Singer quadrangles

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An outline of polar spaces: basics and advances

This paper is an extended version of a series of lectures on polar spaces given during the workshop and conference 'Groups and Geometries', held at the Indian Statistical Institute in Bangalore in December 2012. The aim of this paper is to give an overview of the theory of polar spaces focusing on some research topics related to polar spaces. We survey the fundamental results about polar spaces starting from classical polar spaces. Then we introduce and report on the state of the art on the following research topics: polar spaces of infinite rank, embedding polar spaces in groups and projective embeddings of dual polar spaces

M-systems of polar spaces

AbstractLet P be a finite classical polar space of rank r, with r ⩾ 2. A partial m-system M of P, with 0 ≤ m ≦ r − 1, is any set {π1, π2, …, πk} of k (≠ 0) totally singular m-spaces of P such that no maximal totally singular space containing πi has a point in common with (ν1 ∪ π2 ∪ ⋯ ∪ πk) − πi, i = 1, 2, …, k. In each of the respective cases an upper bound δ for |M| is obtained. If |M| = δ, then M is called an m-system of P. For m = 0 the m-systems are the ovoids of P; for m = r − 1 the m-systems are the spreads of P. Surprisingly δ is independent of m, giving the explanation why an ovoid and a spread of a polar space P have the same size. In the paper many properties of m-systems are proved. We show that with m-systems of three types of polar spaces there correspond strongly regular graphs and two-weight codes. Also, we describe several ways to construct an m′-system from a given m-system. Finally, examples of m-systems are given