New explicit constructions of surfaces of general type

TL;DR

This paper introduces a straightforward method to construct a family of complex surfaces with specific invariants and fundamental group, and uses degeneration techniques and computer calculations to find new fake projective planes.

Contribution

It presents a novel explicit construction of surfaces of general type with particular properties and discovers new fake projective planes using computational methods.

Findings

Constructed a 4-dimensional family of surfaces with $p_g=q=0$, $K^2=3$, cyclic fundamental group $C_{14}$.

Derived explicit equations for six new fake projective planes.

Developed computational techniques potentially applicable to other complex surface classifications.

Abstract

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 surfaces in this family to find (complicated) explicit equations of six new pairs of fake projective planes. Our methods for finding new fake projective planes involve nontrivial computer calculations which we hope will be applicable in other settings.