发表机构
Sun Yat-sen University（中山大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了$n$维良构典范奇点加权射影空间的法诺指数上界，回答了相关猜想，研究了4维加权射影空间法诺指数分布并推测其与终端卡拉比-丘簇指数分布的一致性可推广至更高维度。
AI 中文摘要
西尔维斯特序列由递归式$s_1=2$、$s_i=s_1\boldsymbol{\times}s_2\boldsymbol{\times}\boldsymbol{\neq}s_{i-1}+1$定义。本文证明，$n$维良构、具有典范奇点的加权射影空间的法诺指数上界为$(s_n-1)(2s_n-3)$，这正面回答了程曦（音译）关于加权射影空间及皮卡数为1的$\boldsymbol{Q}$-仿射环面法诺簇的猜想。我们还研究了4维加权射影空间的法诺指数分布，由于$n\boldsymbol{\neq}3$时加权射影空间的法诺指数分布与终端卡拉比-丘（Calabi-Yau）簇的指数分布一致，我们推测该一致性在4维及所有维度均成立。
英文摘要
The Sylvester sequence is defined recursively by $s_1=2$ and $s_i=s_{1}\cdots s_{i-1}+1$. 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 $n\leq 3$, we expect this coincidence to persist also in dimension 4, and more generally, in all dimensions.
Comments15pages, comments are welcome! v2: correct typos and add a link of data and code for Theorem 4.4. v3:correct a small gap in Corollary 2.5
