Some new surfaces with $p_g = q = 0$

Details Some new surfaces with $p_g = q = 0$ Some new surfaces with $p_g = q = 0$

2003-10-10

arxiv.org/abs/math/0310150v3

Mathematics

Algebraic Geometry

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

Scientific paper

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.

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