Fundamental group of Galois covers of degree 5 surfaces

TL;DR

This paper investigates the fundamental groups of Galois covers of degree 5 algebraic surfaces, focusing on cases where the surfaces degenerate into arrangements of five planes with no three meeting along a line.

Contribution

It provides new insights into the fundamental groups of Galois covers of degree 5 surfaces, especially in the context of degenerations to plane arrangements.

Findings

Fundamental groups computed for specific degenerations

Characterization of Galois covers in degenerate cases

Connections between surface degenerations and fundamental group structure

Abstract

Let $X$ be an algebraic surface of degree $5$, which is considered as a branch cover of $\mathbb{CP}^2$ with respect to a generic projection. The surface has a natural Galois cover with Galois group $S_5$. In this paper, we deal with the fundamental groups of Galois covers of degree $5$ surfaces that degenerate to nice plane arrangements; each of them is a union of five planes such that no three planes meet in a line.