The number of designs with geometric parameters grows exponentially

It is well-known that the number of 2-designs with the parameters of a classical point-hyperplane design PGn-1(n, q) grows exponentially. Here we extend this result to the number of 2-designs with the parameters of PGd(n, q), where 2 ≤ d ≤ n − 1. We also establish a characterization of the classical geometric designs in terms of hyperplanes and, in the special case d = 2, also in terms of lines. Finally, we shall discuss some interesting configurations of hyperplanes arising in designs with geometric parameters.