= Finiteness of lines on a smooth surface of degree at least three
For a line $L$ on a degree-$d$ smooth surface in $\mathbb P^3$, the <adjunction formula> gives $L^2=2-d$. If $d\geq3$, all lines are <negative curves>, hence there are only countably many by <countability of negative curves>. Their projective <Fano scheme of lines> cannot have a positive-dimensional complex component, since such a component has uncountably many points. A zero-dimensional projective scheme of finite type has finitely many points, proving the assertion.
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