Blocking sets in $PG(r,q^n)$

Let S be a Desarguesian (n-1)-spread of a hyperplane ∑ of PG(rn, q). Let Ω and B̄ be, respectively, an (n-2)-dimensional subspace of an element of S and a minimal blocking set of an ((r-1)n+1)-dimensional subspace of PG(rn, q) skew to Ω. Denote by K the cone with vertex Ω and base B̄ , and consider the point set B defined by B=( K\∑ ) ∪ {X ∈ S : X ∩ K ≠ Ø} in the Barlotti-Cofman representation of PG(r,qn) in PG(rn, q) associated to the (n-1)-spread S. Generalizing the constructions of Mazzocca and Polverino (J Algebraic Combin, 24(1):61-81, 2006), under suitable assumptions on B̄ , we prove that B is a minimal blocking set in PG(r,qn). 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 qn+2+1 in PG(r,qn), 3 ≤ r ≤ 6 and n ≥ 3, and of size q4+1 in PG(r,q2), 4 ≤ r ≤ 6; actually, in the second case, these blocking sets turn out to be the union of q3 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 q2+2. From these constructions, we also get maximal partial ovoids of the hermitian variety H(4,q2) of size q4 + 1, for any q a power of 3.