On two new classes of semibiplanes
A semibiplane is a connected symmetric finite block design such that every pair of points is incident with 0 or 2 blocks and, dually, every pair of blocks is incident with 0 or 2 points. However, the author is working with a different (though equivalent) definition, where a semibiplane is regarded as a rank 3 geometry. After discussing several classical examples the author describes two new infinite classes of finite semibiplanes. Consider a point \(P\) and some plane \(\pi\) in \(\text{PG}(3,q)\), \(q=2^h\). Let \(O\) be a hyperoval in the star of \(P\) consisting of \(q+2\) lines passing through \(P\) and not incident with \(\pi\). If \(P\) is incident with \(\pi\), consider a dual hyperoval \(O^*\) of \(q+2\) lines in \(\pi\) not passing through \(P\). If \(P\) is \textit{not} incident with \(\pi\), let \(O'\) be the intersection of \(O\) with \(\pi\) and consider the dual hyperoval \(O^*\) consisting of all lines of \(\pi\) which are external to \(O'\). In both cases a semibiplane \(\Gamma(O,O^*)\) arises as a subgeometry of \(\text{PG}(3,q)\), where the points of \(\Gamma(O,O^*)\) are the points on the lines of \(O\) except \(P\), the planes are given by the planes different from \(\pi\) that contain some line of \(O^*\), and the lines of \(\Gamma(O,O^*)\) are those lines of \(\text{PG}(3,q)\) which are incident with two points and two lines of \(\Gamma(O,O^*)\). The paper closes with an investigation on quotients of \(\Gamma(O,O^*)\).
scientific article; zbMATH DE number 1110064