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arXiv · 2404.02495

On Covering Simplices by Dilations in Dimensions 3 and 4

Lei Song·Huanqi Wen·Zhixian Zhu
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Abstract

We propose a conjecture regarding the integrally closedness of lattice polytopes with large lattice lengths. We demonstrate that a lattice simplex in dimension 3 (resp. 4) with lattice length of at least 2 (resp. 3 and no edge has lattice length 5) can be covered by dilated simplices of the form $sQ$, where integer $s\ge 2$ (resp. 3) and $Q$ is a lattice simplex. The covering property implies these simplices are integrally closed. As an application, we obtain a simple criterion for the projective normality of ample line bundles on 3-(resp. 4-) dimensional $\mathbb{Q}$-factorial toric Fano varieties with Picard number one. Along the way, we discover certain unexpected phenomenon.

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Lei Song, Huanqi Wen, Zhixian Zhu. 2026-07-06. On Covering Simplices by Dilations in Dimensions 3 and 4. https://arxiv.org/abs/2404.02495

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