The degree of the bicanonical map of a surface with $p_g=0$

Abstract:

In this note it is shown that, given a smooth minimal complex surface of general type $S$ with $p_g(S)=0$, $K^2_S=3$, for which the bicanonical map $\varphi _{2K}$ is a morphism, the degree of $\varphi _{2K}$ is not 3. This completes our earlier results, showing that if $S$ is a minimal surface of general type with $p_g=0$, $K^2\ge 3$ such that $|2K_S|$ is free, then the bicanonical map of $S$ can have degree 1, 2 or 4.