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PreprintCC BY 4.0arXiv

On Fano indices of weighted projective spaces

  • Haidong Liu
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Abstract

The Sylvester sequence is defined recursively by s1=2 and si=s1⋯ si−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 (sn−1)(2sn−3). This gives an affirmative answer to a conjecture of Chengxi Wang for weighted projective spaces and 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.

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Cite

BibTeXRIS
BibTeX
@misc{liu2026fano,
  title = {{On Fano indices of weighted projective spaces}},
  author = {Haidong Liu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.03434},
  eprint = {2608.03434},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.03434},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On Fano indices of weighted projective spaces
AU  - Haidong Liu
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2608.03434
UR  - https://arxiv.org/abs/2608.03434
AB  - The Sylvester sequence is defined recursively by s_1=2 and s_i=s_1⋯ 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 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.
ER  -