On Z/3-Godeaux surfaces

TL;DR

This paper proves that Godeaux--Reid surfaces with torsion group Z/3 have a fundamental group Z/3, using degenerations to stable surfaces and analyzing their singularities and involutions.

Contribution

It introduces a detailed degeneration analysis of Z/3-Godeaux surfaces, including their stable limits, involutions, and singularities, advancing understanding of their topological and algebraic properties.

Findings

Godeaux--Reid surfaces with torsion Z/3 have fundamental group Z/3.

Constructed degenerations with specific singularities and involutions.

Identified minimal index singularities for stable degenerations.

Abstract

We prove that Godeaux--Reid surfaces with torsion group Z/3 have topological fundamental group Z/3. For this purpose, we describe degenerations to stable KSBA surfaces with one 1/4(1,1) singularity, whose minimal resolution are elliptic fibrations with two multiplicity 3 fibres and one I_4 singular fibre. We study special such degenerations which have an involution, describing the corresponding Campedelli double plane construction. We also find some stable rational degenerations, some of which have more singularities, and one of which has a single 1/9(1,2) singularity, the minimal possible index for such a surface. Finally, we do the analogous study for the Godeaux surfaces with torsion Z/4.