The additive group actions on \(\mathbb Q\)-homology planes.
type: Item
instance of: scholarly article
title: The additive group actions on \(\mathbb Q\)-homology planes.
zbMATH Open document ID: 1085.14054
author: K. Masuda
author: M. Miyanishi
publication date: 2003-06-26T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/116042
review text: A \({\mathbb Q}\)-homology plane is a smooth algebraic complex surface \(X\) such that all the homology groups \(H_i(X;{\mathbb Q})\) with rational coefficients and \(i>0\) are trivial. Such a surface is always affine and rational. In this article, the authors study \({\mathbb Q}\)-homology planes on which there exists a non-trivlal action of the additive group \(({\mathbb C},+)\). The Makar-Limanov invariant of an affine variety \(X\) is the intersection of the kernels of all the locally nilpotent derivations on the ring of polynomial functions on \(X\). The main result of this paper is to show that any \({\mathbb Q}\)-homology plane \(X\) with a trivial Makar-Limanov invariant is the quotient of the hypersurface in \({\mathbb C}^3\) defined by the equation of the form \(xy=z^m-1\) by a suitable action of \({\mathbb Z}/m{\mathbb Z}\), where \(m\) is the order of \(H_1(X;{\mathbb Z})\).
reviewed by: Lucy Moser-Jauslin
Mathematics Subject Classification ID: 14R20
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 14J26
zbMATH DE Number: 1940701
zbMATH Keywords: Makar-Limanov invariant
zbMATH Keywords: \(\mathbb Q\)-homology plane
zbMATH Keywords: additive group actions
MaRDI profile type: Publication
OpenAlex ID: W3033051418
cites work: Q5807681
cites work: Affine surfaces with \(AK(S)=\mathbb C\).
cites work: Pinceaux de droites et automorphismes des surfaces affines.
cites work: On complex affine surfaces with \(\mathbb{C}^ +\)-action
cites work: Q5816127
cites work: On the Jacobian Conjecture for ℚ-homology planes
cites work: \(\mathbb{Q}\)-homology planes are rational. III
cites work: Q4344786
cites work: Étale endomorphisms of algebraic surfaces with \(G_m\)-actions
cites work: Lectures on curves on rational and unirational surfaces
cites work: Q4427538
cites work: Homology planes with quotient singularities
DOI: 10.5802/AIF.1949
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 1940701