Exceptional singular Q-homology planes

TL;DR

This paper classifies certain singular Q-acyclic surfaces with specific singularities, identifying two exceptional cases with unique configurations and Kodaira dimension zero, expanding understanding of their geometric structure.

Contribution

It proves that singular Q-acyclic surfaces with topologically rational singularities are either ruled or one of two exceptional surfaces with Kodaira dimension zero.

Findings

Two exceptional surfaces with Kodaira dimension zero identified.

These surfaces have singularities of types A1 and A2.

Construction from classical line configurations on the projective plane.

Abstract

We consider singular Q-acyclic surfaces with smooth locus of non-general type. We prove that if the singularities are topologically rational then the smooth locus is C^1- or C*-ruled or the surface is up to isomorphism one of two exceptional surfaces of Kodaira dimension zero. For both exceptional surfaces the Kodaira dimension of the smooth locus is zero and the singular locus consists of a unique point of type A1 and A2 respectively. These surfaces can be constructed starting from two classical configurations of lines on the projective plane: the dual Hesse configuration and the complete quadrangle.