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Zbl 1085.14054
Masuda, Kayo; Miyanishi, Masayoshi
The additive group actions on $\Bbb Q$-homology planes. (English)
[J] Ann. Inst. Fourier 53, No. 2, 429-464 (2003). ISSN 0373-0956; ISSN 1777-5310/e

A ${\Bbb Q}$-homology plane is a smooth algebraic complex surface $X$ such that all the homology groups $H_i(X;{\Bbb Q})$ with rational coefficients and $i>0$ are trivial. Such a surface is always affine and rational. In this article, the authors study ${\Bbb Q}$-homology planes on which there exists a non-trivlal action of the additive group $({\Bbb C},+)$. 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 ${\Bbb Q}$-homology plane $X$ with a trivial Makar-Limanov invariant is the quotient of the hypersurface in ${\Bbb C}^3$ defined by the equation of the form $xy=z^m-1$ by a suitable action of ${\Bbb Z}/m{\Bbb Z}$, where $m$ is the order of $H_1(X;{\Bbb Z})$.
[Lucy Moser-Jauslin (Dijon)]

MSC 2000:: *14R20 Group actions on affine varieties
14L30 Group actions on varieties or schemes
14J26 Surfaces (rational and ruled)

Keywords: Makar-Limanov invariant; $\Bbb Q$-homology plane; additive group actions

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