Abstract
We study the Bruck–Bose representation in \(PG{(6,q)}\) of Baer subgeometries \(PG{(3,q)}\) of \(PG{(3,q^{2})}\) meeting the plane at infinity in a line \(PG{(1,q)}\). We construct a family of \(q + 1\) ruled cubic surfaces with a common affine conic and minimal directrices forming a regulus of the Desarguesian spread. We prove that their union, together with the \(q + 1\) corresponding improper points, represents a Baer subgeometry \(PG{(3,q)}\) of \(PG{(3,q^{2})}\) whose intersection with the plane at infinity is exactly a line \(PG{(1,q)}\).