Smooth Rational Curves on Singular Rational Surfaces

Abstract

We classify all complex surfaces with quotient singularities that do not contain any smooth rational curves under the assumption that the canonical divisor of the surface is not pseudo-effective. As a corollary, we show that if $X$ is a log del Pezzo surface such that, for every closed point $p\in X$, there is a smooth curve (locally analytically) passing through $p$, then $X$ contains at least one smooth rational curve.