Very ample sheaves on weighted projective spaces and weighted blowups
Stevell Muller, Erik Paemurru
TL;DR
The paper develops a combinatorial framework of bubbles to address when Veronese subrings $R^{(r)}$ are generated in degree $1$ and when the line bundles $\mathcal{O}(r)$ are very ample on $\mathrm{Proj} R$ for graded rings generated in positive degrees with weights $(w_1,\dots,w_n)$ and $d_{m w}=\mathrm{lcm}(w_1,\dots,w_n)$. It proves sharp generation criteria: $R^{(k d_{m w})}$ is generated in degree $1$ for $k\ge \max(1,n-2)$ (and $\ge \max(1,n-1)$ for Rees rings), with explicit obstructions via $\bm w$-bubbles showing sharpness; pairwise coprime weights guarantee generation in degree $1$. A substantial portion of the work translates these questions into a finite combinatorial problem, characterizing generation and very ampleness via the nonexistence of bubbles, and providing algorithms and tables for small weights to determine when $\mathcal{O}(d_{\bm w})$ is very ample. The study also proves density-type results: for fixed $n$, generation in degree $1$ is typical among large weights, while there exist infinite families of weight vectors that do admit bubbles, impacting weighted blowups and Rees rings. Collectively, the results yield concrete, computable criteria for very ampleness and degree-one generation in weighted settings, with applications to weighted hypersurfaces and well-formed weights.
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 $k \geq \max(1, n-1)$.