On degenerations of $\mathbb Z/2$-Godeaux surfaces

TL;DR

This paper computes equations for a family of $Z/2$-Godeaux surfaces, classifies their degenerations with specific singularities, and relates these degenerations to known surface types like Enriques and elliptic surfaces.

Contribution

It provides explicit equations for $Z/2$-Godeaux surfaces and classifies their degenerations with Wahl singularities, linking them to well-understood surface classes.

Findings

The family of $Z/2$-Godeaux surfaces is at most 7-dimensional.

Degenerations with one Wahl singularity are birational to Enriques or $D_{2,n}$ elliptic surfaces.

Examples are constructed for all classified degeneration types.

Abstract

We compute equations for Coughlan's family of Godeaux surfaces with torsion $\mathbb Z/2$, which we call $\mathbb Z/2$-Godeaux surfaces, and we show that it is (at most) 7 dimensional. We classify non-rational KSBA degenerations $W$ of $\mathbb Z/2$-Godeaux surfaces with one Wahl singularity, showing that $W$ is birational to particular either Enriques surfaces, or $D_{2,n}$ elliptic surfaces, with $n=3,4$ or $6$. We present examples for all possibilities in the first case, and for $n=3,4$ in the second.