Complete non-singular toric varieties with Picard number 4

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Abstract

We classify all complete non-singular toric varieties with Picard number four via a combinatorial framework based on fanlike simplicial spheres and characteristic maps. This classification yields $59$ fanlike seeds with Picard number four, along with all toric manifolds supported by them. As a consequence, we resolve a conjecture of Gretenkort, Kleinschmidt, and Sturmfels by presenting the first known examples of toric manifolds supported by neighborly polytopes. We also answer a question of Batyrev concerning minimal non-faces of such spheres.

Citation

Choi, Suyoung, Hyeontae Jang, and Mathieu Vallée. 2025. “Complete non-singular toric varieties with Picard number 4.” https://arxiv.org/abs/2504.18134

@misc{choi_complete_2025,
  author = {Choi, Suyoung and Jang, Hyeontae and Vallée, Mathieu},
  year = {2025},
  title = {Complete non-singular toric varieties with Picard number 4},
  doi = {10.48550/arXiv.2504.18134},
  url = {http://arxiv.org/abs/2504.18134},
  note = {arXiv:2504.18134 [math]},
  copyright = {Licence Creative Commons Attribution - Pas d’utilisation commerciale - Pas de modification 4.0 International},
  publisher = {arXiv},
}