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GPU Algorithm for Enumerating PL Spheres of Picard Number 4: Application to Toric Topology

Authors Suyoung Choi, Hyeontae Jang, Mathieu Vallée

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Author Details

Suyoung Choi
  • Ajou University, Suwon, South Korea
Hyeontae Jang
  • Ajou University, Suwon, South Korea
Mathieu Vallée
  • LIPN, CNRS UMR 7030, Université Sorbonne Paris Nord, Villetaneuse, France

Funding

This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2019R1A2C2010989).

Acknowledgements

The authors are very grateful to Dr. Axel Bacher, who introduced the last-named author to CUDA programming allowing Algorithm 2 to fit with GPU computing terminate in a reasonable time.

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Suyoung Choi, Hyeontae Jang, and Mathieu Vallée. GPU Algorithm for Enumerating PL Spheres of Picard Number 4: Application to Toric Topology. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 41:1-41:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SoCG.2024.41

Abstract

The fundamental theorem for toric geometry states a toric manifold is encoded by a complete non-singular fan, whose combinatorial structure is the one of a PL sphere together with the set of generators of its rays. The wedge operation on a PL sphere increases its dimension without changing its Picard number. The seeds are the PL spheres that are not wedges. A PL sphere is toric colorable if it comes from a complete rational fan. A result of Choi and Park tells us that the set of toric seeds with a fixed Picard number p is finite. In fact, a toric PL sphere needs its facets to be bases of some binary matroids of corank p with neither coloops, nor cocircuits of size 2. We present and use a GPU-friendly and computationally efficient algorithm to enumerate this set of seeds, up to simplicial isomorphism. Explicitly, it allows us to obtain this set of seeds for Picard number 4 which is of main importance in toric topology for the characterization of toric manifolds with small Picard number. This follows the work of Kleinschmidt (1988) and Batyrev (1991) who fully classified toric manifolds with Picard number ≤ 3.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Enumeration
  • Mathematics of computing → Combinatoric problems
  • Computing methodologies → Shared memory algorithms
  • Computing methodologies → Massively parallel algorithms
  • Theory of computation → Computational geometry
Keywords
  • PL sphere
  • simplicial sphere
  • toric manifold
  • Picard number
  • weak pseudo-manifold
  • characteristic map
  • binary matroid
  • parallel computing
  • GPU programming

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References

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