A Characterization of Inoue Surfaces with $p_g=0$ and $K^2=7$

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.

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