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

Abstract

Inoue constructed the first examples of smooth minimal complex surfaces of general type with $p_g=0$ and $K^2=7$. These surfaces are finite Galois covers of the 4-nodal cubic surface with the Galois group, the Klein group $\mathbb {Z}_2\times \mathbb {Z}_2$. 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 $p_g=0$, $K^2=7$ and having an involution $\sigma$. We show that, if the divisorial part of the fixed locus of $\sigma$ consists of two irreducible components $R_1$ and $R_2$, with $g(R_1)=3, R_1^2=0, g(R_2)=2$ and $R_2^2=-\,1$, then the Klein group $\mathbb {Z}_2\times \mathbb {Z}_2$ acts faithfully on S and S is indeed an Inoue surface.