\(Q\)-homology planes as cyclic covers of \(A^{2}\)
type: Item
instance of: scholarly article
title: \(Q\)-homology planes as cyclic covers of \(A^{2}\)
zbMATH Open document ID: 1218.14054
author: Alok Maharana
published in: Journal of the Mathematical Society of Japan
publication date: 2009-07-01T00:00:00Z
full work available at URL: article
review text: A \(Q\)-homology plane is a smooth affine surface \(S\) satisfying \(H_i(S; Q)= (0)\) for \(i> 0\). These surfaces appear naturally in affine algebraic geometry. Any smooth contractible surface is a \(Q\)-homology plane. Although there are many important papers studying these surfaces, there are very few surfaces known which are hypersufaces in \(\mathbb{C}^3\). In this paper the author completely classifies \(Q\)-homology planes \(S\) which are defined by \(\{z^n- f(x, y)= 0\}\). The affine curve \(\{f= 0\}\) is then smooth, but may be reducible. Using the calculation of topological Euler characteristic in terms of that of \(\{f= 0\}\), the first useful observation is that such a \(Q\)-homology plane contains a curve isomorphic to \(\mathbb{A}^1\). By a result of Miyanishi-Tsunoda, \(\overline k(S)\leq 1\), where \(\overline k\) is the logarithmic Kodaira dimension of \(S\). After this, the author uses the rich structure theory of smooth affine surfaces developed by Japanese mathematicians. The long list of surfaces found by the author will be quite valuable to the workers in affine surfaces theory.
reviewed by: Rajendra Gurjar
Mathematics Subject Classification ID: 14R05
Mathematics Subject Classification ID: 14J25
zbMATH DE Number: 5573644
zbMATH Keywords: \(Q\)-homology planes
zbMATH Keywords: cyclic branch covers
zbMATH Keywords: logarithmic Kodaira dimension
MaRDI profile type: Publication
OpenAlex ID: W1983439133
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DOI: 10.2969/JMSJ/06120393
description: scientific article; zbMATH DE number 5573644