Linear representations of semipartial geometries

Abstract

Semipartial geometries (SPG) were introduced in 1978 by Debroey and Thas. 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. Later on it was shown that if an SPG, with $\alpha>1$, is embedded in affine space it is either a linear representation or $\mathrm{TQ}(4,2^h)$. 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 $\alpha>1$.