On Buekenhout-Metz unitals of even order
A unital is a \(2-(n^ 3+1,n+1,1)\) design. The classical example of a unital is the set of absolute points in a projective plane \(\text{PG}(2,n^ 2)\) with respect to a unitary polarity, blocks being the sets of points on secant lines. A class of unitals, including the classical ones, can be constructed inside \(\text{PG}(2,n^ 2)\) by a method due to Buekenhout and Metz. The unitals obtained in this way are called Buekenhout-Metz unitals. The construction is `parabolic' in a sense, since it can be described almost completely in the affine plane; only one point is on the line at infinity. This point will be denoted by \(P_ \infty\). In this paper the author studies the Buekenhout-Metz unitals embedded in a projective plane of even order. He obtains an algebraic description of these unitals, determines the subgroup of the automorphism group of the unital stabilizing \(P_ \infty\), and determines the (number of) \(\text{PGL}(3,n^ 2)\)-orbits on the set of Buekenhout-Metz unitals in some fixed plane \(\text{PG}(2,n^ 2)\). Furthermore some subconfigurations are analyzed. For example, it is shown, that a Buekenhout-Metz unital does not contain ovals of the plane in which it is embedded.
scientific article; zbMATH DE number 64837
scientific article