Planar arcs

摘要

Let p denote the characteristic of F-q, the finite field with q elements. We prove that if q is odd then an arc of size q + 2 - t in the projective plane over F-q, which is not contained in a conic, is contained in the intersection of two curves, which do not share a common component, and have degree at most t + p(left perpendicularlogp tright perpendicular) , provided a certain technical condition on t is satisfied. This implies that if q is odd then an arc of size at least root q + root q/p + 3 is contained in a conic if q is square and an arc of size at least q - root q + 7/2 is contained in a conic if q is prime. This is of particular interest in the case that q is an odd square, since then there are examples of arcs, not contained in a conic, of size q - root q + 1, and it has long been conjectured that if q not equal 9 is an odd square then any larger arc is contained in a conic.