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Positive del Pezzo Geometry

Nick Early, Alheydis Geiger, Marta Panizzut, Bernd Sturmfels, Claudia He Yun

TL;DR

This work develops a comprehensive positive geometry framework for del Pezzo surfaces and their moduli spaces $Y(3,n)$ by introducing pezzotopes as the Weyl group orbits of their real components. It constructs canonical forms and scattering amplitudes in the CHY and CEGM style, derives Euler characteristics and ML degrees, and uncovers deep connections to Weyl groups $W(E_6)$ and $W(E_7)$, tropicalizations, and Grassmannian approaches. The results unify real, complex, and tropical perspectives through explicit polygonal decompositions, combinatorial pezzotopes, and parametric models, culminating in a positive geometry structure for $Y_+(3,6)$ and conjectural extensions to $Y_+(3,7)$. The work advances the interface of algebraic geometry, combinatorics, and high energy physics by providing concrete canonical forms, amplitudes, and a framework for future exploration of higher del Pezzo moduli and their positive geometric properties.

Abstract

Real, complex, and tropical algebraic geometry join forces in a new branch of mathematical physics called positive geometry. We develop the positive geometry of del Pezzo surfaces and their moduli spaces, viewed as very affine varieties. Their connected components are derived from polyhedral spaces with Weyl group symmetries. We study their canonical forms and scattering amplitudes, and we solve the likelihood equations.

Positive del Pezzo Geometry

TL;DR

This work develops a comprehensive positive geometry framework for del Pezzo surfaces and their moduli spaces by introducing pezzotopes as the Weyl group orbits of their real components. It constructs canonical forms and scattering amplitudes in the CHY and CEGM style, derives Euler characteristics and ML degrees, and uncovers deep connections to Weyl groups and , tropicalizations, and Grassmannian approaches. The results unify real, complex, and tropical perspectives through explicit polygonal decompositions, combinatorial pezzotopes, and parametric models, culminating in a positive geometry structure for and conjectural extensions to . The work advances the interface of algebraic geometry, combinatorics, and high energy physics by providing concrete canonical forms, amplitudes, and a framework for future exploration of higher del Pezzo moduli and their positive geometric properties.

Abstract

Real, complex, and tropical algebraic geometry join forces in a new branch of mathematical physics called positive geometry. We develop the positive geometry of del Pezzo surfaces and their moduli spaces, viewed as very affine varieties. Their connected components are derived from polyhedral spaces with Weyl group symmetries. We study their canonical forms and scattering amplitudes, and we solve the likelihood equations.
Paper Structure (10 sections, 19 theorems, 89 equations, 4 figures, 1 table)

This paper contains 10 sections, 19 theorems, 89 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Blowing Up Four Points
  3. Polygons
  4. Euler Characteristic
  5. Numerical Experiments
  6. Weyl Groups, Roots, and their ML Degrees
  7. Intermezzo from Physics
  8. Combinatorics of Pezzotopes
  9. Geometry of Pezzotopes
  10. Grassmannians, Positive Geometries, and Beyond

Key Result

Theorem \ref{thm:euler}

The Euler characteristic of the complex moduli space $Y(3,n)$ is $32$ for $n=6$ and it is $3600$ for $n= 7$. For $n=8$, a numerical computation gives $4884387$ as a lower bound for the Euler characteristic.

Figures (4)

  • Figure 1: The real moduli space $Y(3,6)$ is glued from $432$ copies of a simple $4$-polytope with $f= (45,90,60,15)$. The picture shows its edge graph. The amplitude is a rational function, given as the sum of $45$ reciprocal monomials, one for each vertex. Singularities correspond to facets: five cubes (in color) and ten associahedra. The Weyl group $W(E_6)$ acts on this data.
  • Figure 2: Subdivision of the del Pezzo surface $\mathcal{S}_5$ into $20$ quadrilaterals (dark) and $16$ pentagons (light). Each blown up point is replaced by a decagon with opposite sides identified.
  • Figure 3: The triangles in these line arrangements are the vertex labels in Figure \ref{['fig:drei']}.
  • Figure 4: Two graphs that reveal the combinatorics of the pezzotopes for $n=6$ (left) and $n=7$ (right). These diagrams are color-enhanced reproductions of those in HKT.

Theorems & Definitions (41)

  • Theorem \ref{thm:euler}
  • Theorem \ref{thm:2polytopes}
  • Theorem \ref{thm:Y3nPosGeom}
  • Proposition 2.1
  • Theorem 3.1
  • proof
  • Example 3.2
  • Remark 3.3
  • Remark 3.4
  • Example 3.5: Clebsch cubic
  • ...and 31 more