Linear representations of semipartial geometries

Semipartial geometries (SPG) were introduced in 1978 by Debroey and Thas [5]. As some of the examples they provided were embedded in affine space it was a natural question to ask whether it was possible to classify all SPG embedded in affine space. In AG(2, q) and AG(3, q) a complete classification was obtained ([6]). Later on it was shown that if an SPG, with α > 1, is embedded in affine space it is either a linear representation or TQ(4, 2h) (see [8],[11]). In this paper we derive general restrictions on the parameters of an SPG to have a linear representation and classify the linear representations of SPG in AG(4, q), hence yielding the complete classification of SPG in AG(4, q), with α > 1.