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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References
J. R. Bastida,Field Extensions and Galois Theory, Addison—Wesley 1984.
H. Crapo andG.-C. Rota,Combinatorial Geometries (prel. ed.) M. I. T. Press, 1970.
A. W. Ingleton andR. A. Main, Non-algebraic matroids exist,Bull. London Math. Soc. 7 (1975), 144–146.
B. Lindström, The non-Pappus Matroid is algebraic,Ars Combinatoria 16 B (1983), 95–96.
B. Lindström, A simple non-algebraic matroid of rank three,Utilitas Mathematica 25 (1984), 95–97.
L. Lovász,private communication.
S. Mac Lane, A lattice formulation for transcendence degrees andp-bases,Duke Math. J. 4 (1938), 455–468.
D. J. A. Welsh,Matroid Theory, Academic Press, 1976.