Complete non-singular toric varieties with Picard number 4
Suyoung Choi, Hyeontae Jang, Mathieu Vallée
TL;DR
The work delivers a complete classification of complete non-singular toric varieties with Picard number $4$ by employing a seed-and-wedge framework that reduces the problem to fanlike simplicial spheres and their characteristic maps. It proves finiteness of seeds at this Picard number, identifies 59 fanlike seeds (with seeds denoted $\mathcal{K}^{n-1}_i$) that admit fan-giving maps, and exhaustively lists all toric manifolds supported by these seeds via explicit $\lambda$ maps and symmetry data, ensuring a full classification via Choi–Park wedge theory. Notably, the paper resolves open conjectures by providing the first known neighborly seeds with toric manifolds (three neighborly seeds $\mathcal{K}^{4}_{24},\mathcal{K}^{4}_{27},\mathcal{K}^{4}_{28}$) that are realized by neighborly polytopes, including both projective and non-projective examples, thereby disproving the conjecture of Gretenkort–Kleinschmidt–Sturmfels. It also establishes an upper bound for the number of minimal non-faces of fanlike spheres (matching $\mathcal{N}(3,p)=(p-1)(p+2)/2$) and constructs higher-dimensional, nontrivial examples where the inequality is strict, advancing understanding of Batyrev’s question. The results are backed by an explicit computational framework, with a public GitHub repository providing the implemented procedures for seed classification and map generation, enabling exact replication and extension of the classification for Picard number $4$.
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.