Rational unicuspidal curves on $\mathbb Q$-homology projective planes whose complements have logarithmic Kodaira dimension $-\infty$

Abstract

Let $S$ be a $\mathbb Q$-homology projective plane, $C$ a rational unicuspidal curve on $S^0 = S - \operatorname{Sing} S$ and $C'$ the proper transform of $C$ with respect to the minimal embedded resolution of $C$. We prove that $S^0 - C$ is affine ruled if and only if $C'^2 \geq -1$ and determine the pairs $(S,C)$ when $\overline{\kappa}(S^0 -C) = -\infty$ and $C'^2 \leq -2$.