Flocks of cones of higher degree
A flock of a cone of PG\((3,q)\) is a partition of the points of the cone different from the vertex into \(q\) disjoint plane sections. The author considers in PG\((3,q)\) for \( q>19 \) and for \(2\leq {d}\leq q^{1/6}\) the cone \(C=\{(1,t,t^d,z): t,z\in \text{GF}(q)\} \cup\{(0,0,1,z):z\in \text{GF}(q)\}\cup\{(0,0,0,1)\}\) and assumes that the planes \(E_i\), \(i=1,\dots,q-\varepsilon, (0,0,0,1)\notin\{E_i\}\), intersect \(C\setminus\{(0,0,0,1)\}\) in pairwise disjoint curves. The author proves that, if \(\varepsilon < [1/d^2q^{1/2}]\), then one can find additional \( \varepsilon\) planes (in a unique way), which extend \(E_i\) to a flock. The paper is based on an earlier paper of the author [J. Comb. Theory, Ser. A 113, No. 4, 698--702 (2006; Zbl 1097.51005)].
scientific article; zbMATH DE number 5134520