Papers › On the Geometry of a Fake Projective Plane with 21 Automorphisms
On the Geometry of a Fake Projective Plane with 21 Automorphisms
Lev Borisov, Mattie Ji, Yanxin Li, Sargam Mondal
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A fake projective plane is a complex surface with the same Betti numbers as ℂ P² but not biholomorphic to it. We study the fake projective plane ℙ_(fake)² = (a = 7, p = 2, ∅, D₃ 2₇) in the Cartwright-Steger classification. In this paper, we exploit the large symmetries given by Aut(ℙ_(fake)²) = C₇ ⋊C₃ to construct an embedding of this surface into ℂ P⁵ as a system of 56 sextics with coefficients in ℚ(√(-7)). For each torsion line bundle T ∈Pic(ℙ_(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.
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