Constructions of rank five geometries for the Mathieu group M 22

We construct nine rank five incidence geometries that are firm and residually connectedand on which the Mathieu group M²² acts flag-transitively. The constructions usemainly objects arising from the Steiner systemS(3, 6, 22).One of these geometries was constructed by Meixner and Pasini in [10]. Three of themare obtained from the geometry of Meixner and Pasini using doubling (see [8] or [12]) or similar constructions. The remaining five are new and four of them have a star diagram. Theselatter four geometries are constructed using special partitions of the 22 points of the Steiner system S(3, 6, 22).