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Rational points of fixed denominator in real toric arrangements

Andrew Hanlon, Davis Painter

TL;DR

This work studies when a rational point of fixed denominator $m$ lies in every stratum of a real toric hyperplane arrangement defined by $A\subset\mathbb{Z}^n$. By viewing strata through lifts to lattice polytopes and proving two elementary lemmas about rescaling and interior lattice points, the authors establish a concrete bound: if $A$ contains a basis of $\\mathbb{R}^n$, and $m$ is a multiple of $D_A$ with $m\\ge (n+1)D_A$, then $L_m$ meets all strata $S\\in\\mathcal{S}_A$. This combinatorial result has toric-geometric consequences: for a smooth toric variety $X_\\Sigma$ with $A$ the primitive generators of $\\Sigma(1)$, the degree $m$ toric Frobenius pushforward $(F_m)_*\\mathcal{O}_{X_\\Sigma}$ contains all possible summands in the Picard group whenever $m=\\ell D_A$ with $\\ell\\ge \dim X_\\Sigma+1$. The paper thus links the stratification of real toric arrangements to Frobenius-splitting phenomena, providing a practical bound and motivating questions about optimality and extensions.

Abstract

We give a sufficient condition on a positive integer $m$ for every stratum of a given real toric hyperplane arrangement to contain a rational point of denominator $m$. As a consequence, we give a sufficient condition on $m$ for the degree $m$ Frobenius pushforward of the structure sheaf on a smooth toric variety to contain all possible summands in the Picard group.

Rational points of fixed denominator in real toric arrangements

TL;DR

This work studies when a rational point of fixed denominator lies in every stratum of a real toric hyperplane arrangement defined by . By viewing strata through lifts to lattice polytopes and proving two elementary lemmas about rescaling and interior lattice points, the authors establish a concrete bound: if contains a basis of , and is a multiple of with , then meets all strata . This combinatorial result has toric-geometric consequences: for a smooth toric variety with the primitive generators of , the degree toric Frobenius pushforward contains all possible summands in the Picard group whenever with . The paper thus links the stratification of real toric arrangements to Frobenius-splitting phenomena, providing a practical bound and motivating questions about optimality and extensions.

Abstract

We give a sufficient condition on a positive integer for every stratum of a given real toric hyperplane arrangement to contain a rational point of denominator . As a consequence, we give a sufficient condition on for the degree Frobenius pushforward of the structure sheaf on a smooth toric variety to contain all possible summands in the Picard group.
Paper Structure (6 sections, 4 theorems, 13 equations, 2 figures)

This paper contains 6 sections, 4 theorems, 13 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Toric hyperplane stratifications
  4. Proof of the main theorem
  5. Application to toric varieties
  6. Examples and further questions

Key Result

Theorem A

Suppose that $A$ contains a basis of $\mathbb R^n$. If $m$ is a multiple of $D_A$ and $m \geq (n+1)D_A$, then $L_m \cap S \neq \emptyset$ for all $S \in \mathcal{S}_A$.

Figures (2)

  • Figure 1: Lifts of the lines determining $\mathcal{S}_A$ for $A = \{e_1, e_2, e_1 + e_2\}$ are depicted on the left with a fundamental domain outlined in red. On the right, the corresponding hyperplane arrangement on the torus is drawn where $L_2$ is red, $L_3$ is blue, and $L_2 \cap L_3$ is green. The top dimensional strata do not contain points in $L_2$.
  • Figure 2: The stratification $\mathcal{S}_A$ on the torus with $A = \{e_1, e_2, e_1 + e_2, -e_1 + 2e_2\}$ is depicted on the left. The zero-dimensional strata are red dots, the one-dimensional strata are solid black lines, and the two-dimensional strata are white regions. On the right, we have the same stratification with $L_{12}$ drawn in blue. There are points in $L_{12}$ in all strata.

Theorems & Definitions (13)

  • Theorem A
  • Corollary B
  • Definition 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • proof : Proof of \ref{['thm:main']}
  • Remark 4.1
  • Remark 4.2
  • ...and 3 more