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Transactions of the American Mathematical Society

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Numerical Godeaux surfaces with an involution

Authors: Alberto Calabri, Ciro Ciliberto and Margarida Mendes Lopes
Journal: Trans. Amer. Math. Soc. 359 (2007), 1605-1632
MSC (2000): Primary 14J29
Published electronically: October 17, 2006
MathSciNet review: 2272143
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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.

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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

Ciro Ciliberto
Affiliation: Dipartimento di Matematica, Università degli Studi di Roma “Tor Vergata”, Via della Ricerca Scientifica, I-00133 Roma, Italy

Margarida Mendes Lopes
Affiliation: Departamento de Matemática, Instituto Superior Técnico, Universidade Técnica de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

Keywords: Godeaux surface, involution, torsion group
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.
Article copyright: © Copyright 2006 American Mathematical Society

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