The ABCT Variety $V(3,n)$ is a Positive Geometry
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| Format: | Preprint |
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2026
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| _version_ | 1866914381041762304 |
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| author | Shen, Dawei Ventura, Emanuele |
| author_facet | Shen, Dawei Ventura, Emanuele |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_09365 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The ABCT Variety $V(3,n)$ is a Positive Geometry Shen, Dawei Ventura, Emanuele Combinatorics High Energy Physics - Theory Algebraic Geometry Primary: 05E14, Secondary: 14J81, 14P10, 14N05, 14N20 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. |
| title | The ABCT Variety $V(3,n)$ is a Positive Geometry |
| topic | Combinatorics High Energy Physics - Theory Algebraic Geometry Primary: 05E14, Secondary: 14J81, 14P10, 14N05, 14N20 |
| url | https://arxiv.org/abs/2603.09365 |