 Solution

The Noether–Lefschetz theorem says that a very general smooth degree-$d$ surface in ${\mathbb{P}}_{\mathbb{C}}^3$, for $d\ge 4$, has Picard group freely generated by its hyperplane class $H$. Here very general means outside a countable union of proper closed loci in the parameter space of surfaces.

If such a surface contained a smooth rational curve $C$, its divisor class would be $mH$ for an integer $m$. Since $H\cdot C=md>0$, one has $m>0$ and therefore $C^2=m^{2}d>0$. Part (a), however, gives $C^2=-2-(d-4)\mathrm{deg}C\le -2$, a contradiction. Thus a very general surface of degree at least four has no smooth rational curve. The statement concerns smooth embedded curves; a singular curve can still have rational normalization.