Nonlinear Representations of Baer Subgeometries of 𝘗𝘎(3,𝘲²)

We study the Bruck–Bose representation in \(PG{(6,q)}\) of Baer subgeometries \(PG{(3,q)}\) of \(PG{(3,q²)}\) meeting the plane at infinity in a line \(PG{(1,q)}\). Unlike the canonical Baer subgeometry, whose representation is a linear \(3\)-space, a Baer subgeometry meeting the plane at infinity in a line cannot be represented by a linear \(3\)-space in \(PG{(6,q)}\). We construct a family of \(q + 1\) ruled cubic surfaces with a common affine conic and linear 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 \(PG{(3,q)}\) whose intersection with the plane at infinity is exactly a line \(PG{(1,q)}\). Moreover, the ruled cubic surfaces represent the \(q + 1\) Baer subplanes of a pencil of planes of this subgeometry, with the common affine conic representing their common affine line.