On multiple blocking sets and blocking semiovals in finite non-Desarguesian planes

Abstract

In this paper we first provide an infinite family of minimal (q−1)-fold blocking sets of size q3 in every affine translation plane of order q2.Next, we focus on the regular Hughes plane H(q2) of order q2. It is well known that π=PG(2,q) is embedded as a Baer subplane in H(q2). Let ℓ be a line of H(q2) having q+1 points in common with π and let us denote by H(q2)ℓ the affine plane obtained from H(q2) by deleting ℓ and all the points incident with ℓ. We exhibit a family of minimal (q−1)-fold blocking sets of size q3 in H(q2)ℓ.Furthermore, we consider the polarity of H(q2) which has q3−q2+q2+1 absolute points and we show that its absolute points form a blocking semioval. Finally, we describe a class of Baer subplanes of H(q2) distinct from π and admitting an automorphism group of order q3(q−1).