Hirzebruch-Kummer covers of algebraic surfaces

TL;DR

This paper demonstrates that certain natural curve arrangements and Hirzebruch-Kummer covers do not produce new ball-quotient surfaces, thus constraining methods for constructing such surfaces in algebraic geometry.

Contribution

It shows that using specific curve arrangements and Hirzebruch-Kummer covers cannot generate new examples of ball-quotients, clarifying limitations in their construction.

Findings

Natural curve arrangements with Hirzebruch-Kummer covers do not yield new ball-quotients.

The approach confirms existing constraints on constructing minimal surfaces of general type.

The results restrict the use of these methods for finding new algebraic surfaces satisfying the Bogomolov-Miyaoka-Yau equality.

Abstract

The aim of this paper is to show that using some natural curve arrangements in algebraic surfaces and Hirzebruch-Kummer covers one cannot construct new examples of ball-quotients, i.e., minimal smooth complex projective surfaces of general type satisfying equality in the Bogomolov-Miyaoka-Yau inequality.