Generalized quadrangles weakly embedded of degree 2 in projective space
In the paper, all generalized quadrangles weakly embedded of degree 2 in projective space are classified. Let \(\Gamma= ({\mathcal P},{\mathcal L},I)\) be any generalized quadrangle and \(V\) any vector space over some skew field. Consider a map \(\pi\) from \({\mathcal P}\) (respectively \({\mathcal L})\) to the set of points (respectively lines) of the projective space \(\text{PG}(V)\), which is injective on points, preserves incidence and for which \({\mathcal P}^\pi\) generates \(\text{PG}(V)\). We call \(\pi\) a weak embedding, if for any two points \(x\), \(y\) with \(y^\pi\) contained in the subspace of \(\text{PG}(V)\) generated by \(\{z^\pi\mid z\) collinear with \(x\) in \(\Gamma\}\), we may conclude that \(y\) is collinear with \(x\) in \(\Gamma\). For a weak embedding \(\pi\), the number of points of \(\Gamma^\pi\) on a line of \(\text{PG}(V)\), which intersects \({\mathcal P}^\pi\) in at least two points (but is not a line of \(\Gamma^\pi\)) is a constant; called the degree. We show that a generalized quadrangle weakly embedded of degree 2 is up to isomorphism: a) an orthogonal quadrangle (arising from an ordinary quadratic form) or a so-called mixed quadrangle (in characteristic 2) and the weak embedding is induced by a semilinear mapping or b) the symplectic quadrangle over \(\text{GF}(2)\) with an exceptional embedding. Together with the authors' results for degree \(>2\) [Forum Math. 11, 139-176 (1999; Zbl 0935.51003)] this completes the classification of all weak embeddings of generalized quadrangles (using the classification of Moufang quadrangles). It also provides new and independent proofs for the full case [\textit{K. J. Dienst}, Arch. Math. (Basel) 35, 177-186 (1980; Zbl 0437.51005)] and the finite case [\textit{J. A. Thas} and \textit{H. Van Maldeghem}, J. Stat. Plann. Inference 73, No. 1-2, 353-361 (1998; Zbl 0946.51009)]. For embeddings more general than weak ones, no classification is possible.
scientific article; zbMATH DE number 1463767