If B is a minimal blocking set of size less than 3(q+1)=2 in PG(2,q), q is a power of the prime p, then Szőnyi’s result states that each line meets B in 1 (mod p) points. It follows that B cannot have bisecants, i.e., lines meeting B in exactly two points. If q >13, then there is only one known minimal blocking set of size 3(q+1)=2 in PG(2, q), the so-called projective triangle. This blocking set is of Rédei type and it has 3(q-1)=2 bisecants, which have a very strict structure. We use polynomial techniques to derive structural results on Rédei type blocking sets from information on their bisecants. We apply our results to point sets of PG(2, q) with few odd-secants.
In particular, we improve the lower bound of Balister, Bollobás, Füredi and Thompson on the number of odd-secants of a (q+2)-set in PG(2, q) and we answer a related open question of Vandendriessche. We prove structural results for semiovals and derive the non existence of semiovals of size q+3 when p≠3 and q>5. This extends a result of Blokhuis who classified semiovals of size q+2, and a result of Bartoli who classified semiovals of size q+3 when q ≤ 17. In the q even case we can say more applying a result of Szőnyi and Weiner about the stability of sets of even type. We also obtain a new proof to a result of Gács and Weiner about (q+t, t)-arcs of type (0, 2, t) and to one part of a result of Ball, Blokhuis, Brouwer, Storme and Szőnyi about functions over GF(q) determining less than (q+3)/2 directions.
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Research supported by the Hungarian National Foundation for Scientific Research, Grant No. K 81310 and by the Italian Ministry of Education, University and Research (PRIN 2012 project Strutture geometriche, combinatoria e loro applicazioni”).
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Csajbók, B. On Bisecants of Rédei Type Blocking Sets and Applications. Combinatorica 38, 143–166 (2018). https://doi.org/10.1007/s00493-016-3442-6
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