Classification of control ensembles of projective points

The aim of this paper is to classify non-ordered sets of complex projective points with control parameter on the line with respect to projective transformations. This problem is equivalent to the problem of classifying binary forms, whose coefficients depend on control parameter, with respect to the action of some pseudogroup. We solve this problem in two steps. Firstly, we consider the action of our pseudogroup on the infinite prolongation of the differential Euler equation and find differential invariant algebra of this action. Secondly, using methods from geometric theory of differential equations, we prove that three dependencies between basic differential invariants and their invariant derivatives uniquely define the equivalent class of binary forms with control parameter.