Alpha-plane ($A_{\ell }=\{W:\ell \subset W\}\cong {\mathbb{CP}}^2$)

An alpha-plane in the compact twistor correspondence is the set of two-planes containing a fixed line $\ell$ in twistor space. Its affine portion has tangent vectors $u^A{\pi }^{A^{\prime }}$ for fixed primed spinor $\pi$, so any two tangent vectors are orthogonal and null. Two distinct compact alpha-planes meet at the unique two-plane spanned by their defining lines; that point can lie at conformal infinity.