eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
41:1
41:15
10.4230/LIPIcs.SoCG.2024.41
article
GPU Algorithm for Enumerating PL Spheres of Picard Number 4: Application to Toric Topology
Choi, Suyoung
1
Jang, Hyeontae
1
Vallée, Mathieu
2
https://orcid.org/0000-0001-6336-9225
Ajou University, Suwon, South Korea
LIPN, CNRS UMR 7030, Université Sorbonne Paris Nord, Villetaneuse, France
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.41/LIPIcs.SoCG.2024.41.pdf
PL sphere
simplicial sphere
toric manifold
Picard number
weak pseudo-manifold
characteristic map
binary matroid
parallel computing
GPU programming