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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.

Godeaux Surfaces with an Enriques Involution and Some Stable Degenerations