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