GPU Algorithm for Enumerating PL Spheres of Picard Number 4: Application to Toric Topology
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| Title: | GPU Algorithm for Enumerating PL Spheres of Picard Number 4: Application to Toric Topology |
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| Authors: | Choi, Suyoung, Jang, Hyeontae, Vallée, Mathieu |
| Contributors: | Suyoung Choi and Hyeontae Jang and Mathieu Vallée |
| Publisher Information: | Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2024. |
| Publication Year: | 2024 |
| Subject Terms: | toric manifold, characteristic map, weak pseudo-manifold, parallel computing, Picard number, PL sphere, binary matroid, simplicial sphere, GPU programming, ddc:004 |
| Description: | 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. |
| Document Type: | Conference object Article |
| File Description: | application/pdf |
| Language: | English |
| DOI: | 10.4230/lipics.socg.2024.41 |
| Access URL: | https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.41 |
| Rights: | CC BY |
| Accession Number: | edsair.dedup.wf.002..cce199e86feaf19fd1fa55a0570433ad |
| Database: | OpenAIRE |
Nájsť tento článok vo Web of Science | 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. |
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| DOI: | 10.4230/lipics.socg.2024.41 |