On Hermitian spreads
Let \(Q^{-}(5,q)\) be the elliptic quadric of PG\((5,q)\) and \(\perp\) its polarity. Assume that \(\mathcal S\) is a locally Hermitian spread of \(Q^{-}(5,q)\) with respect to \(L\in\mathcal S\). Then \(\{\langle L,M\rangle^{\perp}\mid M\in\mathcal S,\;M\not=L\}\cup\{L\}=:\mathcal S\)\(_{\Lambda}\) is a spread of the \(3\)-space \(L^{\perp}=:\Lambda\). Conversely, for each spread \(\mathcal F\) of \(\Lambda\) containing \(L\) there exists a locally Hermitian spread \(\mathcal S(\mathcal F, L)\) of \(Q^{-}(5,q)\) with respect to \(L\) such that \(\mathcal S(\mathcal F, L)\)\(_{\Lambda}=\mathcal F\). The authors prove that the only regular spreads (containing \(L\)) of \(\Lambda\) which give Hermitian spreads of \(Q^{-}(5,q)\) (\(\subset\,Q^{+}(5,q^2)\)) are those whose transversals lie in the two planes of \(Q^{+}(5,q^2)\) through \(L\).
scientific article; zbMATH DE number 2139735