# Explicit equations of a fake projective plane

```@article{Borisov2018ExplicitEO,
title={Explicit equations of a fake projective plane},
author={Lev A. Borisov and JongHae Keum},
journal={arXiv: Algebraic Geometry},
year={2018}
}```
• Published 2018
• Mathematics
• arXiv: Algebraic Geometry
Fake projective planes are smooth complex surfaces of general type with Betti numbers equal to those of the usual projective plane. They come in complex conjugate pairs and have been classified as quotients of the two-dimensional ball by explicitly written arithmetic subgroups. In this paper we find equations of a projective model of a conjugate pair of fake projective planes by studying the geometry of the quotient of such surface by an order seven automorphism.
8 Citations

#### Tables from this paper

Explicit equations of the fake projective plane \$(C20,p=2,\emptyset,D_3 2_7)\$
We study Dolgachev elliptic surfaces with a double and a triple fiber and find explicit equations of the fake projective plane with 21 automorphisms, which is listed as (C20, p = 2, ∅, D327) inExpand
Surfaces with canonical map of maximum degree
We use the Borisov-Keum equations of a fake projective plane and the Borisov-Yeung equations of the Cartwright-Steger surface to show the existence of a regular surface with canonical map of degreeExpand
ALGEBRAIC SURFACES WITH MINIMAL BETTI NUMBERS
• J. Keum
• Mathematics
• Proceedings of the International Congress of Mathematicians (ICM 2018)
• 2019
These are algebraic surfaces with the Betti numbers of the complex projective plane, and are called Q-homology projective planes. Fake projective planes and the complex projective plane are smoothExpand
The Bicanonical map of fake projective planes with an automorphism
• Mathematics
• 2018
We show, for several fake projective planes with nontrivial automorphism group, that the bicanonical map is an embedding.
New explicit constructions of surfaces of general type
• Mathematics
• 2020
We discover a simple construction of a four-dimensional family of smooth surfaces of general type with \$p_g(S)=q(S)=0\$, \$K^2_S=3\$ with cyclic fundamental group \$C_{14}\$. We use a degeneration of theExpand
Examples of surfaces with canonical map of maximal degree
• Mathematics
• 2017
It was shown by Beauville that if the canonical map φ|KM | of a complex smooth projective surface M is generically finite, then deg(φ|KM |) ≤ 36. The first example of a surface with canonical degreeExpand
A journey from the octonionic \$\mathbb P^2\$ to a fake \$\mathbb P^2\$
• Mathematics
• 2020
We discover a family of surfaces of general type with \$K^2=3\$ and \$p=q=0\$ as free \$C_{13}\$ quotients of special linear cuts of the octonionic projective plane \$\mathbb O \mathbb P^2\$. A specialExpand
Explicit equations of the Cartwright-Steger surface
• Mathematics
• 2018
We construct explicit equations of Cartwright-Steger and related surfaces.

#### References

SHOWING 1-10 OF 34 REFERENCES
A fake projective plane with an order 7 automorphism
Abstract A fake projective plane is a compact complex surface (a compact complex manifold of dimension 2) with the same Betti numbers as the complex projective plane, but not isomorphic to theExpand
Toward a geometric construction of fake projective planes
We give a criterion for a projective surface to become a quotient of a fake projective plane. We also give a detailed information on the elliptic fibration of a \$(2,3)\$-elliptic surface that is theExpand
Derived categories of Keum's fake projective planes
• Mathematics
• 2015
Abstract We conjecture that derived categories of coherent sheaves on fake projective n-spaces have a semi-orthogonal decomposition into a collection of n + 1 exceptional objects and a category withExpand
A fake projective plane via 2-adic uniformization with torsion
• Mathematics
• 2017
We adapt the theory of non-Archimedean uniformization to construct a smooth surface from a lattice in PGL3(Q2) that has nontrivial torsion. It turns out to be a fake projective plane, commensurableExpand
Fake projective planes
• Mathematics
• 2007
A fake projective plane is a complex surface different from but has the same Betti numbers as the complex projective plane. It is a complex hyperbolic space form and has the smallest Euler PoincareExpand
Enumeration of the 50 fake projective planes
• Mathematics
• 2010
Abstract Building upon the classification of Prasad and Yeung [Invent. Math. 168 (2007) 321–370], we have shown that there exist exactly 50 fake projective planes (up to homeomorphism; 100 up toExpand
A vanishing theorem on fake projective planes with enough automorphisms
For every fake projective plane \$X\$ with automorphism group of order 21, we prove that \$H^i(X, 2L)=0\$ for all \$i\$ and for every ample line bundle \$L\$ with \$L^2=1\$. For every fake projective planeExpand
A fake projective plane constructed from an elliptic surface with multiplicities (2, 4)
Given any (2, 4)-elliptic surface with nine smooth rational curves, eight (−2)-curves and one (−3)-curve, forming a Dynkin diagram of type [2, 2][2, 2][2, 2][2, 2, 3], we show that a fake projectiveExpand
The Bicanonical map of fake projective planes with an automorphism
• Mathematics
• 2018
We show, for several fake projective planes with nontrivial automorphism group, that the bicanonical map is an embedding.
Quotients of fake projective planes
Recently, Prasad and Yeung classified all possible fundamental groups of fake projective planes. According to their result, many fake projective planes admit a nontrivial group of automorphisms, andExpand