Submission date: 14 March 2007

Abstract:

We classify projective planes in algebraic combinatorial geometries in arbitrary fields of characteristic zero. We investigate the first-order theories of such geometries and pregeometries. Then we classify the algebraic combinatorial geometries of arbitrary field extensions of the transcendence degree >=5 and describe their groups of automorphisms. Our results and proofs extend similar results and proofs by Evans and Hrushovski in the case of algebraically closed fields.

This paper has now appeared, under the title Combinatorial geometries of field extensions, in Bull. Lond. Math. Soc. 40 (2008), no. 5, 789 - 800.

Mathematics Subject Classification: Primary 03C98, 51D20; Secondary 12F20, 05B35

Keywords and phrases: Combinatorial geometry, full algebraic matroid, projective plane, transcendental field extension

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