Wondertopes

TL;DR

The paper introduces wondertopes as log-resolved positive geometries arising from polytopes via sequential blowing up along building sets, and proves that the resulting pair $(X^{\mathcal{B}}, \widetilde{\mathcal{P}}^{\mathcal{B}})$ is a positive geometry with canonical form $\pi_{\mathcal{B}}^*\Omega(\mathbb{P}V, \mathcal{P})$. It develops the foundational semialgebraic framework and the fundamental single-face computation, then extends to sequences of faces using wonderful compactifications, with boundary divisors decomposing as products of smaller wondertopes. The braid arrangement example shows how $M_{0,n+1}$ arises as a complement and yields the curvy associahedron, whose canonical form is the Parke-Taylor form, strengthening the connection between moduli spaces and positive geometry. The work highlights both the geometric structure and combinatorial underpinnings (building sets, nested-set complexes) and lays groundwork for matroidal generalizations of wondertopes. Overall, the results provide a robust method to obtain log-resolutions of positive geometries from polytopes, enriching the toolkit for studying scattering amplitudes and related moduli spaces.

Abstract

Positive geometries were introduced by Arkani-Hamed--Bai--Lam as a method of computing scattering amplitudes in theoretical physics. We show that a positive geometry from a polytope admits a log resolution of singularities to another positive geometry. Our result states that the regions in a wonderful compactification of a hyperplane arrangement complement, which we call wondertopes, are positive geometries. A familiar wondertope is the curvy associahedron, which tiles the moduli space of pointed stable rational curves. Thus our work generalizes the known positive geometry structure on this moduli space.

Figures (10)

• Figure 1: Left: a cube in $\mathbb P^3$ and a $\mathbb P^1$ (green) intersecting the cube along an edge $F$. Right: the cube after blowing-up $\mathbb P^3$ along the $\mathbb P^1$, with $E \cong \mathbb P^1 \times \mathbb P^1$ and $E_{\geq 0} \cong F \times \Delta^1$ (the latter in green). Here, $\Delta^1$ denotes the standard simplex of the projective line (see Proposition \ref{['prop:canonical-form-polytope']}\ref{['prop:canonical-form-general-simplex']} for a definition).
• Figure 2: The lattice of flats $\mathscr L(A_3) \cong \Pi_4$. The flats in the minimal building set $\mathcal{B}^{\textrm{min}}$ are drawn in red.
• Figure 3: A region (a simplex) in $M_{0,5} \cong C(A_3)$ in $\mathbb P^2$ (left) and its wondertope $(\overline{M}_{0,5})_{\geq 0}$ (an associahedron) in $\overline{M}_{0,5}$ (right)
• Figure 4: A triangle with point $p$ (left) and its blow-up (right)
• Figure 5:

Theorems & Definitions (66)

• Definition 1.1
• Definition 1.3
• Theorem 1.4
• Corollary 1.5
• Theorem 1.7
• Definition 2.1
• Proposition 2.2
• Definition 2.3
• Lemma 2.4
• Proposition 2.5