Classification of fundamental groups of Galois covers of surfaces of small degree degenerating to nice plane arrangements (original) (raw)

2011, Contemporary Mathematics

Let X be a surface of degree n, projected onto 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 ≤ 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 CP 1 ×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 CP 1 ×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 CP 1 × CP 1 , which was begun in [23]. Partially supported by the Emmy Noether Research Institute for Mathematics (center of the Minerva Foundation of Germany), the Excellency Center "Group Theoretic Methods in the Study of Algebraic Varieties" of the Israel Science Foundation, and EAGER (EU network, HPRN-CT-2009-00099).