Classification of singular \(\mathbb Q\)-homology planes. II: \(\mathbb C^1\)- and \(\mathbb C^{\ast}\)-rulings
type: Item
instance of: scholarly article
title: Classification of singular \(\mathbb Q\)-homology planes. II: \(\mathbb C^1\)- and \(\mathbb C^{\ast}\)-rulings
zbMATH Open document ID: 1256.14067
author: Karol Palka
published in: Pacific Journal of Mathematics
publication date: 2012-09-19T00:00:00Z
full work available at URL: https://arxiv.org/abs/1201.2463
Mathematics Subject Classification ID: 14R05
Mathematics Subject Classification ID: 14J17
Mathematics Subject Classification ID: 14J26
zbMATH DE Number: 6083927
zbMATH Keywords: rational homology plane
zbMATH Keywords: log Kodaira dimension
zbMATH Keywords: \(\mathbb{A}^1\)-fibration
zbMATH Keywords: \(\mathbb{C}^*\)-fibration
reviewed by: Rajendra Gurjar
MaRDI profile type: Publication
OpenAlex ID: W1542982694
arXiv ID: 1201.2463
DOI: 10.2140/PJM.2012.258.421
review text: A normal quasi-projective surface \(S\) is called a \(\mathbb{Q}\)-homology plane if \(H_i(S; \mathbb{Q})= (0)\) for \(i>0\). Over the last thirty years many algebraic geometers have made important contributions to the study of these surfaces. This paper is the last step in classifying \(S\) such that the log Kodaira dimension, \(\overline k(S\text{-Sing\,}S)\) of \(S\)-\(\text{Sing\,}S\) is at most 1. Such a surface does not always have a \(\mathbb{C}\) or \(\mathbb{C}^*\)-fibration.NEWLINENEWLINE In the first part [``Classification of singular \(\mathbb{Q}\)-homology planes. I. Structure and singularities'', \url{arxiv:0806.3110}], the author had classified \(S\) which have no \(\mathbb{C}\) or \(\mathbb{C}^*\)-fibration. In this paper the author assumes that \(S\) has either a \(\mathbb{C}\) or \(\mathbb{C}^*\)-fibration. In this case he studies the weighted dual graph in a nice compactification. He studies the singular fibers of the \(\mathbb{C}\) or \(\mathbb{C}^*\)-fibration and gives conditions for \(S\) to be a \(\mathbb{Q}\)-homology plane in terms of the singular fibers. As a corollary of the classification the author describes the number of contractible irreducible algebraic curves on \(S\) in case \(\overline k(S\text{-Sing\,}S)= 0\). This was previously done by \textit{A. J. Parameswaran} and the reviewer [J. Math. Kyoto Univ. 35, No. 1, 63--77 (1995; Zbl 0861.14031)] when \(S\) is smooth.NEWLINENEWLINE The proof uses the theory of open algebraic surfaces developed by Japanese mathematicians, particularly T. Fujita.NEWLINENEWLINE The result in this paper will be valuable to affine algebraic geometers.
description: scientific article; zbMATH DE number 6083927