Weak del Pezzo surfaces with global vector fields
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arXiv
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| Format: | Preprint |
| Published: |
2020
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| _version_ | 1866917877574008832 |
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| author | Martin, Gebhard Stadlmayr, Claudia |
| author_facet | Martin, Gebhard Stadlmayr, Claudia |
| contents | We classify smooth weak del Pezzo surfaces with global vector fields over an arbitrary algebraically closed field $k$ of arbitrary characteristic $p \geq 0$. We give a complete description of the configuration of $(-1)$- and $(-2)$-curves on these surfaces and calculate the identity component of their automorphism schemes. It turns out that there are $53$ distinct families of such surfaces if $p \neq 2,3$, while there are $61$ such families if $p = 3$, and $75$ such families if $p = 2$. Each of these families has at most one moduli. As a byproduct of our classification, it follows that weak del Pezzo surfaces with non-reduced automorphism scheme exist over $k$ if and only if $p \in \{2,3\}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2007_03665 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Weak del Pezzo surfaces with global vector fields Martin, Gebhard Stadlmayr, Claudia Algebraic Geometry 14E07, 14J26, 14J50, 14L15 We classify smooth weak del Pezzo surfaces with global vector fields over an arbitrary algebraically closed field $k$ of arbitrary characteristic $p \geq 0$. We give a complete description of the configuration of $(-1)$- and $(-2)$-curves on these surfaces and calculate the identity component of their automorphism schemes. It turns out that there are $53$ distinct families of such surfaces if $p \neq 2,3$, while there are $61$ such families if $p = 3$, and $75$ such families if $p = 2$. Each of these families has at most one moduli. As a byproduct of our classification, it follows that weak del Pezzo surfaces with non-reduced automorphism scheme exist over $k$ if and only if $p \in \{2,3\}$. |
| title | Weak del Pezzo surfaces with global vector fields |
| topic | Algebraic Geometry 14E07, 14J26, 14J50, 14L15 |
| url | https://arxiv.org/abs/2007.03665 |