= Solution
A point $q\in\mathbb P\mathbb T$ 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,
$$
\{W:\ell_q\subset W,\ \dim W=2\}\longleftrightarrow\mathbb P(\mathbb T/\ell_q),\qquad W\longmapsto W/\ell_q.
$$
Since $\dim_{\mathbb C}(\mathbb T/\ell_q)=3$, this is an isomorphism of <Complex projective spaces>. Consequently
$$
\boxed{\widetilde q\cong\mathbb{CP}^2.}
$$
It is the compact <alpha-plane> associated with $q$. In an affine chart with $\pi\ne0$, the solutions of the <twistor incidence relation> are $x^{AA'}=x_0^{AA'}+u^A\pi^{A'}$, 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.
Back to article page
