A geometric construction of a class of twisted field planes by Ree-Tits unitals (Q1904108)

Let \(H(q)\) denote the generalized quadrangle arising from Dickson's simple group \(G_2 (q)\) [see \textit{J. Tits}, Inst. Haut. Etud. Sci. Publ. Math. No. 2, 13-60 (1959; Zbl 0088.37204)]. It is well known that \(H(q)\) admits a polarity if and only if \(q\) is an odd power of 3, and it is assumed \(q = 3^{2h + 1}\) for some nonnegative integer \(h\) throughout this review. In this case it is also known that the set of absolute points forms an ovoid in \(H(q)\); that is, a set of \(q^3 + 1\) points such that every other point is collinear with exactly one of them. This collection of points in \(H(q)\) is called a Ree-Tits unital, whose blocks are defined to be the fixed sets of the involutions in \(\Aut (H(q)) \cong^2 G_2 (q)\). Any two Ree-Tits unitals are isomorphic, and there are \(q^3 (q - 1)\) Ree-Tits unitals containing a particular flag (point-line incident pair). In the paper under review it is shown that if \(h = 0\) (hence \(q = 3)\), then any two distinct Ree-Tits unitals have 0,1 or \(q + 1\) flags in common, while if \(h \geq 1\), then any two distinct Ree-Tits unitals of \(H(q)\) have 0,1,2 or \(q + 1\) flags in common. From this result it is observed that whenever two Ree-Tits unitals meet in \(q + 1\) flags, the points of these flags constitute a block in both unitals. It is then shown how to construct projective plane of order \(q\) from a certain class of Ree-Tits unitals containing a specified flag. This plane turns out to be nondesarguesian if and only if \(h \neq 0\) \((q > 3)\). In fact, these planes are isomorphic to twisted field planes. Moreover, ovals are constructed in all these planes.