On Arcs Sharing the Maximum Number of Points with Ovals in Cyclic Affine Planes of Odd Order

摘要

The sporadic complete 12-arc in PG(2, 13) contains eight points from a conic. In PG(2,q) with q>13 odd, all known complete k-arcs sharing exactly 1/2(q+3) points with a conic C have size at most 1/2(q+3)+2, with only two exceptions, both due to Pellegrino, which are complete (1/2(q+3)+3) arcs, one in PG(2, 19) and another in PG(2, 43). Here, three further exceptions are exhibited, namely a complete (1/2(q+3)+4)-arc in PG(2, 17), and two complete (1/2(q+3)+3)-arcs, one in PG(2, 27) and another in PG(2, 59). The main result is Theorem 6.1 which shows the existence of a (1/2(q(r)+3)+3)-arc in PG(2,q(r)) with r odd and q equivalent to 3 (mod 4) sharing 1/2(q(r)+3) points with a conic, whenever PG(2,q) has a (1/2(q+3)+3)-arc sharing 1/2(q+3) points with a conic. A survey of results for smaller q obtained with the use of the MAGMA package is also presented.