On Fano indices of weighted projective spaces

Abstract

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.