Related papers: Forms of Affine Space

We establish an analytic Hasse principle for linear spaces of affine dimension m on a complete intersection over an algebraic field extension K of Q. The number of variables required to do this is no larger than what is known for the…

In this short note we prove a couple of facts about polynomial count varieties, answering natural questions that they raise. A polynomial count $X$ variety is essentially one for which its number of points over finite fields is given by a…

Let $A$ be a simple algebra over a field $F$. Under a mild cardinality assumption on $F$, we determine the greatest possible dimension for an $F$-affine subspace of $A$ that is included in the group of units $A^\times$, and we describe the…

We show that two varieties X and Y with isomorphic endomorphism semigroups are isomorphic up to field automorphism if one of them is affine and contains a copy of the affine line. A holomorphic version of this result is due to the first…

A class $\mathfrak{M}$ of coarse spaces is called a variety if $\mathfrak{M}$ is closed under formation of subspaces, coarse images and products. We classify the varieties of coarse spaces and, in particular, show that if a variety…

A ringed finite space is a ringed space whose underlying topological space is finite. The category of ringed finite spaces contains, fully faithfully, the category of finite topological spaces and the category of affine schemes. Any ringed…

Let X be a flexible variety of F be an isomorphism of closed one-dimensional subschemes of $X$. We develop criteria which guarantee that F extends to au automorphism of X.

Let $W$ denote the $n$-dimensional affine space over the finite field $\mathbb F_q$. We prove here a Bollob\'as-type upper bound in the case of the set of affine subspaces. We give a construction of a pair of families of affine subspaces,…

We give a necessary and sufficient condition for a one-dimensional regular and Hausdorff topological space definable in a definably complete uniformly locally o-minimal structure of the second kind having definable bounded multiplication…

All isomorphisms of Pl\"ucker spaces on affine spaces with dimensions $\geq 3$ arise from collineations of the underlying affine spaces.

Theorem (uniformization). Let X be a compact Kahler manifold of dimension n with large, residually finite and nonamenable fundamental group. Then its universal covering is a bounded domain in the n-dimensional affine space.

We study the class of 2-dimensional affine k-domains R satisfying ML(R) = k, where k is an arbitrary field of characteristic zero. In particular, we obtain the following result: Let R be a localization of a polynomial ring in finitely many…

The covering of the affine symmetry group, a semidirect product of translations and special linear transformations, in $D \geq 3$ dimensional spacetime is considered. Infinite dimensional spinorial representations on states and fields are…

We characterize finitary coarse spaces $X$ such that every permutation of $X$ is an asymorphism.

In this paper we show that an affine space is determined by the abstract group structure of its group of regular automorphisms in the category of connected affine varieties. To prove this we study commutative subgroups of the group of…

A slip on a paper concerning near-vector spaces is fixed. New characterization of near-vector spaces determined by finite fields is provided and the number (up to the isomorphism) of these spaces is exhibited.

We classify all cubic extensions of any field of arbitrary characteristic, up to isomorphism, via an explicit construction involving three fundamental types of cubic forms. We deduce a classification of any Galois cubic extension of a…

We exhibit a smooth complex rational affine surface with uncountably many nonisomorphic real forms.

We prove that several invariants of a possibly singular complex affine or projective variety of degree $d$ in the affine space $\mathbb{A}^{n}$, or $\mathbb{P}^n$, are bounded by a function of $d$ alone, provided $b_{1}=0$ for a resolution…

Trinomial varieties are affine varieties given by a system of equations consisting of polynomials with three terms. Such varieties are total coordinate spaces of normal varieties with torus action of complexity one. For an affine variety…