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arXiv · 2507.15406

Reflexive polytopes and the Picard ranks of Gorenstein toric Fano varieties

Zhuang He
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Abstract

We prove that the sum of the Picard ranks of a polar pair of Gorenstein toric Fano varieties of dimension $d\geq 3$ is at most the minimum of the number of facets and vertices of the corresponding pair of reflexive polytopes minus $(d-1)$. This is a generalization of Eikelberg's theory of affine dependences describing the Picard groups of toric varieties. The upper bound is achieved if and only if the polar pair is a simple-simplicial pair.

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Zhuang He. 2025-07-21. Reflexive polytopes and the Picard ranks of Gorenstein toric Fano varieties. https://arxiv.org/abs/2507.15406

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