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Strata of toric hyperplane arrangements, zonotope lattice points, and the Bondal-Thomsen collection

Friedrich Bauermeister, Andrew Hanlon, Davis Painter, Sair Shaikh, Benjamin Singer

TL;DR

The authors establish a precise combinatorial bridge between strata of oriented toric hyperplane arrangements and lattice points in a half-open zonotope $Z$, showing a bijection between $\Phi$-strata of $\mathbb{R}^n/\varphi^{-1}(\mathbb{Z}^k)$ and $Z\cap\pi(\mathbb{Z}^k)$, with a dimension relation tying stratum dimension to the minimal face containing the corresponding lattice point. The key tool is the map $\pi\circ\Phi$, whose injectivity and surjectivity are proved using convex-geometric and rank-nullity arguments, yielding $\dim F_p + \dim S = k - |J_S| + \dim(\ker\varphi \cap \mathrm{span}(\widetilde S - u))$. In the toric setting, this combinatorial picture recasts Bondal–Thomsen generators of the derived category as indexed by lattice points in $-Z$, linking the Bondal stratification to the effective cone and GKZ secondary fan. The results provide a purely combinatorial lens on the Bondal–Thomsen construction, with homological implications in the Cox category and potential extensions to toric subvarieties via restricted stratifications and subfans.

Abstract

We show that strata of oriented toric hyperplane arrangements are in bijection with a collection of lattice points in a zonotope. Moreover, we relate the dimension of the stratum and the dimension of the minimal face of the zonotope containing the corresponding lattice point. We discuss how this correspondence is related to toric varieties and the Bondal-Thomsen generators of their derived categories.

Strata of toric hyperplane arrangements, zonotope lattice points, and the Bondal-Thomsen collection

TL;DR

The authors establish a precise combinatorial bridge between strata of oriented toric hyperplane arrangements and lattice points in a half-open zonotope , showing a bijection between -strata of and , with a dimension relation tying stratum dimension to the minimal face containing the corresponding lattice point. The key tool is the map , whose injectivity and surjectivity are proved using convex-geometric and rank-nullity arguments, yielding . In the toric setting, this combinatorial picture recasts Bondal–Thomsen generators of the derived category as indexed by lattice points in , linking the Bondal stratification to the effective cone and GKZ secondary fan. The results provide a purely combinatorial lens on the Bondal–Thomsen construction, with homological implications in the Cox category and potential extensions to toric subvarieties via restricted stratifications and subfans.

Abstract

We show that strata of oriented toric hyperplane arrangements are in bijection with a collection of lattice points in a zonotope. Moreover, we relate the dimension of the stratum and the dimension of the minimal face of the zonotope containing the corresponding lattice point. We discuss how this correspondence is related to toric varieties and the Bondal-Thomsen generators of their derived categories.

Paper Structure

This paper contains 7 sections, 7 theorems, 24 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Combinatorial setup and result
  3. Connection to toric geometry
  4. Acknowledgements
  5. Proof of \ref{['thm:main']}
  6. Faces from restriction
  7. Boundary Bondal-Thomsen elements

Key Result

Theorem A

There is a bijection between the $\Phi$-strata of $\mathbb R^n/\varphi^{-1}(\mathbb Z^k)$ and the lattice points of $Z$. Moreover, the assignment of a minimal face $F_p$ of $Z$ containing the point $p \in Z$ corresponding to a stratum $S$ is entirely determined by $J_S$, is inclusion-reversing with for any lift $\widetilde{S}$ of $S$ to $\mathbb R^n$ and any $u \in \widetilde{S}$.

Figures (2)

  • Figure 1: $A$ from \ref{['ex:hirz2']} depicted on the left. The corresponding toric hyperplane arrangement is depicted in the middle with "hairs" to designate the orientations. The zonotope is depicted on the right with each of the five lattice points of $Z$ colored the same as the corresponding stratum.
  • Figure 2: $A$ from \ref{['ex:blhirz2']} is depicted on the left. The corresponding hyperplane arrangement is depicted in the middle with "hairs" to designate the orientations. The eight strata are depicted on the right with the two dimensional strata in gray, the one dimensional stratum in red, and the zero dimensional stratum in black.

Theorems & Definitions (20)

  • Theorem A
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Example 1.4
  • Example 1.5
  • Corollary B
  • Proposition 2.1
  • proof
  • Lemma 2.2: rockafellar1997convex
  • ...and 10 more