A Characteristic Property of Elliptic Pl\"ucker Transformations

TL;DR

This paper investigates the properties of elliptic Pl"ucker transformations in three-dimensional elliptic spaces, establishing conditions under which certain permutations of lines are characterized as Pl"ucker transformations.

Contribution

It proves that in classical elliptic 3-spaces, any related-line-preserving bijection is a Pl"ucker transformation, and under specific field conditions, injective related-line-preserving maps are also characterized.

Findings

Bijections preserving related lines are Pl"ucker transformations in classical elliptic 3-spaces.

Injective related-line-preserving maps are Pl"ucker transformations if the field admits only surjective monomorphisms.

The paper characterizes transformations based on the properties of the underlying field.

Abstract

We discuss elliptic Pl\"ucker transformations of three-dimensional elliptic spaces. These are permutations on the set of lines such that any two related (orthogonally intersecting or identical) lines go over to related lines in both directions. It will be shown that for "classical" elliptic 3-spaces a bijection of its lines is already a Pl\"ucker transformation, if related lines go over to related lines. Moreover, if the ground field admits only surjective monomorphisms, then "bijection" can be replaced by "injection".