On the geometry of a fake projective plane with 21 automorphisms

Borisov, Lev; Ji, Mattie; Li, Yanxin; Mondal, Sargam

doi:10.2140/involve.2026.19.73

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On the geometry of a fake projective plane with 21 automorphisms

Lev Borisov, Mattie Ji, Yanxin Li and Sargam Mondal

Vol. 19 (2026), No. 1, 73–86

DOI: 10.2140/involve.2026.19.73

Abstract

A fake projective plane is a complex surface with the same Betti numbers as $ℂ ℙ^{2}$ but not biholomorphic to it. In this paper, we study the fake projective plane $(a = 7, p = 2, \emptyset, D_{3} 2_{7})$ in the Cartwright–Steger classification. We exploit the large symmetries given by $Aut (ℙ_{fake}^{2}) = C_{7} ⋊ C_{3}$ to construct an embedding of this surface into $ℂ ℙ^{5}$ as a system of $56$ sextics with coefficients in $ℚ (\sqrt{- 7})$ . For each torsion line bundle $T \in Pic (ℙ_{fake}^{2})$ , we also compute and study the linear systems $| n H + T |$ with small $n$ , where $H$ is an ample generator of the Néron–Severi group.

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Keywords

fake projective plane, explicit equations

Mathematical Subject Classification

Primary: 14E25, 14Q10

Secondary: 14C20, 14J29

Milestones

Received: 2 September 2023

Revised: 26 August 2024

Accepted: 28 August 2024

Published: 25 January 2026

Communicated by Ravi Vakil

Authors

Lev Borisov
Mathematics Department Rutgers University Piscataway, NJ United States

Mattie Ji
Department of Mathematics Brown University Providence, RI United States

Yanxin Li
Department of Mathematics University of Colorado Boulder Boulder, CO United States

Sargam Mondal
Mathematics Department Rutgers University Piscataway, NJ United States

Vol. 19, No. 1, 2026