A desarguesian theorem for algebraic combinatorial geometries |
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Authors: | B Lindström |
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Institution: | (1) Department of Mathematics, University of Stockholm, Box 6701, S-113 85 Stockholm, Sweden |
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Abstract: | The points of an algebraic combinatorial geometry are equivalence classes of transcendentals over a fieldk; two transcendentals represent the same point when they are algebraically dependent overk. The points of an algebraically closed field of transcendence degree two (three) overk are the lines (resp. planes) of the geometry.
We give a necessary and sufficient condition for two coplanar lines to meet in a point (Theorem 1) and prove the converse
of Desargues’ theorem for these geometries (Theorem 2). A corollary: the “non-Desargues” matroid is non-algebraic.
The proofs depend on five properties (or postulates). The fifth of these is a deep property first proved by Ingleton and Main
3] in their paper showing that the Vámos matroid is non-algebraic. |
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Keywords: | 05 B 35 50 D 50 12 F 20 |
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