Toward a geometric construction of fake projective planes

TL;DR

This paper provides a geometric criterion for identifying when a projective surface is a quotient of a fake projective plane, along with detailed analysis of related elliptic fibrations and classifications of certain rational surfaces.

Contribution

It introduces a new geometric criterion for recognizing quotients of fake projective planes and classifies specific rational surfaces with cusps.

Findings

Criterion for a surface to be a quotient of a fake projective plane

Detailed description of elliptic fibrations of related surfaces

Classification of ${f Q}$-homology projective planes with cusps

Abstract

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 minimal resolution of a quotient of a fake projective plane. As a consequence, we give a classification of ${\mathbb Q}$-homology projective planes with cusps only.