Very ample line bundles on weighted projective spaces and weighted blowups

Abstract

We consider line bundles $\mathcal{O}(kd)$ on weighted projective spaces, where $k$ is an integer and $d$ is the least common multiple of the weights. Such line bundles are ample if and only if $k$ is positive. On the other hand, determining which line bundles are very ample is a delicate problem. We give various sharp criteria for very ampleness. As an example, if the weights are pairwise coprime, then $\mathcal{O}(d)$ is always very ample, which implies that general smooth well-formed weighted hypersurfaces of dimension at least two are simply connected. We also treat weighted blowups, relative very ampleness, projective normality, Rees rings and generation in degree 1 of Veronese subrings.

49 pages. Generalised results from normal domains to arbitrary rings. Introduced equivalence relations, which were implicitly given in Lemmas 3.13 and 3.14 in arXiv version 1, to state stronger results in the introduction. Simplified abstract