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Rings and Modules Seminar
~ Abstracts ~

R. Padmanabhan, University of Manitoba
padman(at)cc(dot)umanitoba(dot)ca

Department of Mathematics
University of Manitoba

Tuesday, January 21, 2014

Desargues Configuration as a Cubic Cluster
Abstract:

Hilbert and Cohn-Vossen once declared that the configurations of Desargues and Pappus are by far the most important projective configurations. These two are very similar in many respects: both are regular and self-dual, both could be constructed with ruler alone and hence exist over the rational plane, the final collinearity in both instances are automatic (in the sense that they are theorems over any field). Nevertheless, there is one fundamental difference between these two configurations, viz. while the 9-point Pappus configuration can be realized as nine points on a non-singular cubic curve over the complex plane (in doubly infinite ways), it is impossible to get such a representation for the 10-point Desargues configuration. In fact, the configuration of Desargues can be placed in a projective plane in such a way that its vertices lie on a cubic curve over a field k if and only if k is of characteristic 2 and has at least 16 elements. Moreover, any cubic curve containing the vertices of this configuration must be singular.

  1. Hilbert, D and Cohen-Vossen S. Geometry and Imagination, New York, 1952.
  2. Feld, J.M. Configurations inscriptable in cubic curves, Amer. Math. Monthly, 43 (1946), 413-455.
  3. Mendelsohn, N.S., Padmanabhan, R and Wolk, B. Desargues configuration on a cubic curve, Geom. Dedicata 40 (1991), 165Ð170.

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