Derivation of \(m\)-systems
A partial \(m\)-system \(\mathcal M\) of a polar space \(\mathcal P\) in \(\text{PG}(n, q)\) is a set of \(m\)-dimensional totally singular subspaces \(\pi_1,\dots,\pi_r\) of \(\mathcal P\) such that each generator of \(\mathcal P\) containing an element \(\pi_i\in\mathcal M\) has an empty intersection with \((\pi_1\cup\dots\cup\pi_r)\setminus\pi_i\). Partial \(m\)-systems of polar spaces were introduced by \textit{E. E. Shult} and \textit{J. A. Thas} [J. Comb. Theory, Ser. A 68, 184-204 (1994; Zbl 0824.51004)]. They show that there exists an upper bound, which is independent of \(m\), on the number of elements of a partial \(m\)-system and they call a partial \(m\)-system which meets this upper bound an \(m\)-system. The authors discuss a way to construct new \(m\)-systems from known ones by replacing a set of subspaces of the \(m\)-system by a suitably chosen other set of subspaces, such that the unions of the elements for both sets coincide. This technique is called derivation of \(m\)-systems and it is described in the following theorem: Let \(\mathcal M :=\{\pi_1,\pi_2,\dots,\pi_r\}\) be an \(m\)-system of a finite classical polar space \(\mathcal P_n\) in \(\text{PG}(n, q)\). Suppose that there exists a \(\text{PG}(2m+1, q)\) containing \(1\leqslant s \leqslant r\) elements \(\pi_1,\pi_2,\dots,\pi_r\) of \(\mathcal M\). If in \(\text{PG}(2m+q, q)\cap\mathcal P_n\) there exist \(s\) disjoint \(m\)-dimensional totally singular subspaces \(\alpha_1,\alpha_2,\dots,\alpha_s\) of \(\mathcal P_n\) such that the point sets \(\pi_1\cup\pi_2\cup\dots\cup\pi_s\) and \(\alpha_1\cup\alpha_2\cup\dots\cup\alpha_s\) coincide, then \(\mathcal M' := \{\alpha_1,\alpha_2,\dots,\alpha_s,\pi_{s+1}, \pi_{s+2},\dots,\pi_r\}\) is also an \(m\)-system of \(\mathcal P_n\). It is clear that this theorem can be applied to any locally Hermitian 1-system \(\mathcal M'\) of a non-singular quadric, in which case a regulus can be replaced by its opposite regulus to obtain another 1-system \(\mathcal M'\). In this situation, it is possible to show that the 1-system \(\mathcal M'\) arising in this way, is not isomorphic to the original one. This is an interesting result, because it implies that \(\mathcal M'\) is new. The following result is proved: Suppose that \(\mathcal M\) is a 1-system of a non-singular quadric \(\mathcal Q\) in \(\text{PG}(n, q)\), which is locally Hermitian at some line \(L\in\mathcal M\). Let \(\mathcal M'\) be the 1-system of \(\mathcal Q\), which is obtained by replacing a regulus \(R\) of \(\mathcal M\) through \(L\) by its opposite regulus \(\overline R\). Then \(\mathcal M'\) is not locally Hermitian and so \(\mathcal M\) and \(\mathcal M'\) are not isomorphic. An overview of other known 1-systems which are candidates for the application of the stated above theorem is given. In particular, the semipartial geometries arising from 1-systems of the symplectic polar space \(W_5(q)\) are studied and it is shown that some of the new found 1-systems yield new semipartial geometries.
\(m\)-dimensional totally singular subspaces
scientific article; zbMATH DE number 1891293