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Positive Charts of Toric Varieties

Veronica Calvo Cortes, Simon Telen

TL;DR

This work provides a constructive toric framework for positive geometry by producing positive charts on any smooth projective toric variety X_Sigma. Given a smooth polytope P with Sigma = Sigma_P, the authors build an affine chart Y inside the Cox coordinate space such that Y_{>=0} maps bijectively to the nonnegative part (X_Sigma)_{>=0}, via a positive rational parametrization whose Newton polytope recovers Sigma. The construction uses a unimodular homogenization M built from k Laurent polynomials f_i with nonnegative coefficients, yielding explicit equations f_i^h(y) = 1 that define Y in a Cox-ring setting. They develop associated moment maps mu_{Y,s} that realize Y_{>=0} as a polytope P(s) and relate fibers to scattering/critical-point equations, connecting toric geometry with the algebraic underpinnings of positive geometry and u-equations. The approach is illustrated through numerous examples (including P^1×P^1, pentagons, hexagons, and the permutohedron) and supported by computational implementations, offering a bridge between toric methods and positive-geometric frameworks in physics and moduli theory.

Abstract

We construct affine charts of a smooth projective toric variety which contain its nonnegative points, and which admit a closed embedding into the total coordinate space of Cox's quotient construction. We show that such positive charts arise from smooth subcones of the nef cone. To each positive chart we associate an algebraic moment map, the fibers of which are the critical points of a monomial function in Cox coordinates. This work provides a toric framework for the theory of $u$-equations in positive geometry.

Positive Charts of Toric Varieties

TL;DR

This work provides a constructive toric framework for positive geometry by producing positive charts on any smooth projective toric variety X_Sigma. Given a smooth polytope P with Sigma = Sigma_P, the authors build an affine chart Y inside the Cox coordinate space such that Y_{>=0} maps bijectively to the nonnegative part (X_Sigma)_{>=0}, via a positive rational parametrization whose Newton polytope recovers Sigma. The construction uses a unimodular homogenization M built from k Laurent polynomials f_i with nonnegative coefficients, yielding explicit equations f_i^h(y) = 1 that define Y in a Cox-ring setting. They develop associated moment maps mu_{Y,s} that realize Y_{>=0} as a polytope P(s) and relate fibers to scattering/critical-point equations, connecting toric geometry with the algebraic underpinnings of positive geometry and u-equations. The approach is illustrated through numerous examples (including P^1×P^1, pentagons, hexagons, and the permutohedron) and supported by computational implementations, offering a bridge between toric methods and positive-geometric frameworks in physics and moduli theory.

Abstract

We construct affine charts of a smooth projective toric variety which contain its nonnegative points, and which admit a closed embedding into the total coordinate space of Cox's quotient construction. We show that such positive charts arise from smooth subcones of the nef cone. To each positive chart we associate an algebraic moment map, the fibers of which are the critical points of a monomial function in Cox coordinates. This work provides a toric framework for the theory of -equations in positive geometry.
Paper Structure (8 sections, 9 theorems, 65 equations, 2 figures, 1 algorithm)

This paper contains 8 sections, 9 theorems, 65 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work.
  3. Outline.
  4. Toric preliminaries
  5. Constructing positive charts
  6. Examples
  7. Moment maps
  8. Acknowledgements.

Key Result

Theorem 1.1

Let $\Sigma$ be the normal fan of a smooth $d$-dimensional lattice polytope $P \subset \mathbb{R}^d$. There exists an affine variety $Y\subset \mathbb{C}^{\Sigma(1)} \setminus Z(\Sigma)$ and an affine open subset $U \subset X_\Sigma$ such that

Figures (2)

  • Figure 1: A pentagon and its normal fan.
  • Figure 2: Commutative diagram illustrating the definition of $Y$.

Theorems & Definitions (41)

  • Theorem 1.1
  • Definition 1.2
  • Example 1.3
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • ...and 31 more