On transparent embeddings of point-line geometries

TL;DR

This paper introduces the concept of transparent embeddings in point-line geometries, analyzing their properties in various classical geometries and deriving Chow-like theorems as applications.

Contribution

It defines and investigates the class of transparent embeddings, providing new insights into their structure and applications in classical geometries.

Findings

Plücker embeddings of projective and polar grassmannians exhibit transparency.

Spin embeddings of half-spin geometries are shown to be transparent.

Derived Chow-like theorems for polar grassmannians and half-spin geometries.

Abstract

We introduce the class of transparent embeddings for a point-line geometry $\Gamma = ({\mathcal P},{\mathcal L})$ as the class of full projective embeddings $\varepsilon$ of $\Gamma$ such that the preimage of any projective line fully contained in $\varepsilon({\mathcal P})$ is a line of $\Gamma$. We will then investigate the transparency of Pl\"ucker embeddings of projective and polar grassmannians and spin embeddings of half-spin geometries and dual polar spaces of orthogonal type. As an application of our results on transparency, we will derive several Chow-like theorems for polar grassmannians and half-spin geometries.