Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 59 / 3 / c

The special linear group acts on the twistor correspondence by $g\cdot (\ell ,W)=(g\ell ,gW)$. Since an invertible linear map preserves inclusion, this is again an incident flag. The two projections commute with that action:This is precisely equivariance: projecting a transformed flag gives the same result as transforming its projection. In particular $g$ sends the twistor line of a spacetime point to the line of the transformed point and sends the alpha-plane of a twistor to that of its transformed twistor. The incidence geometry is preserved, even when a transformation takes a point out of the chosen affine Minkowski chart.