Varieties of Combinatorial Geometries

A hereditary class of (finite combinatorial) geometries is a collection of geometries which is closed under taking minors and direct sums. A sequence of universal models for a hereditary class 'S of geometries is a sequence (T„ ) of geometries in ?T with rank Tn = n, and satisfying the universal property: if G is a geometry in 5" of rank n, then G is a subgeometry of T„. A variety of geometries is a hereditary class with a sequence of universal models. We prove that, apart from two degenerate cases, the only varieties of combinatorial geometries are ( 1 ) the variety of free geometries, (2) the variety of geometries coordinatizable over a fixed finite field, and (3) the variety of voltage-graphic geometries with voltages in a fixed finite group.