A characterization of Burniat surfaces with K 2=4 and of non nodal type

Abstract

let $S$ be a minimal surface of general type with $p_{g}(S)=0$ and $K^{2}_{S}=4$./ Assume the bicanonical map $\varphi$ of $S$ is a morphism of degree $4$ such that the image of $\varphi$ is smooth./ Then we prove that the surface $S$ is a Burniat surface with $K^{2}=4$ and of non nodal type.