= Very general surface of degree at least four has no smooth rational curve
The <Noether–Lefschetz theorem> makes any curve class a positive multiple of the hyperplane class, with positive self-intersection. But the <smooth rational-curve self-intersection on a projective hypersurface> formula makes every smooth rational curve have self-intersection at most $-2$ when the surface degree is at least four. Thus none exists. The smoothness condition is essential; this does not prohibit singular curves whose normalizations are rational.
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