Open algebraic surfaces with $\bar{\kappa} = \bar{p}_{g} = 0$ and $\bar{P}_{2} > 0$

Abstract

Let $S$ be a smooth open rational surface with $\bar{\kappa}(S) = \bar{p}_{g}(S) = 0$ and $\bar{P}_{2}(S) > 0$. We construct a certain minimal model of $S$, which is called a strongly minimal model of $S$ in [15], and determine the strongly minimal model in the case where $S$ has non-contractible boundary at infinity. As an application, we classify the log affine surfaces with $\bar{\kappa} = \bar{p}_{g} = 0$ and $\bar{P}_{2} > 0$ under the minimality condition.