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

Let Ω 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 Ω and base B̄ and consider the point set B defined by B=( K\∑ ) ∪ {X ∈ S : X ∩ K ≠ Ø} in the André, Bruck-Bose representation of PG(2,qn) in PG(2n,q) associated to a regular S of PG(2n-1,q). We are interested in finding conditions on B̄ and Ω 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 (Ω, 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 (Ω, B̄ ) with Property α. With this in mind, we deal with the problem in the case Ω 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). For example, for q = 3h , we get large blocking sets of size qn+2 + 1 (5 ≤ n) and of size greater than qn+2+ qn-6 (6 ≤ n). As an application, a characterization of Buekenhout-Metz unitals in PG(2, q2k) is also given.