Classification of singular Q-homology planes. I. Structure and singularities

TL;DR

This paper provides a detailed structure theorem for Q-homology planes with non-general type smooth loci, characterizing their singularities and describing their minimal normal completions.

Contribution

It introduces a classification of Q-homology planes with non-quotient singularities, showing they are quotients of affine cones over projective curves, and details their properties.

Findings

Q-homology planes with non-quotient singularities are quotients of affine cones

Such planes are contractible with negative Kodaira dimension

They have exactly one singular point

Abstract

A Q-homology plane is a normal complex algebraic surface having trivial rational homology. We obtain a structure theorem for Q-homology planes with smooth locus of non-general type. We show that if a Q-homology plane contains a non-quotient singularity then it is a quotient of an affine cone over a projective curve by an action of a finite group respecting the set of lines through the vertex. In particular, it is contractible, has negative Kodaira dimension and only one singular point. We describe minimal normal completions of such planes.