Blocking sets in PG ( 2 , q n ) from cones of PG ( 2 n , q )

Let and B̄ be a subset of = PG(2n − 1, q) and a subset of PG(2n, q) respectively, with ⊂ PG(2n, q) and B̄ ⊂ . Denote by K the cone of vertex and base B̄ and consider the point set B defined by B = (K\ ) ∪ {X ∈ S : X ∩ K = ∅}, in the André, Bruck-Bose representation of PG(2, qn) in PG(2n, q) associated to a regular spread S of PG(2n − 1, q). We are interested in finding conditions on B̄ and in order to force the set B to be a minimal blocking set in PG(2, qn). Our interest is motivated by the following observation. Assume a Property α of the pair ( , B̄) forces B to turn out a minimal blocking set. Then one can try to find new classes of minimal blocking sets working with the list of all known pairs ( , B̄) with Property α. With this in mind, we deal with the problem in the case is a subspace of PG(2n − 1, q) and B̄ a blocking set in a subspace of PG(2n, q); both in a mutually suitable position. We achieve, in this way, new classes and new sizes of minimal blocking sets in PG(2, qn), generalizing the main constructions of [14]. For example, for q = 3h, we get large blocking sets of size qn+2 + 1 (n ≥ 5) and of size greater than qn+2 + qn−6 (n ≥ 6). As an application, a characterization of Buekenhout-Metz unitals in PG(2, q2k) is also given.