Polarized and homogeneous embeddings of dual polar spaces
An embedding \(e: \Gamma \rightarrow PG(V)\) of a dual polar space \(\Gamma\) is \textit{polarized} if, for every point \(x\in \Gamma\), the image \(e(H)_x\) of the hyperplane \(H_x\) of \(\Gamma\), formed by the points at non-maximal distance from \(x\), spans a hyperplane of \(PG(V)\). The main result of the paper under review is the following theorem. Theorem. Let \(\Gamma\) be the dual of a classical polar space and let \(e\) be a projective embedding of \(\Gamma\), defined over a commutative division ring. If \(e\) is \(\Aut(\Gamma)_0\)-homogeneous then it is polarized. Sections 2 and 3 of the paper contain definitions and basics on point-line geometries, dual polar spaces, embeddings and some preliminary results on embeddings of near-polar spaces, on generalized quadrangles and projective spaces. Finally, in Section 4 the main result is proved, and it is also proved a consequence of such a theorem on the absolutely universal embedding of \(\Gamma\).
scientific article; zbMATH DE number 5644671
scientific article