The ABCT variety is the image closure of the rational Veronese map between Grassmannians. It was studied by Arkani-Hamed--Bourjaily--Cachazo--Trnka in the context of tree-level scattering amplitudes arising in planar \mathcal{N} = 4 supersymmetric Yang-Mills theory and Witten's twistor string theory. From this perspective, the ABCT variety is conjectured to be a positive geometry by Lam. We study the combinatorial, analytic, and algebro-geometric aspects of the ABCT variety and its face stratification. We interpret the strata as certain point configurations on \mathbb{P}^2, by the Gelfand-MacPherson correspondence. We construct the canonical forms on the ABCT variety and its strata, and we show that they are positive geometries, proving Lam's conjecture. No backgrounds in physics or positive geometry assumed.