{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/on-the-geometry-of-a-fake-projective-plane","title":"On the Geometry of a Fake Projective Plane with $21$ Automorphisms","arxiv_id":"2308.10429","date":"2023-08-21","proceeding":null,"authors":["Lev Borisov","Mattie Ji","Yanxin Li","Sargam Mondal"],"abstract":"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 N\\'eron-Severi group.","url_abs":"https://arxiv.org/abs/2308.10429v2","url_pdf":"https://arxiv.org/pdf/2308.10429v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"on-the-geometry-of-a-fake-projective-plane","repo_url":"https://github.com/mattie-ji/fpp-21-geometry","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}