Related papers: Projective completions of affine varieties via deg…
We study projective completions of affine algebraic varieties which are given by filtrations, or equivalently, 'degree like functions' on their rings of regular functions. For a quasifinite polynomial map P (i.e. with all fibers finite) of…
Differential algebraic geometry seeks to extend the results of its algebraic counterpart to objects defined by differential equations. Many notions, such as that of a projective algebraic variety, have close differential analogues but their…
This paper deals with properties of filtrations on vector spaces indexed by partially ordered finitely generated abelian groups, which we call multifiltrations. We discuss the usual properties of filtrations, like exhaustivity and…
This paper is devoted to the investigation of selected situations when the computation of projective (and other) equivalences of algebraic varieties can be efficiently solved with the help of finding projective equivalences of finite sets…
This article investigates the splitting problem for finitely generated projective modules $P$ over affine algebras over algebraically closed fields and their polynomial extensions. We then address an open question due to M. Roitman on monic…
In this note we survey recent results on automorphisms of affine algebraic varieties, infinitely transitive group actions and flexibility. We present related constructions and examples, and discuss geometric applications and open problems.
In this paper, we prove that any two birational projective varieties with finite quotient singularities can be realized as two geometric GIT quotients of a non-singular projective variety by a reductive algebraic group. Then, by applying…
We survey recent results on multiple transitivity of automorphism groups of affine algebraic varieties. We consider the property of infinite transitivity of the special automorphism group, which is equivalent to flexibility of the…
An enumerative problem on a variety $V$ is usually solved by reduction to intersection theory in the cohomology of a compactification of $V$. However, if the problem is invariant under a "nice" group action on $V$ (so that $V$ is…
We show that the presentation of affine $\mathbb{T}$-varieties of complexity one in terms of polyhedral divisors holds over an arbitrary field. We also describe a class of multigraded algebras over Dedekind domains. We study how the algebra…
We provide a complete description of normal affine varieties with effective algebraic torus action in terms of what we call proper polyhedral divisors on semiprojective varieties. Our theory extends classical cone constructions of…
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…
We show that the height of a toric variety with respect to a toric metrized line bundle can be expressed as the integral over a polytope of a certain adelic family of concave functions. To state and prove this result, we study the Arakelov…
We describe explicitly the normalization of affine varieties with an algebraic torus action of complexity one in terms of polyhedral divisors. We also provide a description of homogeneous integrally closed ideals of affine T-varieties of…
We consider fibrations by affine lines on smooth affine surfaces obtained as complements of smooth rational curves $B$ in smooth projective surfaces $X$ defined over an algebraically closed field of characteristic zero. We observe that…
Given a finite field $\mathbb{F}_{q}$, we study the distribution of the number of $\mathbb{F}_{q}$-points on (possibly singular) affine curves given by the polynomial equations of the form $C_{f} : y^{m} = f(x)$, where $f$ is randomly…
This thesis is devoted to the study of geometric properties of affine algebraic varieties endowed with an action of an algebraic torus. It comes from three preprints which correspond to the indicated points (1), (2), (3). Let $X$ be an…
It was shown in [S. Kaliman, M. Zaidenberg, Gromov ellipticity of cones over projective manifolds, Math. Res. Lett. (to appear), arXiv:2303.02036 (2023)] that the affine cones over flag manifolds and rational smooth projective surfaces are…
In this paper we introduce a new approach and obtain new results for the problem of studying polynomial images of affine subspaces of finite fields. We improve and generalise several previous known results, and also extend the range of such…
In this article we give a general approach to the following analogue of Shafarevich's conjecture for some polarized algebraic varieties; suppose that we fix a type of an algebraic variety and look at families of such type of varieties over…