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The characterization of $(n-1)$-spheres with $n+4$ vertices having maximal Buchstaber number

Suyoung Choi, Hyeontae Jang, Mathieu Vallée

TL;DR

The paper develops a GPU-enabled framework to enumerate weak pseudo-manifolds and toric colorable seeds, enabling a complete classification of $(n-1)$-dimensional PL spheres with $n+4$ vertices and maximal Buchstaber number at Picard number four. By representing complexes as binary vectors and using ridge-facet incidence kernels over $\mathbb{Z}_2$, the authors derive finite seed sets from which all toric colorable PL spheres can be built via wedge operations, with rigorous isomorphism checks and PL-sphere criteria. They implement a CUDA-based algorithm to manage the combinatorial explosion and provide complete enumerations up to $n\le 11$, including the extreme $n=11$ case, supported by a structural link to binary matroids and mod $2$ characteristic maps. The results yield a precise, finitely generated catalog of seeds, and the work applies these findings to the space of rational curves on toric manifolds, resolving a question of Chen, Fu, and Hwang in the Picard number four setting. Overall, the study advances toric topology by combining algorithmic enumeration with topological and combinatorial invariants to classify toric colorable PL spheres and their geometric implications.

Abstract

We present a computationally efficient algorithm that is suitable for graphic processing unit implementation. This algorithm enables the identification of all weak pseudo-manifolds that meet specific facet conditions, drawn from a given input set. We employ this approach to enumerate toric colorable seeds. Consequently, we achieve a comprehensive characterization of $(n-1)$-dimensional PL spheres with $n+4$ vertices that possess a maximal Buchstaber number. A primary focus of this research is the fundamental categorization of non-singular complete toric varieties of Picard number $4$. This classification serves as a valuable tool for addressing questions related to toric manifolds of Picard number $4$. Notably, we have determined which of these manifolds satisfy equality within an inequality regarding the number of minimal components in their rational curve space. This addresses a question posed by Chen, Fu, and Hwang in 2014 for this specific case.

The characterization of $(n-1)$-spheres with $n+4$ vertices having maximal Buchstaber number

TL;DR

The paper develops a GPU-enabled framework to enumerate weak pseudo-manifolds and toric colorable seeds, enabling a complete classification of -dimensional PL spheres with vertices and maximal Buchstaber number at Picard number four. By representing complexes as binary vectors and using ridge-facet incidence kernels over , the authors derive finite seed sets from which all toric colorable PL spheres can be built via wedge operations, with rigorous isomorphism checks and PL-sphere criteria. They implement a CUDA-based algorithm to manage the combinatorial explosion and provide complete enumerations up to , including the extreme case, supported by a structural link to binary matroids and mod characteristic maps. The results yield a precise, finitely generated catalog of seeds, and the work applies these findings to the space of rational curves on toric manifolds, resolving a question of Chen, Fu, and Hwang in the Picard number four setting. Overall, the study advances toric topology by combining algorithmic enumeration with topological and combinatorial invariants to classify toric colorable PL spheres and their geometric implications.

Abstract

We present a computationally efficient algorithm that is suitable for graphic processing unit implementation. This algorithm enables the identification of all weak pseudo-manifolds that meet specific facet conditions, drawn from a given input set. We employ this approach to enumerate toric colorable seeds. Consequently, we achieve a comprehensive characterization of -dimensional PL spheres with vertices that possess a maximal Buchstaber number. A primary focus of this research is the fundamental categorization of non-singular complete toric varieties of Picard number . This classification serves as a valuable tool for addressing questions related to toric manifolds of Picard number . Notably, we have determined which of these manifolds satisfy equality within an inequality regarding the number of minimal components in their rational curve space. This addresses a question posed by Chen, Fu, and Hwang in 2014 for this specific case.
Paper Structure (15 sections, 20 theorems, 18 equations, 1 figure, 2 tables, 2 algorithms)

This paper contains 15 sections, 20 theorems, 18 equations, 1 figure, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Classification of weak pseudo-manifolds by graphic processing unit computing
  3. Enumerating weak pseudo-manifolds
  4. Generalities about GPU programming
  5. The GPU algorithm for classifying weak pseudo-manifolds
  6. Preparation for applying the algorithm
  7. Finiteness of the problem and seedness
  8. Checking isomorphism using minimal non-faces
  9. Collecting PL spheres among weak pseudo-manifolds
  10. Toric colorable PL spheres of Picard number four
  11. A first intuitive procedure
  12. Enumeration for $n \leq 10$
  13. Enumeration for $n=11$
  14. Toric colorability
  15. Application to the normalized space of rational curves on toric manifolds of Picard number four

Key Result

Theorem 1.1

Up to isomorphism, the number of toric (or $\mathbb{Z}_2^n$-)colorable seeds of dimension $n-1$ and Picard number $p\leq 4$ is as follows: with the empty slots displaying zero.

Figures (1)

  • Figure 1: The two layered parallel structure of a GPU.

Theorems & Definitions (30)

  • Theorem 1.1
  • Corollary 1.2
  • Proposition 2.1
  • Remark 2.3
  • Remark 2.4
  • Proposition 3.1: seedness from minimal non-faces
  • Proposition 3.2
  • proof
  • Proposition 3.3: ewald1996combinatorial, Choi-Park2016
  • Theorem 3.4: CP_wedge_2
  • ...and 20 more