Complete non-singular toric varieties with Picard number 4
Algebraic Geometry
2025-04-28 v1
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 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.
Keywords
Cite
@article{arxiv.2504.18134,
title = {Complete non-singular toric varieties with Picard number 4},
author = {Suyoung Choi and Hyeontae Jang and Mathieu Vallée},
journal= {arXiv preprint arXiv:2504.18134},
year = {2025}
}