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Let us fix two positive integers $d$, and $N$. Can we determine a third integer $n$ such that given $n$ general points $p_1,...,p_n\in\mathbb{P}^N$ there exists a unique rational curve of degree $d$ thorugh $p_1,...,p_n$?

For example, when $d=N$ it is well known that $n = N+3$.

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    $\begingroup$ That's the only case, though. Because if you have, say, 5 general points in P^3, there's no conic through them because they're not coplanar, but there's more than one cubic that go through them because those need 6 points. $\endgroup$ May 28, 2015 at 22:30
  • $\begingroup$ This I know. However, what if we take $8$ general points in $\mathbb{P}^3$, is it true that there is a unique rational curve of degree $4$ passig through them? $\endgroup$
    – user61586
    May 29, 2015 at 0:03
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    $\begingroup$ I think there are $4$ rational curves of degree $4$ passing through $8$ general points in $\mathbb{P}^3$. For a general method for computing these numbers see, for example, section 9 of the article (arxiv.org/abs/alg-geom/9608011). $\endgroup$
    – naf
    May 29, 2015 at 3:55
  • $\begingroup$ Ahh, I missed that you didn't say "normal." As @ulrich noted, it looks like uniqueness might not be doable, but there might be a number where it's finite. $\endgroup$ May 29, 2015 at 9:24

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