  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.
