Defining ideals for rational curves in space

Question

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 $C_{d>3}\subset\mathbb{P}^{n\ge3}$?

Answer 1

In 3-space, isn't this a theorem of Hirschowitz, answering a question of Hartshorne? See the math review MR611384 (82j:14028). He proves that for any degree $d$, a general rational curve $C$ of degree $d$ in 3-space has maximal rank, which implies that it can not be contained in a hyersurface of degree $e$ if $$h^0(\mathcal{O}_{\mathbb{P}^3}(e))=\binom{e+3}{3}\leq ed+1=h^0(\mathcal{O}_C(e))$$. Of course if the inequality is not satisfied, trivially the curve is contained in a hypersurface of degree $e$. If your question is whether we can explicitly write such curves down, I do not think the above answers that.