Séminaire Lotharingien de Combinatoire, 95B.82 (2026), 12 pp.

Dawei Shen and Emanuele Ventura

The ABCT Variety is a Positive Geometry

Abstract. The ABCT variety V(3,n) is the image closure of the rational Veronese map between Grassmannians. It was studied by Arkani-Hamed, Bourjaily, Cachazo, and Trnka in 2011 in the context of MHV and NMHV scattering amplitudes in planar N=4 supersymmetric Yang-Mills theory. From this perspective, V(3,n), along with its totally nonnegative part, is conjectured to be a positive geometry by Lam in 2024.

In this paper, we study the combinatorics and algebraic geometry of V(3,n) and its subvarieties, which arise from iteratively taking the Zariski closures of Euclidean boundaries of the totally nonnegative parts. We draw connections to point configurations on P² via the Gelfand-MacPherson correspondence and prove that V(3,n) is a positive geometry, thereby establishing Lam's conjecture.

Received: November 15, 2025. Accepted: February 15, 2026. Final version: April 1, 2026.

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