Desargues configurations with four self-conjugate points View Full Text


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Article Info

DATE

2018-09

AUTHORS

Aiden A. Bruen, James M. McQuillan

ABSTRACT

In a projective plane over a field F, the diagonal points of a quadrangle are collinear if and only if F has characteristic 2. Such a quadrangle together with its diagonal points and the lines connecting these points form the subplane of order 2, called a Fano plane. Using Desargues configurations and polarities, we provide a similar type of synthetic criterion and construction for characteristic 3 fields. Let F be a field with characteristic not equal to 2. From any quadrangle and one of its diagonal points V, we construct a pair of triangles Δ1,Δ2 in perspective from V, and the resulting Desargues configuration D such that the vertices of Δ1 are self-conjugate under a particular polarity. For this Desargues configuration D, the vertex of perspectivity V of the pair Δ1,Δ2 is a fourth self-conjugate point if and only if F has characteristic 3. If F has characteristic 3, then the 10 points and 10 lines of D together with three additional points and three additional lines yield a projective subplane of order 3 of π. More... »

PAGES

837-844

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/s40879-018-0274-5

DOI

http://dx.doi.org/10.1007/s40879-018-0274-5

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1106023797


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