On the geography of log-surfaces

This survey focuses on the geometric problem of log-surfaces, which are pairs consisting of a smooth projective surface and a reduced non-empty boundary divisor. In the first part, we focus on the geography problem for complex log-surfaces associated with pairs of the form (\mathbb{P}^{2}, C) , where C is an arrangement of smooth plane curves admitting ordinary singularities. Specifically, we focus on the case in which C is an arrangement consisting of smooth rational curves as its irreducible components. In the second part, containing original new results, we study log-surfaces constructed as pairs consisting of a complex projective K3 surface and a rational curve arrangement. In particular, we provide some combinatorial conditions for such pairs to have the log-Chern slope equal to 3 . Our survey is illustrated with many explicit examples of log-surfaces.