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

摘要

In this paper we first provide an infinite family of minimal (q - 1)-fold blocking sets of size q(3) in every affine translation plane of order q(2).Next, we focus on the regular Hughes plane H(q(2)) of order q(2). It is well known that pi = PG(2, q) is embedded as a Baer subplane in H(q(2)). Let l be a line of H(q(2)) having q+1 points in common with pi and let us denote by H(q(2))(l) the affine plane obtained from H(q(2)) by deleting l and all the points incident with l. We exhibit a family of minimal (q - 1)-fold blocking sets of size q(3) in H(q(2))(l).Furthermore, we consider the polarity of H(q(2)) which has q(3)-q/2 + q(2) + 1 absolute points and we show that its absolute points form a blocking semioval. Finally, we describe a class of Baer subplanes of H/(q(2)) distinct from pi and admitting an automorphism group of order q(3)(q - 1). (C) 2019 Elsevier Inc. All rights reserved.