A characterization of Inoue surfaces with p_g= 0 and K²= 7

Inoue constructed the first examples of smooth minimal complex surfaces of general type with p_g= 0 and K²= 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 p_g= 0 , K²= 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,R12=0,g(R2)=2 and R22=-1, then the Klein group Z₂× Z₂ acts faithfully on S and S is indeed an Inoue surface.