Very ample sheaves on weighted projective spaces and weighted blowups
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908574592008192 |
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| author | Muller, Stevell Paemurru, Erik |
| author_facet | Muller, Stevell Paemurru, Erik |
| contents | 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 $k \geq \max(1, n-1)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_03036 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Very ample sheaves on weighted projective spaces and weighted blowups Muller, Stevell Paemurru, Erik Algebraic Geometry 05E14, 13A02, 13A30, 14A25, 14A15 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 $k \geq \max(1, n-1)$. |
| title | Very ample sheaves on weighted projective spaces and weighted blowups |
| topic | Algebraic Geometry 05E14, 13A02, 13A30, 14A25, 14A15 |
| url | https://arxiv.org/abs/2510.03036 |