Related papers: Group Theoretical Characterizations of Rationality

We study the decomposition of a generic element $g \in G$ of a connected reductive complex algebraic group $G$ in the form $g = N(g) B(g) \bar{u} N(g)^{-1}$ where $N: G \dashrightarrow \mathcal{N}_-$ and $B : G \dashrightarrow…

We prove the following theorem: Fibered Power Theorem: Let $X\rar B$ be a smooth family of positive dimensional varieties of general type, with $B$ irreducible. Then there exists an integer $n>0$, a positive dimensional variety of general…

An irreducible algebraic variety $X$ is rigid if it admits no nontrivial action of the additive group of the ground field. We prove that the automorphism group $\text{Aut}(X)$ of a rigid affine variety contains a unique maximal torus…

We classify the maximal algebraic subgroups of Bir(CxPP^1), when C is a smooth projective curve of positive genus.

We explore to what extent the underlying variety of a connected algebraic group or the underlying manifold of a real Lie group determines its group structure.

As in our previous work [1] we address the problem to determine the splitting of the normal bundle of rational curves. With apolarity theory we are able to characterize some particular subvarieties in some Hilbert scheme of rational curves,…

Let G be an algebraic reductive group over a an algebraically closed field of positive characteristic. Choose a parabolic subgroup $P$ in $G$ and denote by $U$ its unipotent radical. Let $X$ be a $G$-variety. The purpose of this paper is to…

Ramanujam's theorem states that any connected finite-dimensional subgroup of the automorphism group $\mathrm{Aut}(X)$ of an irreducible variety $X$ is an algebraic group, in a natural way. In this note, we discuss the notion of dimension…

We call a finite group irrational if none of its elements is conjugate to a distinct power of itself. We prove that those groups are solvable and describe certain classes of these groups, where the above property is only required for…

For a reductive Lie algbera over an algbraically closed field of charasteristic zero,we consider a borel subgroup $B$ of its adjoint group, a Cartan subalgebra contained inthe Lie algebra of $B$ and the closure $X$ of its orbit under $B$ in…

For sets $A, B\subset \mathbb N$, their sumset is $A + B := \{a+b: a\in A, b\in B\}$. If we cannot write a set $C$ as $C = A+B$ with $|A|, |B|\geq 2$, then we say that $C$ is $\textit{irreducible}$. The question of whether a given set $C$…

This paper has two parts. In the first one, we prove that an invariant dp-minimal type is either finitely satisfiable or definable. We also prove that a definable version of the (p,q)-theorem holds in dp-minimal theories of small or medium…

Looking at the finite \'etale congruence covers $X(p)$ of a complex algebraic variety $X$ equipped with a variation of integral polarized Hodge structures whose period map is quasi-finite, we show that both the minimal gonality among all…

The problem of the existence of non-pseudo-$\aleph_1$-compact $\mathbb R$-factorizable groups is studied. It is proved that any such group is submetrizable and has weight larger than $\omega_1$. Closely related results concerning the…

We prove that in a continuous $\aleph_0$-stable theory every type-definable group is definable. The two main ingredients in the proof are: \begin{enumerate} \item Results concerning Morley ranks (i.e., Cantor-Bendixson ranks) from…

Let G be a group acting geometrically on a CAT(0) cube complex X. We prove first that G is hyperbolic relative to the collection P of subgroups if and only if the simplicial boundary of X is the disjoint union of a nonempty discrete set,…

This paper concerns matrix decompositions in which the factors are restricted to lie in a closed subvariety of a matrix group. Such decompositions are of relevance in control theory: given a target matrix in the group, can it be decomposed…

Let $X$ be a projective variety with log terminal singularities and vanishing augmented irregularity. In this paper we prove that if $X$ admits a relatively minimal genus one fibration then it does contain a subvariety of codimension one…

Let X be a normal projective variety of dimension n > 2 admitting the action of the group G := Z^{n-1} such that every non-trivial element of G is of positive entropy. We show: `X is not rationally connected' ==> `X is G-equivariant…

We propose a simple criterion to know if an abelian variety $A$ defined over a finite field $\mathbb{F}_q$ is cyclic, i.e., it has a cyclic group of rational points; this criterion is based on the endomorphism ring End$_{\mathbb{F}_q}(A)$.…