Automorphism groups of real rational quartic del Pezzo surfaces
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866915891046776832 |
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| author | Boitrel, Aurore |
| author_facet | Boitrel, Aurore |
| contents | In this paper we give a complete description of all possible automorphism groups of real $\mathbb{R}$-rational del Pezzo surfaces $X$ of degree $4$, using the description of $X$ as the blow-up of some smooth real quadric surface $Q$ in $\mathbb{P}^{3}_{\mathbb{R}}$. We examine all possible ways to blow up $4$ geometric points on $Q$, illustrate in each case the $\operatorname{Gal}(\mathbb{C}/\mathbb{R})$-action on the conic bundle structures on $X_{\mathbb{C}}$, and use it to give a geometric description of the real automorphism group $\operatorname{Aut}_{\mathbb{R}}(X)$ by generators in terms of automorphisms and birational automorphisms of $Q$. As a consequence, we get which finite subgroups of $\operatorname{Bir}_{\mathbb{C}}(\mathbb{P}^{2})$ can act faithfully by automorphisms on real $\mathbb{R}$-rational del Pezzo surfaces of degree $4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_24447 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Automorphism groups of real rational quartic del Pezzo surfaces Boitrel, Aurore Algebraic Geometry In this paper we give a complete description of all possible automorphism groups of real $\mathbb{R}$-rational del Pezzo surfaces $X$ of degree $4$, using the description of $X$ as the blow-up of some smooth real quadric surface $Q$ in $\mathbb{P}^{3}_{\mathbb{R}}$. We examine all possible ways to blow up $4$ geometric points on $Q$, illustrate in each case the $\operatorname{Gal}(\mathbb{C}/\mathbb{R})$-action on the conic bundle structures on $X_{\mathbb{C}}$, and use it to give a geometric description of the real automorphism group $\operatorname{Aut}_{\mathbb{R}}(X)$ by generators in terms of automorphisms and birational automorphisms of $Q$. As a consequence, we get which finite subgroups of $\operatorname{Bir}_{\mathbb{C}}(\mathbb{P}^{2})$ can act faithfully by automorphisms on real $\mathbb{R}$-rational del Pezzo surfaces of degree $4$. |
| title | Automorphism groups of real rational quartic del Pezzo surfaces |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2603.24447 |