Combinatorial geometries of field extensions

TL;DR

This paper classifies the algebraic combinatorial geometries of field extensions with transcendence degree over 4 and describes their automorphism groups, extending previous results to broader classes of fields.

Contribution

It provides a comprehensive classification of combinatorial geometries for arbitrary field extensions and describes their automorphism groups, generalizing earlier work to non-algebraically closed fields.

Findings

Classified algebraic combinatorial geometries for field extensions with transcendence degree > 4

Described automorphism groups of these geometries

Extended classification results to arbitrary fields of characteristic zero

Abstract

We classify the algebraic combinatorial geometries of arbitrary field extensions of transcendence degree greater than 4 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. The classification of projective planes in algebraic combinatorial geometries in arbitrary fields of characteristic zero will also be given.