Schemes of relations of the finite projective plane, and their extensions

There are several schemes (coherent configurations) associated with a finite projective plane $\mathcal P$. In the paper, a new scheme is constructed, which, in a sense, contains all of them. It turns out that this scheme coincides with the 2-extension of the nonhomogeneous scheme of $\mathcal P$, and is uniquely determined up to similarity by the order $q$ of $\mathcal P$. Moreover, for $q\ge 3$ the rank of the scheme does not depend on $q$ and equals 416. The results obtained have interesting applications in the theory of multidimensional extensions of schemes and similarities.