Classification of del {P}ezzo surfaces of rank one. III. Height 3
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866908494628651008 |
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| author | Palka, Karol Pełka, Tomasz |
| author_facet | Palka, Karol Pełka, Tomasz |
| contents | This article is a part of a series aimed at classifying normal del Pezzo surfaces of Picard rank one over an algebraically closed field of arbitrary characteristic, up to an isomorphism. The key invariant guiding our classification is the height, defined as the minimal number $h$ such that the minimal resolution of singularities admits a $\mathbb{P}^1$-fibration whose fiber meets the exceptional divisor $h$ times. It is expected that every singular del Pezzo surface of rank one is of height $h\leq 4$, with minor exceptions in characteristics $2$ and $3$. Having settled the case $h\leq 2$ in our previous article arXiv:2412.21174, we now give a classification in case the height equals $3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_13609 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Classification of del {P}ezzo surfaces of rank one. III. Height 3 Palka, Karol Pełka, Tomasz Algebraic Geometry 14J10 (Primary) 14D06, 14J45, 14R05 (Secondary) This article is a part of a series aimed at classifying normal del Pezzo surfaces of Picard rank one over an algebraically closed field of arbitrary characteristic, up to an isomorphism. The key invariant guiding our classification is the height, defined as the minimal number $h$ such that the minimal resolution of singularities admits a $\mathbb{P}^1$-fibration whose fiber meets the exceptional divisor $h$ times. It is expected that every singular del Pezzo surface of rank one is of height $h\leq 4$, with minor exceptions in characteristics $2$ and $3$. Having settled the case $h\leq 2$ in our previous article arXiv:2412.21174, we now give a classification in case the height equals $3$. |
| title | Classification of del {P}ezzo surfaces of rank one. III. Height 3 |
| topic | Algebraic Geometry 14J10 (Primary) 14D06, 14J45, 14R05 (Secondary) |
| url | https://arxiv.org/abs/2508.13609 |