Classification of del Pezzo surfaces of rank one. I. Height 1 and 2. II. Descendants with elliptic boundaries
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913997006045184 |
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| author | Palka, Karol Pełka, Tomasz |
| author_facet | Palka, Karol Pełka, Tomasz |
| contents | This is the first article in a series aimed at classifying normal del Pezzo surfaces of Picard rank one over algebraically closed fields of arbitrary characteristic up to an isomorphism. Our guiding invariant is the height of a del Pezzo surface, defined as the minimal intersection number of the exceptional divisor of the minimal resolution and a fiber of some $\mathbb{P}^1$-fibration. The geometry of del Pezzo surfaces gets more constrained as the height grows; in characteristic $0$ no example of height bigger than $4$ is known.
In this article, we classify del Pezzo surfaces of Picard rank one and height at most $2$; in particular we describe the non-log terminal ones. We also describe a natural class of del Pezzo surfaces which have descendants with elliptic boundary, i.e. whose minimal resolution has a birational morphism onto a canonical del Pezzo surface of rank one mapping the exceptional divisor to an anti-canonical curve. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_21174 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Classification of del Pezzo surfaces of rank one. I. Height 1 and 2. II. Descendants with elliptic boundaries Palka, Karol Pełka, Tomasz Algebraic Geometry 14J10 (Primary) 14D06, 14J45, 14R05 (Secondary) This is the first article in a series aimed at classifying normal del Pezzo surfaces of Picard rank one over algebraically closed fields of arbitrary characteristic up to an isomorphism. Our guiding invariant is the height of a del Pezzo surface, defined as the minimal intersection number of the exceptional divisor of the minimal resolution and a fiber of some $\mathbb{P}^1$-fibration. The geometry of del Pezzo surfaces gets more constrained as the height grows; in characteristic $0$ no example of height bigger than $4$ is known. In this article, we classify del Pezzo surfaces of Picard rank one and height at most $2$; in particular we describe the non-log terminal ones. We also describe a natural class of del Pezzo surfaces which have descendants with elliptic boundary, i.e. whose minimal resolution has a birational morphism onto a canonical del Pezzo surface of rank one mapping the exceptional divisor to an anti-canonical curve. |
| title | Classification of del Pezzo surfaces of rank one. I. Height 1 and 2. II. Descendants with elliptic boundaries |
| topic | Algebraic Geometry 14J10 (Primary) 14D06, 14J45, 14R05 (Secondary) |
| url | https://arxiv.org/abs/2412.21174 |