= Twistor correspondence
{title2=$\operatorname{Gr}(2,\mathbb C^4)\leftarrow F_{1,2}(\mathbb C^4)\rightarrow\mathbb{CP}^3$}
The <flag manifold> of a line $\ell$ contained in a two-plane $W$ has projections $(\ell,W)\mapsto W$ and $(\ell,W)\mapsto\ell$. Their fibers are respectively $\mathbb{CP}^1$ and $\mathbb{CP}^2$. This is the flat <twistor correspondence>: <complexified compactified Minkowski space> is the <Grassmannian> of two-planes in <twistor space>, while <projective twistor space> is its projective space of lines.
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