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

Let Ω and be a subset of Σ = PG(2n−1,q) and a subset of PG(2n,q) respectively, with Σ ⊂ PG(2n,q) and . Denote by K the cone of vertex Ω and base and consider the point set B defined by in the André, Bruck-Bose representation of PG(2,qⁿ) in PG(2n,q) associated to a regular spread of PG(2n−1,q). We are interested in finding conditions on 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, ) 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, ) 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 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 = 3^h, we get large blocking sets of size q^{n + 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,q^2k) is also given.