The geometry of arithmetic noncommutative projective lines

Let k   be a perfect field and let K/kK/k be a finite extension of fields. An arithmetic noncommutative projective line is a noncommutative space of the form ProjSK(V), where V be a k-central two-sided vector space over K   of rank two and SK(V)SK(V) is the noncommutative symmetric algebra generated by V over K defined by M. Van den Bergh [26]. We study the geometry of these spaces. More precisely, we prove they are integral, we classify vector bundles over them, we classify them up to isomorphism, and we classify isomorphisms between them. Using the classification of isomorphisms, we compute the automorphism group of an arithmetic noncommutative projective line.