Placement of the Desargues configuration on a cubic curve

Autor: Barry Wolk, N. S. Mendelsohn, R. Padmanabhan
Rok vydání: 1991
Předmět:
Zdroj: Geometriae Dedicata. 40
ISSN: 1572-9168
0046-5755
DOI: 10.1007/bf00145912
Popis: 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’ and both could be regarded as self-inscribed and self-circumscribed p9lygons (see [1, p. 128]). Nevertheless, there is one fundamental difference between these two configurations, viz. while the Pappus-Brianchon 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 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.
Databáze: OpenAIRE
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