Complete non-singular toric varieties with Picard number 4

Suyoung Choi; Hyeontae Jang; Mathieu Vallée

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arXiv · 2504.18134

Complete non-singular toric varieties with Picard number 4

Suyoung Choi·Hyeontae Jang·Mathieu Vallée

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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.

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Suyoung Choi, Hyeontae Jang, Mathieu Vallée. 2025-04-25. Complete non-singular toric varieties with Picard number 4. https://arxiv.org/abs/2504.18134

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