A twisted bicanonical system with base point

In this talk I will present a joint work with Roberto Pignatelli about a family of minimal surfaces of general type with $p_g=q=0$ and $K^{^2}=3$, whose general member $S$ is birational to a surface $\Sigma$ of degree $10$. The order of $\pi_1(S)$ is the (conjectured) maximum possible and the birational map from $S$ to $\Sigma$ is not a morphism and it is given by a twisted bicanonical system. For both these properties, the surfaces in this family are the only known examples and answer some open questions on surfaces of general type. In the first part of the talk I will recall some results and conjectures around these topics, whereas in the second part I will describe the construction of the family.