Surfaces with $p_g = 0$, $K^2 = 5$ and bicanonical maps of degree 4

Let $S$ be a minimal surface of general type with $p_g(S) = 0, K_S^2 = 5$ and bicanonical map of degree 4. Denote by $\Sigma$ the bicanonical image. If $\Sigma$ is smooth, then $S$ is a Burniat surface; and if $\Sigma$ is singular, then we reduced $\Sigma$ to one case and described it, furthermore $S$ has at most one $(-2)$-curve.

Categories: math.AG