The additive group actions on \(\mathbb Q\)-homology planes.
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 \(H_1(X;{\mathbb Z})\).
scientific article; zbMATH DE number 1940701