Classification of Fundamental Groups of Galois Covers of Surfaces of Small Degree Degenerating to Nice Plane Arrangements

TL;DR

This paper surveys the fundamental groups of Galois covers of small degree surfaces degenerating to plane arrangements, providing new computations and completing classifications for certain embeddings.

Contribution

It offers a comprehensive survey of fundamental groups for surfaces of degree up to 4 degenerating to nice plane arrangements, including new calculations and classifications.

Findings

Fundamental groups computed for all small degree surfaces degenerating to plane arrangements.

Includes classical and new examples such as cubic Hirzebruch surfaces and Cayley cubic.

Completes classification of embeddings of P^1 P^1 in this context.

Abstract

Let $X$ be a surface of degree $n$, projected onto $\mathbb{CP}^2$. The surface has a natural Galois cover with Galois group $S_n.$ It is possible to determine the fundamental group of a Galois cover from that of the complement of the branch curve of $X.$ In this paper we survey the fundamental groups of Galois covers of all surfaces of small degree $n \leq 4$, that degenerate to a nice plane arrangement, namely a union of $n$ planes such that no three planes meet in a line. We include the already classical examples of the quadric, the Hirzebruch and the Veronese surfaces and the degree $4$ embedding of $\mathbb{CP}^1 \times \mathbb{CP}^1,$ and also add new computations for the remaining cases: the cubic embedding of the Hirzebruch surface $F_1$, the Cayley cubic (or a smooth surface in the same family), for a quartic surface that degenerates to the union of a triple point and a plane not through the triple point, and for a quartic $4$-point. In an appendix, we also include the degree $8$ surface $\mathbb{CP}^1\times \mathbb{CP}^1$ embedded by the $(2,2)$ embedding, and the degree $2n$ surface embedded by the $(1,n)$ embedding, in order to complete the classification of all embeddings of $\mathbb{CP}^1 \times \mathbb{CP}^1,$ which was begun in \cite{15}.