Spreads and ovoids translation with respect to disjoint flags (Q1877366)

The points of the split hexagon \(H(q)\) may be seen as the points of a nondegenerate quadric \(P_6\) in a natural way embedded into the projective space PG\((6,q)\), whereas the lines of \(H(q)\) form a certain subset of the set of lines of \(P_6\). A spread of lines of \(H(q)\) is a translation spread \(\mathcal S\) with respect to a flag \(\{L,u\}\), where \(L\) is a line of \(\mathcal S\) and \(u\) is point on \(L\), if there exists a subgroup \(G_{\{L,u\}}\) of the collineation group of \(H(q)\) fixing the spread \(\mathcal S\) and acting transitively on each set of lines of \({\mathcal S}^+ ={\mathcal S} \setminus \{L\}\) which have the distance 4 from \(M\), where \(M\) is a line through \(u\). The authors prove that a spread \(\mathcal S\) of a finite split Cayley hexagon is either a Hermitian or a Ree-Tits spread if \(\mathcal S\) is a translation spread with respect to two disjoint flags. To obtain this result they introduce the notion of local polarity of ovoid-spread pairings and show that if an ovoid-spread pairing is locally polar at each of its elements then it arises from a polarity.