Spreads and ovoids of generalized quadrangles of order \((s,s\pm 2)\) (Q2716674)

Let \({\mathcal G} = PG(3,q)\), \(q = 2^h\), \(h\geq 2\), and let \(\pi\) be a (desarguesian) plane embedded in \(\mathcal G\) with \(O\) a hyperoval of \(\pi\) (\(q+2\) points of \(\pi\) with no three on a line). (Note: In the first part of the paper under review wherever the author uses the word `oval', he means `hyperoval'.) Let \(T_2^*(O)\) be the pointline incidence structure defined as follows: points of \(T^*_2(O)\) are the points of \({\mathcal G}\setminus \pi\); lines of \(T_2^*(O)\) are the lines of \(\mathcal G\) which meet \(\pi\) in a single point of \(O\). Then \(T^*_2(O)\) is a GQ of order \((q-1, q+1)\). This GQ has the property that any two intersecting lines are contained in a unique subquadrangle with parameters \((q-1,1)\), i.e., a \(q\times q\) grid.NEWLINENEWLINENEWLINEIn the Ph.D. dissertation of \textit{B. De Bruyn} (Univ. Ghent (2000)) it is shown that if \(\mathcal S\) is a generalized quadrangle (GQ) of order \((s,t) = (s,s+2)\) with each pair of intersecting lines contained in a unique subquadrangle with parameters \((s,1)\), then \(\mathcal S\) is obtained from a GQ \({\mathcal S}'\) with parameters \((s+1,s+1)\) having a very special kind of points. This GQ of order \((s+1)\) is what is known as an amalgamation of two projective planes. The author shows that if the two planes are desarguesian and if a mild additional hypothesis is satisfied, then the GQ \(\mathcal S\) is isomorphic to \(T_2^*(O)\) for some hyperoval \(O\). However, this still leaves open the general question of whether or not a GQ which is the amalgamation of two desarguesian planes must be related to an oval in some way.NEWLINENEWLINENEWLINEFor the second part of the paper, let \(A\) and \(B\) be any two points of the hyperoval \(O^+\) of \(\pi\), and put \(O^-=O^+ \setminus \{A, B\}\). Let \(L\) be the line through \(A\) and \(B\), let \({\mathcal O}_{AB}\) (respectively, \({\mathcal O}_{BA}\)) be the set of planes in \(\mathcal G\) which contain \(A\) but not \(B\) (respectively, \(B\) but not \(A\)). Finally, let \({\mathcal P}_{AB}\) be the set of planes in \(\mathcal G\) other than \(\pi\) which contain the line \(L\). Then there is a GQ \(P(T_2(O),L)\) with parameters \((q+1,q-1)\) defined as follows: points are the points of \({\mathcal G}\setminus \pi\) together with the planes of \({\mathcal O}_{AB}\) and the planes of \({\mathcal O}_{BA}\); lines are the lines of \(\mathcal G\) which meet \(\pi\) in a single point of \(O^-\). Incidence is that inherited from incidence in \(\mathcal G\). The points of a plane in \({\mathcal P}_{AB}\) not in \(\pi\) form an ovoid of the GQ, as do the elements of \({\mathcal O}_{AB}\) (respectively, the elements of \({\mathcal O}_{BA}\)). This gives a collection (called a fan) of \(q+2\) ovoids of \(P(T_2(O),L)\) that partition its points. Each ovoid of the fan is associated with an affine plane. The author identifies a property (``grid-like'') of the fan that guarantees that any two of these planes are isomorphic. He observes that when the original oval \(O\) is a translation oval and \(A\) and \(B\) are carefully chosen, then the corresponding fan is grid-like.