Numerical Godeaux surfaces with an involution
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- by Alberto Calabri, Ciro Ciliberto and Margarida Mendes Lopes PDF
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Abstract:
Minimal algebraic surfaces of general type with the smallest possible invariants have geometric genus zero and $K^2=1$ and are usually called numerical Godeaux surfaces. Although they have been studied by several authors, their complete classification is not known. In this paper we classify numerical Godeaux surfaces with an involution, i.e. an automorphism of order 2. We prove that they are birationally equivalent either to double covers of Enriques surfaces or to double planes of two different types: the branch curve either has degree 10 and suitable singularities, originally suggested by Campedelli, or is the union of two lines and a curve of degree 12 with certain singularities. The latter type of double planes are degenerations of examples described by Du Val, and their existence was previously unknown; we show some examples of this new type, also computing their torsion group.References
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Additional Information
- Alberto Calabri
- Affiliation: Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Università degli Studi di Padova, via Trieste 63, I-35131 Padova, Italy
- MR Author ID: 676879
- Email: calabri@dmsa.unipd.it
- Ciro Ciliberto
- Affiliation: Dipartimento di Matematica, Università degli Studi di Roma “Tor Vergata”, Via della Ricerca Scientifica, I-00133 Roma, Italy
- MR Author ID: 49480
- Email: cilibert@mat.uniroma2.it
- Margarida Mendes Lopes
- Affiliation: Departamento de Matemática, Instituto Superior Técnico, Universidade Técnica de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
- Email: mmlopes@math.ist.utl.pt
- Received by editor(s): January 19, 2005
- Published electronically: October 17, 2006
- Additional Notes: This research has been carried out in the framework of the EAGER project financed by the EC, project n. HPRN-CT-2000-00099. The first two authors are members of G.N.S.A.G.A.-I.N.d.A.M., which generously supported this research. The third author is a member of the Center for Mathematical Analysis, Geometry and Dynamical Systems, IST, and was partially supported by FCT (Portugal) through program POCTI/FEDER and Project POCTI/MAT/44068/2002.
- © Copyright 2006 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 359 (2007), 1605-1632
- MSC (2000): Primary 14J29
- DOI: https://doi.org/10.1090/S0002-9947-06-04110-9
- MathSciNet review: 2272143