Seshadri constants on blow-ups of Hirzebruch surfaces
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866917815109287936 |
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| author | Hanumanthu, Krishna Jacob, Cyril J. N., Suhas B. Singh, Amit Kumar |
| author_facet | Hanumanthu, Krishna Jacob, Cyril J. N., Suhas B. Singh, Amit Kumar |
| contents | Let $e,r \ge 0$ be integers and let $\mathbb{F}_e : = \mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(-e))$ denote the Hirzebruch surface with invariant $e$. We compute the Seshadri constants of an ample line bundle at an arbitrary point of the $r$-point blow-up of $\mathbb{F}_e$ when $r \leq e-1$ and at a very general point when $r=e$ or $r=e+1$. We also discuss several conjectures on linear systems of curves on the blow-up of $\mathbb{F}_e$ at $r$ very general points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_14555 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Seshadri constants on blow-ups of Hirzebruch surfaces Hanumanthu, Krishna Jacob, Cyril J. N., Suhas B. Singh, Amit Kumar Algebraic Geometry 14C20, 14E05, 14J26 Let $e,r \ge 0$ be integers and let $\mathbb{F}_e : = \mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(-e))$ denote the Hirzebruch surface with invariant $e$. We compute the Seshadri constants of an ample line bundle at an arbitrary point of the $r$-point blow-up of $\mathbb{F}_e$ when $r \leq e-1$ and at a very general point when $r=e$ or $r=e+1$. We also discuss several conjectures on linear systems of curves on the blow-up of $\mathbb{F}_e$ at $r$ very general points. |
| title | Seshadri constants on blow-ups of Hirzebruch surfaces |
| topic | Algebraic Geometry 14C20, 14E05, 14J26 |
| url | https://arxiv.org/abs/2312.14555 |