Deformations of rational curves in positive characteristic

We study deformations of rational curves and their singularities in positive characteristic. We use this to prove that if a smooth and proper surface in positive characteristic $p$ is dominated by a family of rational curves such that one member has all $\delta$-invariants (resp. Jacobian numbers) strictly less than $(p-1)/2$ (resp. $p$), then the surface has negative Kodaira dimension. We also prove similar, but weaker results hold for higher dimensional varieties. Moreover, we show by example that our result is in some sense optimal. On our way, we obtain a sufficient criterion in terms of Jacobian numbers for the normalization of a curve over an imperfect field to be smooth.