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.
From: Lei Song [view email]
[v1]
Wed, 3 Apr 2024 06:20:51 UTC (523 KB)
[v2]
Sat, 14 Dec 2024 08:26:20 UTC (523 KB)
[v3]
Mon, 6 Jul 2026 13:35:50 UTC (343 KB)
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