A survey on Zariski pairs

Abstract

As Zariski pointed out in the Introduction of [130], this question was first considered by Enriques and the problem is reduced to finding the fundamental group of the complement of the given curve (the word complement is understood and often omitted for short). Zariski considered some explicit cases and proved important results. Here we detail some of the most relevant: (Z1) If two curves lie in a connected family of equisingular curves, then they have isomorphic fundamental groups. (Z2) If a continuous family {Ct}t∈[0,1] is equisingular for t ∈ (0, 1] and C0 is reduced, then there is a natural epimorphism π1(P\C0, p0) π1(P\Ct, pt), where the base point pt (t ∈ [0, 1]) depends on t continuously. (Z3) The fundamental group of an irreducible curve of order n, possessing ordinary double points only, is cyclic of order n ([130, Theorem 7]), see Remark 1. (Z4) Consider the projection from the general cubic surface in P onto P, centered at a general point outside the surface. Its branch locus is a sextic C6 with six cusps whose fundamental group is isomorphic to Z/2Z ∗ Z/3Z. (Z5) He noted that the six cusps of any sextic described in (Z4) satisfy the extra condition of lying on a conic –without decreasing the dimension of their family. Moreover, if C6 is a sextic with six cusps and its fundamental group has a representation onto the symmetric group of three letters, then C6 is the branch curve of a cubic surface and its six cusps lie on a conic. In particular if a sextic C′ 6 with six cusps not on a conic exists, then π1(P C6, po) 6∼= π1(P C′ 6, po).