On the fundamental groups of some open rational surfaces (Q1923263)

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.