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.
The first author is a member of the Center for Mathematical Analysis, Geometry and Dynamical Systems of Instituto Superior Técnico, Universidade de Lisboa. The second author is a member of G.N.S.A.G.A.–I.N.d.A.M. This research was partially supported by FCT (Portugal) through program POCTI/FEDER and Projects PTDC/MAT-GEO/0675/2012 and EXCL/MAT-GEO/0222/2012 and by MIUR (Italy) through PRIN 2010–11 “Geometria delle varietà algebriche”.
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Acknowledgements
We are grateful to the editors of this volume for inviting us to contribute to it. We hope that, although the topic is not directly related to the work of Corrado Segre, the influence of the classical italian tradition of algebraic geometry that pervades the paper makes it a suitable addition to this project.
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Lopes, M.M., Pardini, R. (2016). Godeaux Surfaces with an Enriques Involution and Some Stable Degenerations. In: Casnati, G., Conte, A., Gatto, L., Giacardi, L., Marchisio, M., Verra, A. (eds) From Classical to Modern Algebraic Geometry. Trends in the History of Science. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-32994-9_12
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DOI: https://doi.org/10.1007/978-3-319-32994-9_12
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