A characterization of Inoue surfaces with $$p_g=0$$ p g = 0 and $$K^2=7$$ K 2 = 7

Abstract

Inoue constructed the first examples of smooth minimal complex surfaces of general type with pg = 0 and K 2 = 7. These surfaces are finite Galois covers of the 4-nodal cubic surface with the Galois group, the Klein group Z2 × Z2. For such a surface S, the bicanonical map of S has degree 2 and it is composed with exactly one involution in the Galois group. The divisorial part of the fixed locus of this involution consists of two irreducible components: one is a genus 3 curve with self-intersection number 0 and the other is a genus 2 curve with self-intersection number −1.Conversely, assume that S is a smooth minimal complex surface of general type with pg = 0, K 2 = 7 and having an involution σ. We show that, if the divisorial part of the fixed locus of σ consists of two irreducible components R1 and R2, with g(R1) = 3, R 2 1 = 0, g(R2) = 2 and R 2 2 = −1, then the Klein group Z2 × Z2 acts faithfully on S and S is indeed an Inoue surface.