Symmetric homology planes
type: Item
instance of: scholarly article
title: Symmetric homology planes
zbMATH Open document ID: 0681.14017
DOI: 10.1007/BF01453569
author: Tammo tom Dieck
published in: Mathematische Annalen
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164631
review text: \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.
Mathematics Subject Classification ID: 14J10
Mathematics Subject Classification ID: 14F45
Mathematics Subject Classification ID: 14J50
Mathematics Subject Classification ID: 32J99
zbMATH DE Number: 4114783
zbMATH Keywords: Steiner quartic
zbMATH Keywords: complex plane
zbMATH Keywords: homology planes
zbMATH Keywords: algebraic automorphisms
MaRDI profile type: Publication
cites work: Q3490124
cites work: Q3658113
cites work: On the rationality of complex homology 2-cells: I
cites work: Some examples of algebraic surfaces
cites work: Q3952229
cites work: Algebraic automorphisms of smooth affine surfaces
description: scientific article; zbMATH DE number 4114783