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

Abstract

Let ? and $${\bar B}$$ be a subset of ? = PG(2n?1,q) and a subset of PG(2n,q) respectively, with ? ? PG(2n,q) and $${{\bar B}\not\subset \Sigma}$$ . Denote by K the cone of vertex ? and base $${\bar B}$$ and consider the point set B defined by $$B=\big(K{\setminus}\Sigma\big) \cup \{X\in \S\, : \, X\cap K\neq \emptyset\},$$ in the Andre, Bruck-Bose representation of PG(2,qn) in PG(2n,q) associated to a regular spread $${\cal S}$$ of PG(2n?1,q). We are interested in finding conditions on $${\bar B}$$ 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 ? of the pair (?, $${\bar B}$$ ) 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 (?, $${\bar B}$$ ) with Property ?. With this in mind, we deal with the problem in the case ? is a subspace of PG(2n?1,q) and $${\bar B}$$ 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.