On ( k, pe)-arcs in Desarguesian planes

Abstract

In this paper we improve Szőnyi's embeddability result on ( k, p)-arcs, in PG(2, q), q= p h , p prime. Szőnyi proved that for k> qp− q+ p− ε, ε⩽ 1 2 q 4 , a ( k, p)-arc can be embedded in a maximal arc. Our main theorem is that this result can be extended for ε⩽ 1 4 q , furthermore it can be generalized to ( k, p e )-arcs, p e< q . This and the result of Ball, Blokhuis and Mazzocca on the non-existence of maximal arcs for p>2, yields an upper bound on the size of a ( k, p e )-arc. In the particular case p=2, Segre showed that when k>q+1− q , any k-arc can be extended to a hyperoval. This result is sharp, since there are complete arcs of size q+1− q . A new proof for Segre's theorem is also presented.