Some subsets of the Hermitian curve

Three types of subsets of PG(2,q2) of type (0,1,2,q+1) are defined, namely CF-sets, K-sets and H-sets. These subsets may also be obtained as the intersection of the Hermitian curve H with Baer subpencils of lines with vertex not on H. It is shown that under the Bruck–Bose representation these subsets correspond to three-dimensional elliptic quadrics, quadratic cones and hyperbolic quadrics, respectively contained in suitable hyperplanes of PG(4,q).