On Fano indices of weighted projective spaces

2026/08/04 by Haidong Liu
Mathematics · #math.AG

paper · pdf

15pages, comments are welcome!

arxiv created 2026/08/04 · arxiv updated 2026/08/05

Abstract

The Sylvester sequence is defined recursively by s₁=2 and s_i=s₁⋯ s_i-1+1. In this paper, we prove that the Fano index of an n-dimensional well-formed weighted projective space with canonical singularities is bounded above by (s_n-1)(2s_n-3). This gives an affirmative answer to a conjecture of Chengxi Wang for weighted projective spaces and \mathbb Q-factorial toric Fano varieties with Picard number one. We also investigate the distribution of Fano indices among 4-dimensional weighted projective spaces. As the distribution of Fano indices of weighted projective spaces coincides with that of indices of terminal Calabi--Yau varieties in dimension n≤ 3, we expect this coincidence to persist also in dimension 4, and more generally, in all dimensions.

Citations

• A Kawamata--Miyaoka type inequality for Fano varieties of arbitrary Picard number
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• Complete classification of reflexive polyhedra in four dimensions
• The indices of log canonical singularities
• Lattice points in lattice polytopes