Let $X \subseteq \mathbb{P}^n$ ba a projective variety, and $\mathbb{P}^1 \to X$ be a rational curve.
Why it is claimed that there are countably many families of rational curves on $X$ (certainly, one just need to show there are countably many families of rational curves of fixed degree with certain very ample line bundle of $X$)?
First of all, what does "family" mean in this setting.