Blocking Sets in PG(r, q n )

Abstract

Let $$\mathcal S$$ be a Desarguesian (n --- 1)-spread of a hyperplane Σ of PG(rn, q). Let ? and $${\bar B}$$ be, respectively, an (n --- 2)-dimensional subspace of an element of $$\mathcal S $$ and a minimal blocking set of an ((r --- 1)n + 1)-dimensional subspace of PG(rn, q) skew to ?. Denote by K the cone with vertex ? and base $${\bar B}$$ , and consider the point set B defined by $$B=\left(K\setminus\Sigma\right)\cup \{X\in \mathcal S\, : \, X\cap K\neq \emptyset\}$$ in the Barlotti---Cofman representation of PG(r, q n ) in PG(rn, q) associated to the (n --- 1)-spread $$\mathcal S$$ . Generalizing the constructions of Mazzocca and Polverino (J Algebraic Combin, 24(1):61---81, 2006), under suitable assumptions on $${\bar B}$$ , we prove that B is a minimal blocking set in PG(r, q n ). In this way, we achieve new classes of minimal blocking sets and we find new sizes of minimal blocking sets in finite projective spaces of non-prime order. In particular, for q a power of 3, we exhibit examples of r-dimensional minimal blocking sets of size q n+2 + 1 in PG(r, q n ), 3 ? r ? 6 and n ? 3, and of size q 4 + 1 in PG(r, q 2), 4 ? r ? 6; actually, in the second case, these blocking sets turn out to be the union of q 3 Baer sublines through a point. Moreover, for q an even power of 3, we construct examples of minimal blocking sets of PG(4, q) of size at least q 2 + 2. From these constructions, we also get maximal partial ovoids of the hermitian variety H(4, q 2) of size q 4 + 1, for any q a power of 3.