On the Number of Directions Determined by a Set of Points in an Affine Galois Plane

In this paper the number of directions determined by a set ofq−npoints ofAG(2, q) is studied. To such a set we associate a curve of degreenand show that its linear components correspond to points that can be added to the set without changing the set of determined directions. The existence of linear components is guaranteed by Weilʹs theorem concerning the number of GF(q)-rational points of an absolutely irreducible curve, ifnis small enough.