Related papers: Image closure of symmetric wide-matrix varieties

We prove that the universal cover of a normal complex algebraic variety admitting a faithful complex representation of its fundamental group is an analytic Zariski open subset of a holomorphically convex complex space. This is a non-proper…

Given an integer $N \geq 3$, we prove that for any ring $R$ and any finite locally free $R$-group scheme $G$ which is fppf-locally (over $R$) isomorphic the $N$-torsion subscheme of some elliptic curve $E/R$, there is a smooth affine curve…

The article demonstrates the procedure how to compute the Zariski closure of an orbit by an algebraic action of finitely generated group on the affine plane. First half of the algorithm is about deciding whether given finitely generated…

Let $Z$ be a closed subscheme of a smooth complex projective variety $Y\subseteq \Ps^N$, with $\dim\,Y=2r+1\geq 3$. We describe the intermediate N\'eron-Severi group (i.e. the image of the cycle map $A_r(X)\to H_{2r}(X;\mathbb{Z})$) of a…

We prove a "Generic Equivalence Theorem which says that two affine morphisms $p: S \to Y$ and $q: T \to Y$ of varieties with isomorphic (closed) fibers become isomorphic under a dominant etale base change $\phi: U \to Y$. A special case is…

Let $Z$ be an affine algebraic variety and $X$ be a smooth flexible variety. We develop some criteria under which $Z$ admits a closed embedding into $X$. In particular, we show that if $X$ is isomorphic (as an algebraic variety) to a…

Let $\Gamma$ be a discrete subgroup of a simply connected, solvable Lie group~$G$, such that $\Ad_G\Gamma$ has the same Zariski closure as $\Ad G$. If $\alpha \colon \Gamma \to \GL_n(\real)$ is any finite-dimensional representation…

We exhibit an algorithm to compute the strongest polynomial (or algebraic) invariants that hold at each location of a given affine program (i.e., a program having only non-deterministic (as opposed to conditional) branching and all of whose…

We study two families of $g$-dimensional abelian varieties, induced by distinct rational maps defined on a common variety $\overline{\mathcal A}$ and mapping to two bases $\overline{S}_1$ and $\overline{S}_2$. Two non-torsion sections…

Let R be an associative ring with identity. We study an elementary generalization of the classical Zariski topology, applied to the set of isomorphism classes of simple left R-modules (or, more generally, simple objects in a complete…

Let $X:=\mathrm{Spec}(R)$ be an affine Noetherian scheme, and $\mathcal{M} \subset \mathcal{N}$ be a pair of finitely generated $R$-modules. Denote their Rees algebras by $\mathcal{R}(\mathcal{M})$ and $\mathcal{R}(\mathcal{N})$. Let…

We work out the notion of mirror symmetry for abelian varieties and study its properties. Our construction are based on the correspondence between two $Q$--algebraic groups. One is the Hodge (or special Mumford--Tate) group. The second…

In the preprint arXiv:2511.07900 we proved that there exists a localizing ring $A_M$ for $A$ an associative ring with unit, and $M=\oplus_{i=1}^rM_i$ a direct sum of $r\geq 1$ simple right $A$-modules. For a homomorphism of associative…

Let $\mathcal K$ be a complete quasivariety of completely regular universal topological algebras of continuous signature $\mathcal E$ (which means that $\mathcal K$ is closed under taking subalgebras, Cartesian products, and includes all…

In this paper we give different compactifications for the domain and the codomain of an affine rational map $f$ which parametrizes a hypersurface. We show that the closure of the image of this map (with possibly some other extra…

Following Serre's initial work, a number of authors have considered twists of quadratic forms on a scheme Y by torsors of a finite group G, together with formulas for the Hasse-Witt invariants of the twisted form. In this paper we take the…

We prove a decomposition theorem for the equivariant K-theory of actions of affine group schemes G of finite type over a field on regular separated noetherian algebraic spaces, under the hypothesis that the actions have finite geometric…

Andr\'{e} and Maulik--Poonen proved that for any smooth proper family $X\to B$ of varieties over an algebraically closed field of characteristic $0$, there is a closed fiber whose N\'{e}ron-Severi group has the same rank as that of the…

A typical large complex-structure limit for mirror symmetry consists of toric varieties glued to each other along their toric boundaries. Here we construct the mirror large volume limit space as a Weinstein symplectic manifold. We prove…

The main result asserts: Let $G$ be a reductive, affine algebraic group and let $(\rho ,V)$ be a regular representation of $G$. Let $X$ be an irreducible $\mathbb{C}^{ \times } G$ invariant Zariski closed subset such that $G$ has a closed…