The Klein correspondence and the ubiquity of certain translation planes (Q1076338)

Let V be a 4-dimensional vector space over a field F. The Klein correspondence is an isomorphism between the projective geometry P(V) and the orthogonal geometry \(O^+_ 6(F)\) on \(V\wedge V\). If \(\pi\) is a translation plane of dimension two over its kernel F, then \(\pi\) corresponds to a spread in V. The Klein correspondence induces a bijection between the spreads in V and the ovoids in \(V\wedge V\). Hence questions about translation planes can be answered by looking at ovoids in \(V\wedge V\). The authors claim that ovoids are easier to handle than spreads. For this they treat the isomorphism-problem for translation planes over GF(p) possessing a group of collineations K such that p does not divide the order of K. In Math. Z. 156, 59-71 (1977; Zbl 0343.50003), \textit{T. G. Ostrom} proved that the maximal possible group K is a certain nonsplit extension of an extraspecial group of order 32 by \(A_ 5\). The authors determine for \(p\leq 23\) all isomorphism classes of such translation planes admitting this maximal collineation group. The method they use is to determine the possible ovoids. In fact, for \(p\leq 11\) the first author and \textit{T. G. Ostrom} determined all isomorphism classes in Geom. Dedicata 17, 307-322 (1985; Zbl 0566.51020) using the spread- approach. Also this result has been redone in the paper under review, but now using ovoids, which proves to a certain degree that this approach is easier than using spreads. Though there are 14 isomorphism classes for \(p=23\) all calculations are done by hand. Unfortunately there is no general system visible. The authors claim that the number of classes goes to infinity with increasing p. But it is not clear whether this number must be always at least 1. For \(p\leq 23\) this is true.