Godeaux Surfaces with an Enriques Involution and Some Stable Degenerations

Abstract

We give an explicit description of the Godeaux surfaces S (minimal surfaces of general type with \( K_{S}^{2} = \chi ({\mathcal{O}}_{S} ) = 1 \)) that admit an involution σ such that S/σ 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; in particular we show that one of the families of stable Gorenstein Godeaux surfaces classified in Franciosi et al. (in preparation) consists of smoothable surfaces.