The tangent splash in $\PG(6,q)$

Abstract

Let B be a subplane of PG(2,q^3) of order q that is tangent to $\ell_\infty$. Then the tangent splash of B is defined to be the set of q^2+1 points of $\ell_\infty$ that lie on a line of B. In the Bruck-Bose representation of PG(2,q^3) in PG(6,q), we investigate the interaction between the ruled surface corresponding to B and the planes corresponding to the tangent splash of B. We then give a geometric construction of the unique order-$q$-subplane determined by a given tangent splash and a fixed order-$q$-subline.