Linear representations of semipartial geometries (Q2471468)

According to \textit{I. Debroey} and \textit{J. A. Thas} [J. Comb. Theory, Ser. A 25, 242--250 (1978; Zbl 0399.05012)] a semipartial geometry spq\((s,t\alpha,\mu)\) is a connected partial linear space of order \( (s,t)\) such that (1) for any anti-flag \((p,L)\) there are either \(0\) or \(\alpha > 0\) points collinear with \(p\) and incident with \(L\), and (2) for any two non-collinear points there are \(\mu > 0 \) points collinear with both of them. A complete classification of semipartial geometries that are embeddable in some affine plane AG\((2,q)\) or some affine space AG\((3,q)\) has been obtained by the authors in [Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 64, 147--151 (1978; Zbl 0425.51005)]. In 2004 \textit{F. de Clerck} and \textit{M. Delanote} [Des. Codes Cryptography 32, No.~1--3, 103--110 (2004; Zbl 1056.51005)] and a year later \textit{N. de Feyter} [Adv. Geom. 5, No.~2, 279--292 (2005; Zbl 1076.51002)] showed that any semipartial geometry with \(\alpha > 1\) embeddable into some affine space AG\((n,q\)) is either a linear representation or a known example. In the manuscript under review the author derives conditions on the parameters of a semipartial geometry that ensure it has a linear representation. Moreoever, all linear representations of semipartial geometries in AG\((4,q)\) are determined. This gives a complete classification of semipartial geometries in AG\((4,q)\) for \(\alpha > 1\).