Embeddings of Line-grassmannians of Polar Spaces in Grassmann Varieties

TL;DR

This paper introduces a generalized concept of embeddings for point-line geometries, focusing on Grassmann embeddings related to polar spaces, and explores their properties and applications to generalized quadrangles.

Contribution

It proposes a comprehensive definition of embeddings including various cases and studies Grassmann embeddings of polar spaces and generalized quadrangles.

Findings

Characterization of sets of points in PG(V∧V) for totally singular lines

New insights into Grassmann embeddings of generalized quadrangles

Extension of embedding concepts to broader classes of geometries

Abstract

An embedding of a point-line geometry \Gamma is usually defined as an injective mapping \epsilon from the point-set of \Gamma to the set of points of a projective space such that \epsilon(l) is a projective line for every line l of \Gamma, but different situations have lately been considered in the literature, where \epsilon(l) is allowed to be a subline of a projective line or a curve. In this paper we propose a more general definition of embedding which includes all the above situations and we focus on a class of embeddings, which we call Grassmman embeddings, where the points of \Gamma are firstly associated to lines of a projective geometry PG(V), next they are mapped onto points of PG(V\wedge V) via the usual projective embedding of the line-grassmannian of PG(V) in PG(V\wedge V). In the central part of our paper we study sets of points of PG(V\wedge V) corresponding to lines of PG(V) totally singular for a given pseudoquadratic form of V. Finally, we apply the results obtained in that part to the investigation of Grassmann embeddings of several generalized quadrangles.