Fundamental group of Galois covers of degree $6$ surfaces

TL;DR

This paper investigates the fundamental groups of Galois covers of degree 6 algebraic surfaces, revealing patterns of triviality and non-triviality across different degenerations and computing their Chern numbers and signatures.

Contribution

It provides the first comprehensive computation of fundamental groups for degree 6 Galois covers and proposes a conjecture on their structure based on degeneration analysis.

Findings

8 degeneration types have non-trivial fundamental groups

20 degeneration types have trivial fundamental groups

All surfaces have negative signatures

Abstract

In this paper we consider the Galois covers of algebraic surfaces of degree 6, with all associated planar degenerations. We compute the fundamental groups of those Galois covers, using their degeneration. We show that for 8 types of degenerations the fundamental group of the Galois cover is non-trivial and for 20 types it is trivial. Moreover, we compute the Chern numbers of all the surfaces with this type of degeneration and prove that the signatures of all their Galois covers are negative. We formulate a conjecture regarding the structure of the fundamental groups of the Galois covers based on our findings. With an appendix by the authors listing the detailed computations and an appendix by Guo Zhiming classifying degree 6 planar degenerations.