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Aug 19, 2012 at 21:18 comment added Jason Starr @Artie: I like your answer.
Aug 19, 2012 at 20:27 comment added user5117 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.)
Aug 19, 2012 at 20:03 history asked jaz CC BY-SA 3.0