Analysing 2- (v, k, 2) designs admitting a flag-transitive almost simple automorphism group with socle PSL (2, q) by means of conics and hyperovals of PG (2, q)

The classification of the 2-designs with =2 admitting a flag-transitive automorphism groups with socle PSL (2, q) is completed by settling the two open cases in ABDT. The result is achieved by using conics and hyperovals of PG (2, q).