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Do you know calculate the dimension of the space of rational curves of degree m, through d given points, contained in some projective variety?

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The question seems a bit broad, since there is a large body of work on this topic. Anyway, the short answer is "no": for instance, it is not even known in general how to prove that lines on a Fano hypersurface (through d=0 points) have the expected dimension. But you can look at papers of Beheshti, Landsberg, Starr and others to find positive results in this direction. (Hopefully the last-named author will be along presently to give you a more complete answer.) – user5117 Aug 19 '12 at 20:27
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@Artie: I like your answer. – Jason Starr Aug 19 '12 at 21:18

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