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Consider a projective complex K3 surface $X$, then $\lvert D\rvert$ contains only finitely many rational curves for any divisor $D$ on $X$.

What is the original reference for this result?

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    $\begingroup$ What do you mean by $\lvert D \rvert$ contains only finitely many rational curves? That only finitely members of $D$ are rational curves, or only finitely many members of $D$ contain a rational curve [as a component]? $\endgroup$
    – R. van Dobben de Bruyn
    Commented Jan 26, 2023 at 19:59
  • $\begingroup$ @R.vanDobbendeBruyn I mean contains a rational curve; but I think there are also only finite many members of $D$ contain a rational curve as a component (it follows from the same argument in Dori Bejleri's answer). $\endgroup$
    – user493108
    Commented Jan 26, 2023 at 20:05

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I'm not sure about a reference but here is the reason. If $|D|$ contained infinitely many rational curves, they would move in a positive dimensional family. Since $X$ is a surface this implies it's covered by rational curves but then $X$ is uniruled contradicting that $\omega_X = \mathcal{O}_X$.

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