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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.11656 |
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| _version_ | 1866908669114843136 |
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| author | Dionne, Chris Roth, Mike |
| author_facet | Dionne, Chris Roth, Mike |
| contents | In this paper we compute the $r$-point Seshadri constant on $\mathbb{P}^1\times\mathbb{P}^1$ for those line bundles where the answer might be expected to be governed by $(-1)$-curves. As a consequence we obtain explicit formulas for the symplectic packing problem for $\mathbb{P}^1\times\mathbb{P}^1$. Some exact values of the Seshadri constant outside the region governed by Mori's cone theorem are also given. These latter results use a useful new "reflection method".
In the analysis there is a striking difference between the cases when $r$ is odd and when $r$ is even. When $r$ is even the problem admits an infinite order automorphism, and there are infinitely many $(-1)$-curves to consider. In contrast, when $r$ is odd only a finite number (usually $4$) types of $(-1)$-curves are relevant to our answer. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_11656 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Seshadri constants on $\mathbb{P}^1\times\mathbb{P}^1$, and applications to the symplectic packing problem Dionne, Chris Roth, Mike Algebraic Geometry Symplectic Geometry [2020] 14C20, 14J42 In this paper we compute the $r$-point Seshadri constant on $\mathbb{P}^1\times\mathbb{P}^1$ for those line bundles where the answer might be expected to be governed by $(-1)$-curves. As a consequence we obtain explicit formulas for the symplectic packing problem for $\mathbb{P}^1\times\mathbb{P}^1$. Some exact values of the Seshadri constant outside the region governed by Mori's cone theorem are also given. These latter results use a useful new "reflection method". In the analysis there is a striking difference between the cases when $r$ is odd and when $r$ is even. When $r$ is even the problem admits an infinite order automorphism, and there are infinitely many $(-1)$-curves to consider. In contrast, when $r$ is odd only a finite number (usually $4$) types of $(-1)$-curves are relevant to our answer. |
| title | Seshadri constants on $\mathbb{P}^1\times\mathbb{P}^1$, and applications to the symplectic packing problem |
| topic | Algebraic Geometry Symplectic Geometry [2020] 14C20, 14J42 |
| url | https://arxiv.org/abs/2406.11656 |