Split semi-biplanes in antiregular generalized quadrangles (Q1281178)

Let \(GQ=({\mathcal P}, {\mathcal L},I)\) be a generalized quadrangle with all points antiregular. This means that if \(x,y,z\) is any triad (three points of \(GQ\) with no two collinear), then there are exactly 0 or 2 points collinear with all three points of the triad. Now let \(p,q,r,s\) be distinct collinear points of \(GQ\) with connecting line \(l\). Then if \({\mathcal P}^l_p\) denotes the set of points collinear with \(p\) but not on \(l\), let \(GQ_{p,q,r,s}:= ({\mathcal P}_{p,q,r,s},{\mathcal L}_{p,q,r,s}, \sim):=({\mathcal P}^l_p \cup{\mathcal P}^l_q, {\mathcal P}_r^l \cup {\mathcal P}^l_s, \sim)\). Here \(x\sim y\) denotes that \(x\) and \(y\) are collinear. The authors investigate at some length just when \(GQ_{p,q,r,s}\) is a semibiplane, i.e., any two distinct points of \({\mathcal P}_{p,q,r,s}\) are incident with 0 or 2 blocks of \({\mathcal L}_{p,q,r,s}\) and any two distinct blocks of \({\mathcal L}_{p,q, r,s}\) are incident with 0 or 2 points of \({\mathcal P}_{p,q,r,s}\). Their interesting results have as a corollary the fact that when \(GQ\) is the classical example \(Q(4, K)\) derived from a nonsingular quadric in \(PG(4,K)\), where \(K\) is a finite field of odd order or \(K={\mathcal R}\), then \(p,q,r,s\) may be chosen so that \(GQ_{p,q,r, s}\) is a semi-biplane (of a special type called a split semi-biplane). For \(K\) finite of odd order, this is also Theorem 11 of \textit{P. Wild} [J. Geom. 25, 121-130 (1985; Zbl 0576.51011)]. Several results of P. Wild were in fact a major inspiration for current work of the authors.