, Volume 139, Issue 1, pp 49-55
Date: 21 Nov 2008

Campedelli surfaces with fundamental group of order 8

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Abstract

Let S be a Campedelli surface (a minimal surface of general type with p g  = 0, K 2 = 2), and \({\pi\colon Y\to S}\) an etale cover of degree 8. We prove that the canonical model \({\overline {Y}}\) of Y is a complete intersection of four quadrics \({\overline {Y}=Q_{1}\cap Q_{2}\cap Q_{3}\cap Q_{4}\subset\mathbb{P}^{6}}\) . As a consequence, Y is the universal cover of S, the covering group G = Gal(Y/S) is the topological fundamental group π 1 S and G cannot be the dihedral group D 4 of order 8.

The first author is a member of the Centre for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, Lisboa. The second is a member of G.N.S.A.G.A.–I.N.d.A.M.