Placement of the Desargues configuration on a cubic curve (Q1179061)

The Pappus configuration can be realized as nine points on a nonsingular cubic curve over the complex plane. In contrast, the Desargues configuration can be placed in a projective plane such that its vertices lie on a cubic curve over a field \(k\) if and only if \(k\) has characteristic 2 and contains at least 16 elements. Moreover, any cubic curve containing the vertices of the Desargues configuration must be singular. An example for such a curve is the cuspidal cubic curve \(y^ 2-x^ 3=0\) over the field \(GF(2^ 4)\). As a by-product, the authors obtain the result that the Fano configuration can be placed on a singular cubic curve over \({\mathbb{Z}_ 2}\).