Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 59 / 2 / b

A point $q\in \mathbb{PT}$ specifies a one-dimensional subspace ${\ell }_q\subset \mathbb{T}$. Its correspondence fiber consists of two-planes $W$ containing ${\ell }_q$. Passing to the quotient gives a bijection, with inverse taking the inverse image of a line,Since ${\dim }_{\mathbb{C}}(\mathbb{T}/{\ell }_q)=3$, this is an isomorphism of Complex projective spaces. ConsequentlyIt is the compact alpha-plane associated with $q$. In an affine chart with $\pi \ne 0$, the solutions of the twistor incidence relation are $x^{A{A}^{\prime }}=x_0^{A{A}^{\prime }}+u^A{\pi }^{A^{\prime }}$, where $u^A$ supplies two complex coordinates. That affine ${\mathbb{C}}^2$ is only an open subset of the compact ${\mathbb{CP}}^2$; distinguishing them is essential to the printed compact correspondence.