On \((0,{\alpha})\)-geometries and dual semipartial geometries fully embedded in an affine space
A \((0,\alpha)\)-geometry \(\mathcal{S}\) is a connected point-line geometry with the property that two distinct lines meet in at most one point, the number of points on a line (lines through a point) is constant, and every point \(x\) is collinear with \(0\) or \(\alpha\) points on a given line \(L\) not through \(x\) (and both values occur). If the points and lines of \((0,\alpha)\)-geometry \(\mathcal{S}\) with \(q\) points on each line, are some points and some lines, respectively, of a \(d\)-dimensional affine space AG\((d,q)\), with natural incidence, then \(\mathcal{S}\) is said to be fully embedded in AG\((d,q)\). A full embedding is, by definition, a linear representation if for every line \(L\) of \(\mathcal{S}\), every line of AG\((d,q)\) parallel to \(L\) also belongs to \(\mathcal{S}\). The main result of the paper under review states that a fully embedded \((0,\alpha)\)-geometry \(\mathcal{S}\) in AG\((d,q)\) is a linear representation if two intersecting lines never generate a sub-\((0,2)\)-geometry with \(2\) lines through each point. In particular, this is true whenever \(\alpha>2\). This might be seen as a first major step towards a classification of affine full embeddings of \((0,\alpha)\)-geometries, in particular of semipartial geometries. But the authors consider a classification of linear representations out of reach. However, if \(\mathcal{S}\) is the dual of a (automatically proper) semipartial geometry, i.e., the number of lines meeting two given nonconcurrent lines is a constant, then the authors show that \(\mathcal{S}\) cannot be fully embedded in any (finite) affine space unless \(\alpha=1\), and in this case, if the embedding is a linear representation, then the set of points at infinity of the affine lines of \(\mathcal{S}\) is not an arc.
scientific article; zbMATH DE number 2091508