Seshadri constants on $\mathbb{P}^1\times\mathbb{P}^1$, and applications to the symplectic packing problem
| Autor: | Dionne, Chris, Roth, Mike |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: v1: 46 pages; many (similar looking) figures; comments welcome. v2 : Minor wording changes to better indicate contributions of the article |
| Databáze: | arXiv |
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