Affine lines on logarithmic \(Q\)-homology planes
type: Item
instance of: scholarly article
title: Affine lines on logarithmic \(Q\)-homology planes
zbMATH Open document ID: 0757.14022
DOI: 10.1007/BF01934336
author: M. Miyanishi
author: Rajendra Gurjar
published in: Mathematische Annalen
publication date: 1993-01-12T00:00:00Z
full work available at URL: https://eudml.org/doc/165012
review text: The following result (in a slightly general form) is proved: Theorem: Let \(X\) be a normal affine surface\(/\mathbb{C}\) having at most quotient singularities and with \(H_ i(X;\mathbb{Z})=(0)\) for \(i\geq 1\). Then (1) If \(\bar k(X)=2\), then \(X\) contains no algebraic curve homeomorphic to \(\mathbb{A}^ 1\). (2) If \(\bar k(X)=1\), then \(X\) contains a unique algebraic curve homeomorphic to \(\mathbb{A}^ 1\). It is actually isomorphic to \(\mathbb{A}^ 1\) and contains at most one (cyclic) quotient singular point of \(X\). This result in the smooth case was proved by Miyanishi, Tsunoda and Sugie. Using recent work of Peter Russell, M. Koras, Hanspeter Kraft and the authors on the linearization problem of a \(\mathbb{C}^*\) action on \(\mathbb{C}^ 3\), from the above theorem it follows that if \(\mathbb{C}^ 3//\mathbb{C}^*\) contains a contractible algebraic curve, then the \(\mathbb{C}^*\) action is linearizable.
Mathematics Subject Classification ID: 14J26
Mathematics Subject Classification ID: 14H45
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 20G20
zbMATH DE Number: 86205
zbMATH Keywords: affine lines on logarithmic homology planes
zbMATH Keywords: linearizable \(\mathbb{C}^*\)- action
MaRDI profile type: Publication
cites work: Q3658113
cites work: On the rationality of complex homology 2-cells. II
cites work: Q4193585
cites work: The cone of curves of algebraic varieties
cites work: Q4005981
cites work: Q3972944
cites work: Non-complete algebraic surfaces
cites work: Homology planes with quotient singularities
cites work: Q3728130
cites work: Absence of the affine lines on the homology planes of general type
cites work: Q3784181
cites work: Structure of open algebraic surfaces. I
OpenAlex ID: W2022265826
description: scientific article; zbMATH DE number 86205