Godeaux surfaces with an Enriques involution and some stable degenerations

TL;DR

This paper explicitly characterizes Godeaux surfaces with an involution leading to an Enriques surface quotient, explores their moduli space, and constructs examples of non-normal stable Godeaux surfaces through degenerations.

Contribution

It provides an explicit description of Godeaux surfaces with Enriques involutions and analyzes their moduli space, also constructing non-normal stable degenerations.

Findings

Godeaux surfaces with Enriques involutions form a 6-dimensional unirational subset.

Enriques surfaces birational to Godeaux quotients form a 5-dimensional unirational subset.

Degenerations produce examples of non-normal stable Godeaux surfaces.

Abstract

We give an explicit description of the Godeaux surfaces that admit an involution such that the quotient surface is birational to an Enriques surface; these surfaces give a 6-dimensional unirational irreducible subset of the moduli space of surfaces of general type. In addition, we describe the Enriques surfaces that are birational to the quotient of a Godeaux surface by an involution and we show that they give a 5-dimensional unirational irreducible subset of the moduli space of Enriques surfaces. Finally, by degenerating our description we obtain some examples of non-normal stable Godeaux surfaces.