Embeddings of Line-grassmannians of Polar Spaces in Grassmann Varieties

An embedding of a point-line geometry $\Gamma$ is usually defined as an injective mapping $\varepsilon$ from the point-set of $\Gamma$ to the set of points of a projective space such that $\varepsilon(l)$ is a projective line for every line $l$ of $\Gamma$. However, different situations are considered in the literature, where $\varepsilon(l)$ 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 Grassmann embeddings, where the points of $\Gamma$ are firstly associated to lines of a projective geometry $\mathrm{PG}(V)$, next they are mapped onto points of $\mathrm{PG}(V\wedge V)$ via the usual projective embedding of the line-grassmannian of $\mathrm{PG}(V)$ in $\mathrm{PG}(V\wedge V)$. In the central part of our paper we study sets of points of $\mathrm{PG}(V\wedge V)$ corresponding to lines of $\mathrm{PG}(V)$ totally singular for a given alternating, hermitian or quadratic form of $V$. Finally, we apply the results obtained in that part to the investigation of Grassmann embeddings of several generalized quadrangles.