Some new surfaces with $p_g=q=0$

Abstract

Motivated by a question by D. Mumford : can a computer classify all surfaces with $p_g=0$ ? we try to show the complexity of the problem. We restrict it to the classification of the minimal surfaces of general type with $p_g=0,K^2=8$ which are constructed by the Beauville construction, namely, which are quotients of a product of curves by the free action of a finite group G acting separately on each component. We think that man and computer will soon solve this classification problem. In the paper we classify completely the 5 cases where the group G is abelian. For these surfaces, we describe the moduli space (sometimes it is just a real point), and the first homology group. We describe also 5 examples where the group G is non abelian. Three of the latter examples had been previously described by R. Pardini.

23 pages, to appear in the Proceedings of the Fano Conference (Torino, 2002) Volume, Bull. U.M.I. Example 5.3 corrected

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