On (0,α)-Geometries and Dual Semipartial Geometries Fully Embedded in an Affine Space

The (0,α)-geometries fully embedded in a projective space are up to a few open cases classified. For (0,α)-geometries fully embedded in an affine space AG(n,q), less is known. The most important model is the so-called linear representation T _n-1* (k) of a set k of type {0,1,α +1} with respect to lines in the hyperplane at infinity. We will give a characterization of this model. We also investigate the case where the (0,α)-geometry, fully embedded in AG(n,q), is the dual of a semipartial geometry.