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

Dawei Shen, Emanuele Ventura

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.

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

Abstract

The ABCT variety is the image closure of the rational Veronese map from the Grassmannian to the Grassmannian . It was studied by Arkani-Hamed--Bourjaily--Cachazo--Trnka in the context of tree-level scattering amplitudes arising in planar supersymmetric Yang-Mills theory and Witten's twistor string theory. From this perspective, is conjectured to be a positive geometry by Lam. In this paper, we study the combinatorial and algebraic geometry aspects of and its subvarieties induced by iteratively taking analytic boundaries of the totally nonnegative part. We interpret these subvarieties as point configurations on by the Gelfand-MacPherson correspondence. We construct a top-degree meromorphic form on and show that it is a positive geometry, proving Lam's conjecture.
Paper Structure (26 sections, 45 theorems, 180 equations, 2 figures, 1 table)

This paper contains 26 sections, 45 theorems, 180 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Positive Geometries, Positive Grassmannians, and Positroid Varieties
  3. Positive geometries
  4. Positive Grassmannians and Positroid Varieties
  5. Positroids, Affine permutations, and Grassmann necklaces
  6. The Positive Grassmannian is a Positive Geometry
  7. Actions by $S_n$, $C_n$, $D_n$, and $(\mathbb{C}^*)^{n-1}$
  8. The ABCT Variety
  9. Candidate Canonical Form by Pushing-forward
  10. Pushforwards of differentials
  11. Pushforward of the canonical form on $\mathop{\mathrm{Gr}}\nolimits(2,n)$ to $V(3,n)$ via the rational map $\vartheta$
  12. Boundaries of $V(3,n)$
  13. Decomposition of $(G_{\geq0}\setminus G_{>0})\cap V$
  14. The analytic Boundary of $V(3,n)_{\geq0}$
  15. Two notions of positivity agree
  16. ...and 11 more sections

Key Result

Theorem 1.1

The following properties of the ABCT variety $V(3,n)$ hold true:

Figures (2)

  • Figure 1:
  • Figure 2:

Theorems & Definitions (129)

  • Theorem 1.1
  • Remark 1.2
  • Definition 2.1: Iterated boundary components
  • Definition 2.2: Residue of a form
  • Example 2.3
  • Definition 2.4: Normal Positive Geometry
  • Remark 2.5
  • Definition 2.6: Total Nonnegativity on Grassmannians
  • Theorem 2.7
  • Example 2.8
  • ...and 119 more