On the fundamental groups of some open rational surfaces
type: Item
instance of: scholarly article
title: On the fundamental groups of some open rational surfaces
zbMATH Open document ID: 0853.14010
DOI: 10.1007/BF01445240
published in: Mathematische Annalen
publication date: 1997-01-06T00:00:00Z
full work available at URL: https://eudml.org/doc/165443
review text: 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.
Mathematics Subject Classification ID: 14F35
Mathematics Subject Classification ID: 14J26
Mathematics Subject Classification ID: 14M20
zbMATH DE Number: 931990
zbMATH Keywords: finiteness of fundamental groups of open rational surfaces
zbMATH Keywords: fiber of morphism
author: Rajendra Gurjar
author: De-Qi Zhang
MaRDI profile type: Publication
cites work: Rationale Singularitäten komplexer Flächen
cites work: A Note Concerning Fox's Paper on Fenchel's Conjecture
cites work: Q5816127
cites work: Q3658113
cites work: Q3952228
cites work: Q4193585
cites work: On compact analytic surfaces. III
cites work: ON THE STRUCTURE OF COMPACT COMPLEX ANALYTIC SURFACES
cites work: The topology of normal singularities of an algebraic surface and a criterion for simplicity
cites work: Zariski's conjecture and related problems
cites work: Logarithmic Enriques surfaces
description: scientific article; zbMATH DE number 931990