On the fundamental groups of some open rational surfaces
This paper is concerned with the following two conjectures: Conjecture 1: Let \(S\) be a projective rational surface/\(\mathbb{C}\) with at most one singular point, a quotient singularity. Then \(\pi_1(S-\)Sing \(S)\) is finite. Conjecture 2: Let \(X\) be a smooth projective, rational surface/\(\mathbb{C}\) and \(C \subset X\) a smooth irreducible curve. Then \(\pi_1 (X - C)\) is finite. The two main results are: Theorem 1. Conjecture 1 is true if \(\overline \kappa (S-\)Sing \(S) \leq 1\). Theorem 2. Conjecture 2 is true if \(\overline \kappa (X - C) \leq 1\). We also have some more partial results in support of conjecture 2. The following is a striking consequence if Conjecture 2 is true. Conjecture 3: Let \(f : X \to \mathbb{P}^1\) be a morphism from a smooth projective, rational surface with connected fibers. Then \(f\) has at most one multiple fiber. The authors do not know if this is true.
finiteness of fundamental groups of open rational surfaces
scientific article; zbMATH DE number 931990
scientific article