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

Abstract

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 $p^2$ and $C\subset S$ has the same rational homology as $p^1$. We will prove results concerning smooth rational curves on $S$ and the singularities of $S$ such that $\overline{\kappa }(S^0)=1$ and $\overline{\kappa }(S-C)\ne -\infty$. We end with an example of such pairs.

13 Pages