An axiomatic projective plane is a point-line incidence structure with the following axioms:

1. any two distinct points are collinear (via a unique line);
2. any two distinct lines meet in a unique point;
3. there exists a 4-gon.

Now consider $P = \mathrm{Proj}(k[x,y,z])$, a projective plane over a commutative ring $k[x,y,z]$ with $k$ a field. (This is just an example, I am in fact interested in any algebro-geometric projective plane over a graded commutative ring.)

Can we detect an axiomatic projective plane in $P$ in a natural way ? In other words: does there arise an axiomatic projective plane in some way ?

An axiomatic projective plane is a point-line incidence structure with the following axioms:

1. any two distinct points are collinear (via a unique line);
2. any two distinct lines meet in a unique point;
3. there exists a 4-gon.

Now consider $P = \mathrm{Proj}(k[x,y,z])$, a projective plane over a commutative ring $k[x,y,z]$ with $k$ a field. Then if we consider the $k$-rational points together with the $k$-rational lines, we obtain an axiomatic projective plane.

Can we also detect an axiomatic projective plane in a general algebro-geometric plane $P = \mathrm{Proj}(A[x,y,z])$, with $A$ a commutative ring, in a natural way ? In other words: does there arise an axiomatic projective plane in some way ?