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arXiv · 1303.5609

Embeddings of Line-grassmannians of Polar Spaces in Grassmann Varieties

Ilaria Cardinali·Antonio Pasini
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Abstract

An embedding of a point-line geometry Γis usually defined as an injective mapping εfrom the point-set of Γto the set of points of a projective space such that ε(l) is a projective line for every line l of Γ, but different situations have lately been considered in the literature, where ε(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 Γare 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.

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Ilaria Cardinali, Antonio Pasini. 2013-03-22. Embeddings of Line-grassmannians of Polar Spaces in Grassmann Varieties. https://arxiv.org/abs/1303.5609

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