On the structure of open del Pezzo surfaces
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866908493746798592 |
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| author | Palka, Karol Pełka, Tomasz |
| author_facet | Palka, Karol Pełka, Tomasz |
| contents | Let $(X,D)$ be an open log del Pezzo surface of rank one, that is, $X$ is a normal projective surface of Picard rank one, the boundary $D$ is a reduced nonzero divisor on $X$, and the anti-log canonical divisor $-(K_X+D)$ is ample. We show that, up to well described exceptions in characteristics 2, 3 and 5, the smooth part of $X\setminus D$ admits an $\mathbb{A}^1$- or an $\mathbb{A}^{1}_{*}$-fibration, which extends to a $\mathbb{P}^1$-fibration of the minimal log resolution of $(X,D)$. In characteristic 0 this improves a well-known structure theorem of Miyanishi-Tsunoda. Within the proof, we classify rational anti-canonical curves contained in smooth loci of canonical del Pezzo surfaces of rank one. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_07458 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the structure of open del Pezzo surfaces Palka, Karol Pełka, Tomasz Algebraic Geometry 14J10 (Primary) 14D06, 14J45, 14R05 (Secondary) Let $(X,D)$ be an open log del Pezzo surface of rank one, that is, $X$ is a normal projective surface of Picard rank one, the boundary $D$ is a reduced nonzero divisor on $X$, and the anti-log canonical divisor $-(K_X+D)$ is ample. We show that, up to well described exceptions in characteristics 2, 3 and 5, the smooth part of $X\setminus D$ admits an $\mathbb{A}^1$- or an $\mathbb{A}^{1}_{*}$-fibration, which extends to a $\mathbb{P}^1$-fibration of the minimal log resolution of $(X,D)$. In characteristic 0 this improves a well-known structure theorem of Miyanishi-Tsunoda. Within the proof, we classify rational anti-canonical curves contained in smooth loci of canonical del Pezzo surfaces of rank one. |
| title | On the structure of open del Pezzo surfaces |
| topic | Algebraic Geometry 14J10 (Primary) 14D06, 14J45, 14R05 (Secondary) |
| url | https://arxiv.org/abs/2412.07458 |