Rigidity in conical flocks
\textit{J. A. Thas} [Eur. J. Comb. 8, 441-452 (1987; Zbl 0646.51019)] completely characterizes the flocks of a quadratic cone whose planes share a common point. The authors establish a fundamental relationship between \(s\)-locally rigid groups and intersections of the planes of the partial flock. The planes of a partial flock \(P\) of \(s\)-conics of a quadratic cone in \(PG(3,q)\) share a common point if and only if there exists a nontrivial linear \(s\)-locally rigid group on \(P\). It is considered the possibility that an \(s\)-locally rigid group might exist but contain no nontrivial linear subgroups (subgroups of \(PGL(4,q))\). If there exists an \(s\)-locally rigid group on a partial flock of \(s\)-conics in \(PG(3,q)\) and \(s > 2q^{1/2} + 1\) then the planes of the partial flock share a common point. It is obtained a group theoretic characterization of the linear flock and the semifield flocks of Knuth type. A flock \(F\) of a quadratic cone on \(PG(3,q)\) is linear or a semifield flock of Knuth type if and only if \(F\) admits a nontrivial rigid automorphism group. These are automorphism groups of flocks which fix at least \(s\) planes of the flock. Apart from fundamental results, the most striking result is the characterization of the Fisher flocks by \(s\)-rigid groups. If \(F\) is a nonlinear flock in \(PG(3,q)\) that admits an \(s\)-rigid automorphism group of order \(> 2\) for \(s \geq (q - 1) \geq 2\) then \(F\) is a Fisher flock. Using methods of rigid groups, the authors improve the result of Thas for even order flocks. If \(F\) is a flock of a quadratic cone in \(PG(3,q)\) for \(q\) even such that there are \(\geq q/2\) planes of the flock that share a common point then \(F\) is linear. As an application it is investigated the maximal situation possible where there is a flock and a set of \(s\)-planes containing the conics of the flock that share a common point. Let \(F\) be a flock of a quadratic cone in \(PG(3,q)\) and let at least \(q-1\) of the planes that contains the conics of \(F\) share a common point. Then all such planes share a common point and the flock is linear or a semified flock of Knuth type.
scientific article; zbMATH DE number 842830