$\mathbb{Q}$-Homology Plane pairs with Logarithmic Kodaira dimension 1

A pair $(S,C)$ is called a singular $\mathbb{Q}$-homology plane pair if $S$ is a singular projective surface with only quotient singularities having the same rational homology as $\mathbb{p}^2$ and $C \subset S$ has the same rational homology as $\mathbb{p}^1$. We will prove results concerning smooth rational curves on $S$ and the singularities of $S$ such that $\overline \kappa(S^0)=1$ and $\overline \kappa(S-C) \neq -\infty$. We end with an example of such pairs.