\(Q\)-homology planes as cyclic covers of \(A^{2}\)
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.
scientific article; zbMATH DE number 5573644
scientific article