Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Canonical Lattices and Integer Relations Associated to Rational Fans

Rizwan Jahangir

TL;DR

This work develops a canonical local-to-global lattice theory for rational fans by defining the ray lattice $L_{\mathrm{rays}}(\Sigma)$ and the relation lattice $L_{\mathrm{rel}}(\Sigma)$ as functorial invariants under fan isomorphisms. It introduces star-local lattices via localized quotient fans and a codimension filtration $F_k L_{\mathrm{rel}}(\Sigma)$, culminating in the Local Generation Theorem which states that the global relation lattice is generated by local relations supported on stars of codimension $\ge 1$ cones. The paper demonstrates the filtration's sensitivity to the facial structure, shows monotonic behavior under subdivision, and provides concrete examples illustrating the depth and topology dependent nature of the invariants. Together, these results offer a coordinate-free, canonical framework for understanding toric relations that refines classical invariants such as the divisor class group and toric ideals, with potential implications for subdivision strategies and singularity resolution in toric geometry.

Abstract

We propose a canonical local-to-global lattice theory for rational fans. We define the $\textit{ray lattice } L_{\mathrm{rays}}(Σ)$ and the $\textit{relation lattice } L_{\mathrm{rel}}(Σ)$ as invariants functorial under fan isomorphisms. We introduce $\textit{star-local relation lattices}$, defined via the relation lattice of the localized quotient fan, which capture the linear dependencies visible within local neighborhoods. We define a $\textit{codimension filtration}$ on the global relation lattice and prove a generation theorem: the global lattice is generated by local relations supported on the stars of cones of codimension at least 1. This filtration is sensitive to the facial structure of $Σ$; explicit examples and a conjecture suggest that subdivisions can only preserve or lower filtration depths, distinguishing fans with different combinatorial topologies.

Canonical Lattices and Integer Relations Associated to Rational Fans

TL;DR

This work develops a canonical local-to-global lattice theory for rational fans by defining the ray lattice and the relation lattice as functorial invariants under fan isomorphisms. It introduces star-local lattices via localized quotient fans and a codimension filtration , culminating in the Local Generation Theorem which states that the global relation lattice is generated by local relations supported on stars of codimension cones. The paper demonstrates the filtration's sensitivity to the facial structure, shows monotonic behavior under subdivision, and provides concrete examples illustrating the depth and topology dependent nature of the invariants. Together, these results offer a coordinate-free, canonical framework for understanding toric relations that refines classical invariants such as the divisor class group and toric ideals, with potential implications for subdivision strategies and singularity resolution in toric geometry.

Abstract

We propose a canonical local-to-global lattice theory for rational fans. We define the and the as invariants functorial under fan isomorphisms. We introduce , defined via the relation lattice of the localized quotient fan, which capture the linear dependencies visible within local neighborhoods. We define a on the global relation lattice and prove a generation theorem: the global lattice is generated by local relations supported on the stars of cones of codimension at least 1. This filtration is sensitive to the facial structure of ; explicit examples and a conjecture suggest that subdivisions can only preserve or lower filtration depths, distinguishing fans with different combinatorial topologies.
Paper Structure (12 sections, 4 theorems, 24 equations)

This paper contains 12 sections, 4 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Notation
  3. The Ray and Relation Lattices
  4. Local Decompositions and Intrinsic Structure
  5. A complete proof of Local Generation
  6. Structure and Functoriality
  7. Illustrative Examples and Comparison
  8. Example 1: The Projective Plane
  9. Explicit example: $\mathbb{P}^2\times \mathbb{P}^1$ and a relation in $F_2\setminus F_1$
  10. Example 3: Nontrivial Filtration Sensitivity
  11. Comparison with Known Invariants
  12. Conclusion

Key Result

Lemma 3.1

$\operatorname{rank}(L_{\mathrm{rays}}(\Sigma)) \leq \operatorname{rank}(N)$. Equality holds if $\Sigma$ is a complete fan.

Theorems & Definitions (11)

  • Lemma 3.1: Rank Bound
  • proof
  • Proposition 3.2: Canonical Nature
  • Definition 4.1: Codimension Filtration
  • Remark 4.1
  • Theorem 4.2: Local Generation for complete fans
  • proof
  • Conjecture 5.1: Filtration monotonicity under refinement
  • Theorem 5.1
  • proof
  • ...and 1 more