Del Pezzo surfaces of degree $5$ over perfect fields
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2023
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| Acceso en línea: | |
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| _version_ | 1866911459595780096 |
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| author | Boitrel, Aurore |
| author_facet | Boitrel, Aurore |
| contents | In this paper we study the classification of del Pezzo surfaces $X$ of degree $5$ over any perfect field $\mathbf{k}$ in explicit geometric terms. More precisely, in each case we use the Petersen graph to illustrate the $\operatorname{Gal}(\overline{\mathbf{k}}/\mathbf{k})$-action on the $(-1)$-curves of $X$ and we describe explicitly its group of automorphisms, $\operatorname{Aut}_{\mathbf{k}}(X)$. For the cases when $X$ is not minimal, we describe how to realize it as the blow-up of $\mathbb{P}^{2}$, or of a (minimal) quadric in $\mathbb{P}^{3}$, and classify them up to $\mathbf{k}$-isomorphism. In all cases, the elements of the group $\operatorname{Aut}_{\mathbf{k}}(X)$ are described geometrically. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_05328 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Del Pezzo surfaces of degree $5$ over perfect fields Boitrel, Aurore Algebraic Geometry In this paper we study the classification of del Pezzo surfaces $X$ of degree $5$ over any perfect field $\mathbf{k}$ in explicit geometric terms. More precisely, in each case we use the Petersen graph to illustrate the $\operatorname{Gal}(\overline{\mathbf{k}}/\mathbf{k})$-action on the $(-1)$-curves of $X$ and we describe explicitly its group of automorphisms, $\operatorname{Aut}_{\mathbf{k}}(X)$. For the cases when $X$ is not minimal, we describe how to realize it as the blow-up of $\mathbb{P}^{2}$, or of a (minimal) quadric in $\mathbb{P}^{3}$, and classify them up to $\mathbf{k}$-isomorphism. In all cases, the elements of the group $\operatorname{Aut}_{\mathbf{k}}(X)$ are described geometrically. |
| title | Del Pezzo surfaces of degree $5$ over perfect fields |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2304.05328 |