On transparent embeddings of point-line geometries

We introduce the class of transparent embeddings for a point-line geometry $\mathrm{\Gamma }=(\mathcal{P},\mathcal{L})$ as the class of full projective embeddings $\epsilon$ of $\mathrm{\Gamma }$ such that the preimage of any projective line fully contained in $\epsilon (\mathcal{P})$ is a line of $\mathrm{\Gamma }$. 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.