Primitive block designs with automorphism group PSL(2,Q)

We present the results of a research which aims to determine, up to isomorphism and complementation, all primitive block designs with the projective line Fq∪{∞} as the set of points and PSL(2,q) as an automorphism group. The obtained designs are classified by the type of a block stabilizer. The results are complete, except for the designs with block stabilizers in the fifth Aschbacher\u27s class. In particular, the problem is solved if q is a prime. We include formulas for the number of such designs with q=p2α3β, α,β nonnegative integers