Homomorphisms of multiplicative groups of fields preserving algebraic dependence
We study homomorphisms of multiplicative groups of fields preserving algebraic dependence and show that such homomorphisms give rise to valuations.
math.AG↗
arXiv subjects
Marat Rovinsky
Publications and source records attributed to Marat Rovinsky.
Collineation group as a subgroup of the symmetric group
Let $Ψ$ be the projectivization (i.e., the set of one-dimensional vector subspaces) of a vector space of dimension $\ge 3$ over a field. Let $H$ be a closed (in the pointwise convergence topology) subgroup of the permutation group $\mathfrak{S}_Ψ$ of the set $Ψ$. Suppose that $H$ contains the projective group and an arbitrary self-bijection of $Ψ$ transforming a triple of collinear points to a non-collinear triple. It is well-known from \cite{KantorMcDonough} that if $Ψ$ is finite then $H$ contains the alternating subgroup $\mathfrak{A}_Ψ$ of $\mathfrak{S}_Ψ$. We show in Theorem \ref{density} below that $H=\mathfrak{S}_Ψ$, if $Ψ$ is infinite.
math.GR↗
Remarks on endomorphisms and rational points
Let X be a variety over a number field and let f: X --> X be an "interesting" rational self-map with a fixed point q. We make some general remarks concerning the possibility of using the behaviour of f near q to produce many rational points on X. As an application, we give a simplified proof of the potential density of rational points on the variety of lines of a cubic fourfold (originally obtained by Claire Voisin and the first author in 2007).
math.AG↗