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

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  • $\begingroup$ Take an arbitrary defining ideal $I$ and replace it with $I^m$ with $m\gg 0$. This will not have generators of low degree. $\endgroup$
    – Sándor Kovács
    Commented Mar 26, 2012 at 20:01
  • $\begingroup$ No, but I assume the curve to be reduced, and define $I$ to be all the polynomials vanishing on $C_d$. So, I'm asking for rational curves that do not lie on hypersurfaces of small degrees. $\endgroup$
    – Dmitry Kerner
    Commented Mar 26, 2012 at 20:04
  • $\begingroup$ Then you should say you are looking for "generators of the ideal", not "defining ideals" $\endgroup$
    – Sándor Kovács
    Commented Mar 26, 2012 at 20:35
  • $\begingroup$ Take an arbitrarily high degree curve in some projective space and take a general projection to a plane. $\endgroup$
    – Sándor Kovács
    Commented Mar 26, 2012 at 20:37
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    $\begingroup$ qui-vadis: I recommend you look at the article of Eisenbud and Van de Ven. They consider rational curves such that $I/I^2$ has a particular structure. That should be quite close to what you are looking for. There was later work by Clemens on this as well. $\endgroup$
    – Jason Starr
    Commented Mar 26, 2012 at 21:48

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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.

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  • $\begingroup$ Great! That's almost the solution! Thanks a lot! (Being ignorant I thought that for any degree the generic rational curve lies on quadrics or cubics.) Still, probably there exist some more general/stronger results? So, let me wait with accepting your answer as the official answer to the question. $\endgroup$
    – Dmitry Kerner
    Commented Mar 27, 2012 at 11:27

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