Subplanes of $PG\left(\mathrm{2,}{q}^3\right)$ and the Ruled Varieties $V_2^5$ of $PG\left(\mathrm{6,}q\right)$

In this note we study subplanes of order q of the projective plane $\Pi =PG\left(\mathrm{2,}{q}^3\right)$ and the ruled varieties $V_2^5$ of $\Sigma =PG\left(\mathrm{6,}q\right)$ using the spatial representation of Π in Σ, by fixing a hyperplane ${\Sigma }^{\prime }$ with a regular spread of planes. First are shown some configurations of the affine q-subplanes. Then to prove that a variety $V_2^5$ of Σ represents a non-affine subplane of order q of Π, after having shown basic incidence properties of it, such a variety $V_2^5$ is constructed by choosing appropriately the two directrix curves in two complementary subspaces of Σ. The result can be translated into further incidence properties of the affine points of $V_2^5$