Symmetric homology planes
\textit{T. Petrie} conjectured in Invent. Math. 95, No.2, 355-378 (1989; Zbl 0669.14009) that homology planes (i.e. smooth affine surfaces over the complex numbers which have \(H_ i(V;{\mathbb{Z}})=0\) for \(i>0)\) which are not isomorphic to the standard plane \({\mathbb{C}}^ 2\) have no algebraic automorphisms of finite order. In this paper infinite families of homology planes with algebraic \({\mathbb{Z}}/3\)-action and \({\mathbb{Z}}/2\)-action are constructed. The construction is based on blow up processes applied to plane algebraic curves with very special intersection properties. The family with \({\mathbb{Z}}/3\)-action is constructed from the Steiner quartic together with its bi-tangent.
scientific article; zbMATH DE number 4114783