Rational surfaces with $K^2>0$

The main but not all of the results in this paper concern rational surfaces $X$ for which the self-intersection $K_X^2$ of the anticanonical class $-K_X$ is positive. In particular, it is shown that no superabundant numerically effective divisor classes occur on any smooth rational projective surface $X$ with $K_X^2>0$. As an application, it follows that any 8 or fewer (possibly infinitely near) points in the projective plane $\mathbf{P}^2$ are in good position. This is not true for 9 points, and a characterization of the good position locus in this case is also given. Moreover, these results are put into the context of conjectures for generic blowings up of $\mathbf{P}^2$. All results are proven over an algebraically closed field of arbitrary characteristic.