On Buekenhout-Metz unitals in \(PG(2, q^ 2)\), \(q\) even
The classical example of a Hermitian arc (also called an embedded unital) in \(PG(2,q^ 2)\) is a non-singular Hermitian curve. Using an idea of F. Buekenhout, R. Metz exhibited some non-classical unitals which are known as parabolic Buekenhout-Metz unitals, or briefly parabolic BM unitals [\textit{R. Metz}, Geom. Dedicata 8, 125-126 (1979; Zbl 0403.51007)]; for an alternative approach see [\textit{L. Maria Abatangelo}, Matematiche 39, No. 1/2, 101-110 (1984; 632.51005)]. So for collineation groups fixing parabolic BM unitals have been studied in the odd order case [cf. the author and \textit{B. Larato}, Boll. Unione Mat. Ital., VII. Ser., A5, No. 2, 195-206 (1991; Zbl 0746.51013) and \textit{R. D. Baker} and \textit{G. L. Ebert}, On Buekenhout-Metz unitals of odd order, preprint; see also J. Comb. Theory, Ser. A60, No. 1, 67-84 (1992)]. In this paper we show that the linear collineation group \(\Gamma\) fixing a parabolic BM unital \({\mathcal U}\) of \(PG(2,q^ 2)\), \(q\) even, has the following properties: (1.1) \(\Gamma\) fixes the special point \(A\) of \(\mathcal U\). (1.2) \(\Gamma\) has a normal subgroup \(S\) acting on the non-special points of \(\mathcal U\) as a sharply transitive group. (1.3) The one-point stabilizer in \(\Gamma\) has order \(q-1\) for any non- special point of \(\mathcal U\). The main result is that the above properties characterize the parabolic BM unitals among the non-classical unitals of \(PG(2,q^ 2)\), \(q\) even.
scientific article; zbMATH DE number 119850