Campedelli surfaces with fundamental group of order 8

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.