On Covering Simplices by Dilations in Dimensions 3 and 4

2024/04/03 by Lei Song, Song, Lei, Huanqi Wen +3
Computer Science · Engineering · Mathematics · #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Graph Labeling and Dimension Problems #Mathematics and Applications #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2404.02495

openalex publication_date 2024/04/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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≥ 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 ℚ-factorial toric Fano varieties with Picard number one. Along the way, we discover certain unexpected phenomenon.

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