Let S/R be a localization of this extension which is unramified. Using a generalized cyclic crossed product construction it is shown that certain reflexive fractional ideals of T with trivial norm give rise to Azumaya R-algebras that are split by S. Sufficient conditions on T/A are derived under which this construction can be reversed and the relative Brauer group of S/R is shown to fit into the exact sequence of Galois cohomology associated to the ramified covering T/A. Many examples of affine algebraic varieties are exhibited for which all of the computations are carried out.

The relation type of affine algebras and algebraic varieties

We introduce the notion of relation type of an affine algebra and prove that it is well defined by using the Jacobi–Zariski exact sequence of André–Quillen homology. In particular, the relation type is an invariant of an affine algebraic variety. Also as a consequence of the invariance, we show that in order to calculate the relation type of an ideal in a polynomial ring one can reduce the problem to trinomial ideals. When the relation type is at least two, the extreme equidimensional components play no role. This leads to the non-existence of affine algebras of embedding dimension three and relation type two.

Linearly reductive and unipotent actions of affine groups

We present a generalized version of classical geometric invariant theory à la Mumford where we consider an affine algebraic group G acting on a specific affine algebraic variety X. We define the notions of linearly reductive and of unipotent action in terms of the G fixed point functor in the category of (G, 𝕜[X])-modules. In the case that X = {⋆} we recuperate the concept of linearly reductive and of unipotent group. We prove in our "relative" context some of the classical results of GIT such as: existence of quotients, finite generation of invariants, Kostant–Rosenlicht's theorem and Matsushima's criterion. We also present a partial description of the geometry of such linearly reductive actions.

On the character varieties of finitely generated groups

We establish three results dealing with the character varieties of finitely generated groups. The first two are concerned with the behavior of $\dim X_n(\Gamma)$ as a function of $n$, and the third addresses the problem of realizing a $\mathbb{Q}$-defined complex affine algebraic variety as a character variety.

Numerical Decomposition of Affine Algebraic Varieties

Abstract An irreducible algebraic decomposition ∪ i = 0 d X i = ∪ i = 0 d ( ∪ j = 1 d i X ij ) $ \cup _{i = 0}^d X_i = \cup _{i = 0}^d ({\cup _{j = 1}^{d_i } X_{ij} } )$ of an affine algebraic variety X can be represented as a union of finite disjoint sets ∪ i = 0 d W i = ∪ i = 0 d ( ∪ j = 1 d i W ij ) $\cup _{i = 0}^d W_i = \cup _{i = 0}^d ({\cup _{j = 1}^{d_i } W_{ij} } )$ called numerical irreducible decomposition (cf. [14],[15],[18],[19],[20],[22],[23],[24]). The Wi correspond to the pure i-dimensional components Xi , and the Wij present the i-dimensional irreducible components Xij . The numerical irreducible decomposition is implemented in Bertini (cf. [3]). The algorithms use homotopy continuation methods. We modify this concept using partially Gröbner bases, triangular sets, local dimension, and the so-called zero sum relation. We present in this paper the corresponding algorithms and their implementations in Singular (cf. [8]). We give some examples and timings, which show that the modified algorithms are more efficient if the number of variables is not too large. For a large number of variables Bertini is more efficient*.

Comparison of probabilistic algorithms for analyzing the components of an affine algebraic variety

Systems of polynomial equations arise throughout mathematics, engineering, and the sciences. It is therefore a fundamental problem both in mathematics and in application areas to find the solution ...

On some varieties associated with trees

This article considers some affine algebraic varieties attached to finite trees and closely related to cluster algebras. Their definition involves a canonical coloring of vertices of trees into three colors. These varieties are proved to be smooth and to admit sometimes free actions of algebraic tori. Some results are obtained on their number of points over finite fields and on their cohomology.

The Cramer varieties [formula omitted

In this paper we study quasi-homogeneous affine algebraic varieties, that is, varieties obtained as closures of orbits of suitable group representations. We also discuss one interesting case that has links with the Orthogonal Grassmannian OGr(5,10). The main aim is to write the tangent bundle and the canonical class of quasi-homogeneous affine algebraic varieties in terms of group representations.

Comparison of probabilistic algorithms for analyzing the components of an affine algebraic variety

Systems of polynomial equations arise throughout mathematics, engineering, and the sciences. It is therefore a fundamental problem both in mathematics and in application areas to find the solution sets of polynomial systems. The focus of this paper is to compare two fundamentally different approaches to computing and representing the solutions of polynomial systems: numerical homotopy continuation and symbolic computation. Several illustrative examples are considered, using the software packages Bertini and Singular.

A-hypergeometric functions in transcendental questions of algebraic geometry

We generalize known constructions of A-hypergeometric functions. In particular, we show that the periods of middle dimension on affine or projective complex algebraic varieties are A-hypergeometric functions of the coefficients of polynomial equations of these varieties.

Projective completions of affine varieties via degree-like functions

We introduce and study a class of projective completions of affine algebraic varieties which generalize the construction of toric varieties from convex rational polytopes.

On restriction of roots on affine $$T$$ T -varieties

Let \(X\) be a normal affine algebraic variety with a regular action of a torus \(\mathbb {T}\) and \(T\subset \mathbb {T}\) be a subtorus. We prove that each root of \(X\) with respect to \(T\) can be obtained by restriction of some root of \(X\) with respect to \(\mathbb {T}\). This allows to give an elementary proof of the description of roots of the affine Cremona group. Several results on restriction of roots in the case of a subtorus action on an affine toric variety are obtained.

On Additive invariants of actions of additive and multiplicative groups

AbstractLet X be an algebraic variety with an action of either the additive or multiplicative group. We calculate the additive invariants of X in terms of the additive invariants of the fixed point set, using a formula of Białynicki-Birula. The method is also generalized to calculate certain additive invariants for Chow varieties. As applications, we obtain results on the Hodge polynomial of Chow varieties in characteristic zero and the number of points for Chow varieties over finite fields. As applications, we obtain the l-adic Euler-Poincaré characteristic for the Chow varieties of certain projective varieties over a field of arbitrary characteristic. Moreover, we show that the virtual Hodge (p,0) and (0,q)-numbers of the Chow varieties and affine algebraic group varieties are zero for all p,q positive.