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

TL;DR

This paper introduces an efficient GPU algorithm to classify certain PL spheres with $n+4$ vertices, advancing the understanding of toric varieties with Picard number 4 and answering a 2014 question.

Contribution

The paper provides a novel GPU-based algorithm for enumerating weak pseudo-manifolds and characterizes $(n-1)$-spheres with maximal Buchstaber number, focusing on toric varieties with Picard number 4.

Findings

Complete classification of $(n-1)$-spheres with $n+4$ vertices and maximal Buchstaber number.

Identification of toric varieties satisfying specific inequalities related to rational curve components.

Resolution of a 2014 question on non-singular complete toric varieties with Picard number 4.

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.