Department of Mathematics
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It is well-known that the familiar 9-point Pappus configuration can be realized as nine points on a nonsingular cubic curve over the complex field, but here we prove that the equally familiar 10-point Desargues configuration cannot be so represented. In fact, the Desargues configuration can be realized as ten points on a plane cubic curve if and only if the cubic is singular, the underlying field has characteristic 2 and the field has at least 16 elements. In that case, the curve also contains copies of the 7-point Fano configuration, the only other classically known forbidden design. Notice that the Fano configuration is a (finite) projective plane. Here we prove that no finite projective plane can be so realized on an elliptic curve over the complex field.

These results were obtained in collaboration with N.S. Mendelsohn and Barry Wolk.