Smooth space filling curve over algebraically closed field

Question

Let $X$ be a smooth projective geometrically integral variety of dimension $m \geq 1$ over $\mathbb{F}_q$ and a finite subset of points $F \subseteq X$. Then there is a way to construct a smooth projective geometrically integral curve $Y \subseteq X$ s.t. $F \subseteq Y$. My question is: Does this also work over an algebraically closed field (or more specifically over the complex numbers)? I know the concept of space filling curves as some kind of limit of curves that hit every single point in a higher dimensional space, however these will in general not be smooth. Does the hypothesis that $F$ be a finite set give us anything in this context? Does anoyone know of any research regarding this question?