Counting 7-Arcs in Projective Planes over Finite Fields

Given a collection of points in the plane, classifying which subsets are collinear is a natural problem and is related to classical geometric constructions. We consider collections of points in a projective plane over a finite field such that no three are collinear. This is a finite set and its size is both combinatorially interesting and has deeper topological consequences. We count the number of such collections classified by the algebraic symmetries of the finite field. Variations of this problem have been considered by Glynn, Bergvall, Das, O'Connor et al. We obtain the counts for 7 points over fields of characteristic 2. These new counts are governed by the existence and classification of a configuration of points called the Fano plane.