# 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} }

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.

#### Paper Mentions

#### 8 Citations

Explicit equations of the fake projective plane $(C20,p=2,\emptyset,D_3 2_7)$

- Mathematics
- 2021

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) in… Expand

Surfaces with canonical map of maximum degree

- Mathematics
- 2019

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 degree… Expand

ALGEBRAIC SURFACES WITH MINIMAL BETTI NUMBERS

- 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 smooth… Expand

The Bicanonical Map of Fake Projective Planes with an Automorphism

- Mathematics
- International Mathematics Research Notices
- 2018

We show, for several fake projective planes with a nontrivial group of automorphisms, 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 the… Expand

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 degree… Expand

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 special… Expand

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

- Mathematics
- 2005

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 the… Expand

Toward a geometric construction of fake projective planes

- Mathematics
- 2010

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 the… Expand

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 with… Expand

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, commensurable… Expand

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 Poincare… Expand

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 to… Expand

A vanishing theorem on fake projective planes with enough automorphisms

- Mathematics
- 2014

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 plane… Expand

A fake projective plane constructed from an elliptic surface with multiplicities (2, 4)

- Mathematics
- 2011

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 projective… Expand

The Bicanonical Map of Fake Projective Planes with an Automorphism

- Mathematics
- International Mathematics Research Notices
- 2018

We show, for several fake projective planes with a nontrivial group of automorphisms, that the bicanonical map is an embedding.

Quotients of fake projective planes

- Mathematics
- 2008

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, and… Expand