The group PGL(2,q) has an embedding into PGL(3,q) such that it acts as
the group fixing a nonsingular conic in PG(2,q). This action affords a
coherent configuration R(q) on the set L(q) of non-tangent lines of the
conic. We show that the relations can be described by using the cross-ratio.
Our results imply that the restrictions R+β(q) and Rββ(q) to the sets
L+β(q) of secant lines and to the set Lββ(q) of exterior lines,
respectively, are both association schemes; moreover, we show that the elliptic
scheme Rββ(q) is pseudocyclic.
We further show that the coherent configuration R(q2) with q even allow
certain fusions. These provide a 4-class fusion of the hyperbolic scheme
R+β(q2), and 3-class fusions and 2-class fusions (strongly regular graphs)
of both schemes R+β(q2) and $R_{-}(q^2). The fusion results for the
hyperbolic case are known, but our approach here as well as our results in the
elliptic case are new.Comment: 33 page