Classifying the complete arcs in small projective planes
Presented by Prof. Kris COOLSAET
Type: Keynote Lecture
An arc is a set of points in a projective plane with the property that no three of them are on the same line. An arc is called complete if no point can be added to it so that it still remains an arc. The conic serves as the standard example of a complete arc, but many other constructions are known. Although arcs are relatively small (at most q+2 points in a finite projective plane of order q and at most q+1 if q is odd), only for really small values of q (say, q < 10) it is possible to find all complete arcs in a plane by hand, and classify them up to isomorphism. Until five years ago, a full (computer) classification of the complete arcs in Desarguesian projective planes was only known for q<20. We managed to extend this bound to q=29. The algorithm we have used is based on the well-known process of 'canonical augmentation', introduced in the 1980's by Brendan McKay. In this talk I will give a sketch of the algorithm as it was applied to our problem. Instead of generating all complete arcs, it is also of interest to focus on certain special cases, for example, arcs that have a 'large' number of points (q-1 in particular). It was proved that for q>89, q odd, an arc of size q-1 can always be extended to a conic, and except for two sporadic examples (q=7,13), it is believed that this property holds for all (odd) values of q. I shall describe an algorithm that might be used to settle the remaining cases within reasonable time, by classifying the larger arcs without having to generate all the smaller arcs as well.