= Solution
The <Noether–Lefschetz theorem> says that a <very general> smooth degree-$d$ surface in $\mathbb P^3_{\mathbb C}$, for $d\geq4$, 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 $C^2=m^2d>0$. Part (a), however, gives $C^2=-2-(d-4)\deg C\leq-2$, a contradiction. Thus \b[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.
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