Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 14J29

Retrieve articles in all journals with MSC (2000): 14J29


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
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
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

DOI: http://dx.doi.org/10.1090/S0002-9947-06-04110-9
PII: S 0002-9947(06)04110-9
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