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

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 [Formula presented] +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).