Desarguesian Klingenberg planes

Abstract:

Klingenberg planes are generalizations of Hjelmslev planes. If R is a local ring, one can construct a projective Klingenberg plane ${\textbf {V}}(R)$ and a derived affine Klingenberg plane ${\textbf {A}}(R)$ from R. If V is a projective Klingenberg plane, if ${R_1}, {R_2}$ and ${R_3}$ are local rings, if ${s_1}, {s_2}$ and ${s_3}$ are the sides of a nondegenerate triangle in V, and if each of the derived affine Klingenberg planes $\mathcal {a}\left ( {V, {s_i}} \right )$ is isomorphic to ${\textbf {A}}({R_i}),$, $i = 1, 2, 3$, then the rings ${R_1}, {R_2}$ and ${R_3}$ are isomorphic, and V is isomorphic to ${\textbf {V}}({R_1});$; also, if g is a line of V, then the derived affine Klingenberg plane $\mathcal {a}({V, g})$ is isomorphic to $\textbf {A}({R_1})$. Examples are given of projective Klingenberg planes V, each of which has the following two properties: (1) V is not isomorphic to ${\textbf {V}}(R)$ for any local ring R; and (2) there is a flag $(B, b)$ of V, and a local ring S such that each derived affine Klingenberg plane $\mathcal {a}({V, m})$ is isomorphic to ${\textbf {A}}(S)$ whenever $m = b$, or m is a line through B which is not neighbor to b.