Smooth plane curves with a unique outer Galois point and their automorphism groups
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918412789219328 |
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| author | Badr, Eslam Harui, Takeshi |
| author_facet | Badr, Eslam Harui, Takeshi |
| contents | We consider smooth plane curves $\mathcal{X}$ of degree $d\geq4$, defined over an algebraically closed field of characteristic $0$, that possess a unique outer Galois point. This geometric condition forces the curve to be a cyclic covering of the projective line, and ensures that its automorphism group fits into a specific theoretical framework. For each possible non-cyclic reduced automorphism group $\operatorname{Aut}_{\operatorname{red}}(\mathcal{X})$, we fully characterize the defining equation of $\mathcal{X}$ and the precise structure of its full automorphism group $\operatorname{Aut}(\mathcal{X})$.
This comprehensive analysis not only identifies the exact form of the equation for each automorphism type but also establishes the detailed criteria under which these scenarios can occur, thereby offering a complete classification of defining equations for smooth plane curves with a unique outer Galois point and a non-cyclic reduced automorphism group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_26180 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Smooth plane curves with a unique outer Galois point and their automorphism groups Badr, Eslam Harui, Takeshi Algebraic Geometry 14H37, 14H45, 14H50, 14H10 We consider smooth plane curves $\mathcal{X}$ of degree $d\geq4$, defined over an algebraically closed field of characteristic $0$, that possess a unique outer Galois point. This geometric condition forces the curve to be a cyclic covering of the projective line, and ensures that its automorphism group fits into a specific theoretical framework. For each possible non-cyclic reduced automorphism group $\operatorname{Aut}_{\operatorname{red}}(\mathcal{X})$, we fully characterize the defining equation of $\mathcal{X}$ and the precise structure of its full automorphism group $\operatorname{Aut}(\mathcal{X})$. This comprehensive analysis not only identifies the exact form of the equation for each automorphism type but also establishes the detailed criteria under which these scenarios can occur, thereby offering a complete classification of defining equations for smooth plane curves with a unique outer Galois point and a non-cyclic reduced automorphism group. |
| title | Smooth plane curves with a unique outer Galois point and their automorphism groups |
| topic | Algebraic Geometry 14H37, 14H45, 14H50, 14H10 |
| url | https://arxiv.org/abs/2603.26180 |