Very ample sheaves on weighted projective spaces and weighted blowups

Abstract

We consider graded rings $R$ generated by $n$ homogeneous elements of positive integer degrees $w_1, \ldots, w_n$ that have least common multiple $d$. We show that for every integer $k \geq \max(1, n-2)$, the $kd$th Veronese subring $R^{(kd)}$ is generated in degree 1, which implies that the line bundle $\mathcal{O}(kd)$ is very ample on the scheme $\operatorname{Proj}(R)$. This statement is sharp for every $n \geq 4$. We show that if all the weights $w_i$ are less than 15, then $R^{(d)}$ is generated in degree 1. This bound is sharp for all $n \geq 4$. We prove that if the weights are pairwise coprime, then $R^{(d)}$ is always generated in degree 1. We show that for almost all of the vectors $(w_1, \ldots, w_n)$, $R^{(d)}$ is generated in degree 1. Finally, we show that there exist 14 fundamental vectors such that if all the weights are less than 42 and $R^{(d)}$ is not generated in degree 1, then up to permutation, a subsequence of $(w_1, \ldots, w_n)$ is equal to a fundamental vector. We prove similar statements for Rees rings and the line bundle $\mathcal{O}(kd)$ on the weighted blowup of the affine $n$-space with weights $(w_1, \ldots, w_n)$, with the inequality $k \geq \max(1, n-2)$ replaced by $k \geq \max(1, n-1)$.