It is known that a Hilbert scheme of degree $d$ curves in $\mathbb P^3$ can have dimension more than $4d$. But, does it imply that for some types of curves there are such a curve through any, say, $2d+1$ points? (or $3d$)



Curve of degree $d$ through $2d+1$ points in $\mathbb P^3$
Take complete intersections of degree $(4,4)$: there is a pencil of quartics passing through $2\times 16+1=33$ general points. More generally, complete intersections of degree $(d,d)$ will pass through any $n(d).d$ points, with $n(d)\rightarrow\infty$ with $d$.


