Desargues configurations with four self-conjugate points

Abstract

In a projective plane over a field F, the diagonal points of a quadrangle are collinear if and only if F has characteristic 2. Such a quadrangle together with its diagonal points and the lines connecting these points form the subplane of order 2, called a Fano plane. Using Desargues configurations and polarities, we provide a similar type of synthetic criterion and construction for characteristic 3 fields. Let F be a field with characteristic not equal to 2. From any quadrangle and one of its diagonal points V, we construct a pair of triangles $\Delta _1,\Delta _2$ in perspective from V, and the resulting Desargues configuration D such that the vertices of $\Delta _1$ are self-conjugate under a particular polarity. For this Desargues configuration D, the vertex of perspectivity V of the pair $\Delta _1,\Delta _2$ is a fourth self-conjugate point if and only if F has characteristic 3. If F has characteristic 3, then the 10 points and 10 lines of D together with three additional points and three additional lines yield a projective subplane of order 3 of $\pi$.