Two-transitive ovals in generalized twisted field planes
Let \(\pi\) be a finite affine plane whose projective extension contains an oval \(\Omega\) which intersects the line at infinity in a single point \((\infty)\), i.e. the line at infinity is tangent to \(\Omega\) at \((\infty) \). In this case \(\Omega\) is said to be a parabolic oval of \(\pi\). If there is a collineation group \(G\) of \(\pi\) which leaves \(\Omega\) invariant and acts 2-transitively on \(\Omega\setminus \{(\infty)\}\), we call \(\Omega\) a 2-transitive parabolic oval. The authors remark that, if \(\pi\) is a finite translation plane of odd order admitting a 2-transitive parabolic oval, then \(\pi\) is a commutative twisted field plane. Moreover, in [\textit{M. J. Ganley}, Geom. Dedicata 1, 103-116 (1972; Zbl 0241.50024)], it has been proved that commutative twisted field planes of odd order admit a 2-transitive parabolic oval. In this paper the authors prove that the only generalized twisted field plane of even order admitting a 2-transitive parabolic oval is the Desarguesian one.
scientific article; zbMATH DE number 1837333