Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 59 / 2 / c / ii / Solution

For the compact alpha-plane $A_{\ell }$ and beta plane $B_H$, an intersection point must satisfy $\ell \subset W\subset H$. If $\ell \not\subset H$, that is impossible. If $\ell \subset H$, intermediate two-planes correspond to lines $W/\ell$ in the two-dimensional quotient $H/\ell$. ThereforeThe nonempty intersection is a projective null geodesic. In an affine chart its tangent must be both $u^A{\pi }^{A^{\prime }}$ and ${\lambda }^A{v}^{A^{\prime }}$; these conditions force a multiple of ${\lambda }^A{\pi }^{A^{\prime }}$. Thus its affine portion is a null line, with the indicated projective compactification.