Partial flocks of non-singular quadrics in PG\((2r+1,q)\)
The authors generalize the notions of flocks and partial flocks, known for cones in PG\((3,q)\) and for non-singular quadrics in PG\((3,q)\), to non-singular quadrics in PG\((2r+1,q)\). This generalization for quadratic cones in PG\((2r+1,q)\) with a point vertex was already done by \textit{Christine M. O'Keefe} and \textit{J. A. Thas} [J. Algebr. Comb. 6, No. 4, 377--392 (1997; Zbl 0897.51007)]. Now the analogous results for non-singular quadrics in PG\((2r+1,q)\) are presented. Let \(Q_{2r+1}\) be a non-singular quadric of elliptic or hyperbolic character in PG\((2r+1,q)\). A partial flock of \(Q_{2r+1}\) of cardinality \(s\) is a set of \(s\) hyperplanes of PG\((2r+1,q)\) intersecting this quadric in non-singular parabolic quadrics, and such that two distinct such hyperplanes intersect in a \((2r-1)\)-dimensional space sharing an elliptic quadric with \(Q_{2r+1}\). Among the results presented are: upper bounds on the size of such partial flocks, a generalization of a construction of Thas, and a characterization of linear partial flocks. Moreover, partial flocks of \(Q_{2r+1}\) of size larger than the size of flocks of the non-singular quadrics in PG\((3,q)\) are constructed.
scientific article; zbMATH DE number 2146019