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

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}.