On the Geometry of a Fake Projective Plane with $21$ Automorphisms
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866908808486322176 |
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| author | Borisov, Lev Ji, Mattie Li, Yanxin Mondal, Sargam |
| author_facet | Borisov, Lev Ji, Mattie Li, Yanxin Mondal, Sargam |
| contents | A fake projective plane is a complex surface with the same Betti numbers as $\mathbb{C} P^2$ but not biholomorphic to it. We study the fake projective plane $\mathbb{P}_{\operatorname{fake}}^2 = (a = 7, p = 2, \emptyset, D_3 2_7)$ in the Cartwright-Steger classification. In this paper, we exploit the large symmetries given by $\operatorname{Aut}(\mathbb{P}_{\operatorname{fake}}^2) = C_7 \rtimes C_3$ to construct an embedding of this surface into $\mathbb{C} P^5$ as a system of $56$ sextics with coefficients in $\mathbb{Q}(\sqrt{-7})$. For each torsion line bundle $T \in \operatorname{Pic}(\mathbb{P}_{\operatorname{fake}}^2)$, we also compute and study the linear systems $|nH + T|$ with small $n$, where $H$ is an ample generator of the Néron-Severi group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_10429 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the Geometry of a Fake Projective Plane with $21$ Automorphisms Borisov, Lev Ji, Mattie Li, Yanxin Mondal, Sargam Algebraic Geometry 14E25, 14Q10 (Primary) 14C20, 14J29 (Secondary) A fake projective plane is a complex surface with the same Betti numbers as $\mathbb{C} P^2$ but not biholomorphic to it. We study the fake projective plane $\mathbb{P}_{\operatorname{fake}}^2 = (a = 7, p = 2, \emptyset, D_3 2_7)$ in the Cartwright-Steger classification. In this paper, we exploit the large symmetries given by $\operatorname{Aut}(\mathbb{P}_{\operatorname{fake}}^2) = C_7 \rtimes C_3$ to construct an embedding of this surface into $\mathbb{C} P^5$ as a system of $56$ sextics with coefficients in $\mathbb{Q}(\sqrt{-7})$. For each torsion line bundle $T \in \operatorname{Pic}(\mathbb{P}_{\operatorname{fake}}^2)$, we also compute and study the linear systems $|nH + T|$ with small $n$, where $H$ is an ample generator of the Néron-Severi group. |
| title | On the Geometry of a Fake Projective Plane with $21$ Automorphisms |
| topic | Algebraic Geometry 14E25, 14Q10 (Primary) 14C20, 14J29 (Secondary) |
| url | https://arxiv.org/abs/2308.10429 |