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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

The classification of minimal product-quotient surfaces with $ p_g=0$.


Authors: I. Bauer and R. Pignatelli
Journal: Math. Comp. 81 (2012), 2389-2418
MSC (2010): Primary 14J10, 14J29, 14Q10; Secondary 14J25, 20F99
Published electronically: April 20, 2012
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Abstract: A product-quotient surface is the minimal resolution of the singularities of the quotient of a product of two curves by the action of a finite group acting separately on the two factors. We classify all minimal product-quotient surfaces of general type with geometric genus 0: they form 72 families. We show that there is exactly one product-quotient surface of general type whose canonical class has positive selfintersection which is not minimal, and describe its $ (-1)$-curves. For all of these surfaces the Bloch conjecture holds.


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Additional Information

I. Bauer
Affiliation: Lehrstuhl Mathematik VIII, Mathematisches Institut der Universität Bayreuth, NW II, Universitätsstr. 30, D-95447 Bayreuth, Germany

R. Pignatelli
Affiliation: Dipartimento di Matematica della Università di Trento, Via Sommarive 14, I-38123 Trento (TN), Italy

DOI: http://dx.doi.org/10.1090/S0025-5718-2012-02604-4
PII: S 0025-5718(2012)02604-4
Received by editor(s): June 16, 2010
Received by editor(s) in revised form: April 1, 2011, and July 26, 2011
Published electronically: April 20, 2012
Additional Notes: The present work took place in the realm of the DFG Forschergruppe 790 “Classification of algebraic surfaces and compact complex manifolds”, in particular the visit of the second author to Bayreuth was supported by the DFG. The second author is a member of G.N.S.A.G.A. of I.N.d.A.M. We are very grateful to Fritz Grunewald from whom we learned a lot about group theory, mathematics and life. Fritz passed away on March 21, 2010; we lost a very close friend, a great mathematician and a wonderful person. We are grateful to the referee for many useful comments which helped to improve the exposition substantially. We are also indebted to D. Frapporti for pointing out an error in a previous version of the program.
Dedicated: This article is dedicated to the memory of our dear friend and collaborator Fritz Grunewald
Article copyright: © Copyright 2012 American Mathematical Society