A rational normal curve $C_d\subset\mathbb{P}^d$ is defined by quadrics. I guess, for the generic projection $\mathbb{P}^d\stackrel{\pi}{\to}\mathbb{P}^n$ the image $\pi(C_d)$ is still defined by quadrics. But choosing non-generic projections one can obtain rather "exotic" defining ideals, i.e. with $h^0(I_{\pi(C)/\mathbb{P}^n}(j))=0$ for small $j$. e.g. the image of $t\to(t^k,t^{4k+1},t^{17k+1})$.

What is known about the possible defining ideals of rational curves? I'm especially interested in the cases when the ideal does not have generators of low degree. Any systematic way to obtain such monsters? Maybe even some (partial) classification? At least for rational curves in $\mathbb{P}^3$?

upd.: By the defining ideal I mean all the polynomials vanishing of $C$. So the condition is: looking for rational curves that do not lie on hypersurfaces of small degrees.

upd2: As Mohan points out: the $\underline{generic}$ rational curve in $\mathbb{P}^3$ does not lie on surfaces of low degree!!! This solves at least part of the question. The open part (yet): what else is known about the defining ideals of rational curves $C_{d>3}\subset\mathbb{P}^{n\ge3}$?