The ABCT Variety $V(3,n)$ is a Positive Geometry

TL;DR

This paper proves that the ABCT variety V(3,n), arising from Grassmannians and relevant in scattering amplitudes, is a positive geometry by analyzing its algebraic and combinatorial structure and constructing a suitable meromorphic form.

Contribution

The paper confirms Lam's conjecture that V(3,n) is a positive geometry by studying its boundary structures and constructing a top-degree form.

Findings

V(3,n) is a positive geometry.

Constructed a top-degree meromorphic form on V(3,n).

Connected subvarieties to point configurations on projective plane.

Abstract

The ABCT variety $V(3,n)$ is the image closure of the rational Veronese map from the Grassmannian $\operatorname{Gr}(2,n)$ to the Grassmannian $\operatorname{Gr}(3,n)$. 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, $V(3,n)$ is conjectured to be a positive geometry by Lam. In this paper, we study the combinatorial and algebraic geometry aspects of $V(3,n)$ and its subvarieties induced by iteratively taking analytic boundaries of the totally nonnegative part. We interpret these subvarieties as point configurations on $\mathbb{P}^2$ by the Gelfand-MacPherson correspondence. We construct a top-degree meromorphic form on $V(3,n)$ and show that it is a positive geometry, proving Lam's conjecture.