CYCLIC ARCS IN PG(2, Q)

B.C. Kestenband [91, J.C. Fisher, J.WR Hirschfeld, and J.A. Thas [31, E. Boros, and T. Szonyi [1] constructed complete (q2 - q + 1)-arcs in PG(2, q2), q greater-than-or-equal-to 3. One of the interesting properties of these arcs is the fact that they are fixed by a cyclic projective group of order q2 - q + 1. We investigate the following problem: What are the complete k-arcs in PG(2, q) which are fixed by a cyclic projective group of order k? This article shows that there are essentially three types of those arcs, one of which is the conic in PG(2, q), q odd. For the other two types, concrete examples are given which shows that these types also occur.