Positivity results for spaces of rational curves

Let X be a very general hypersurface of degree d in Pⁿ . We investigate positivity properties of the spaces R_e (X) of degree e rational curves in X. We show that for small e, R_e (X) has no rational curves meeting the locus of smooth embedded curves. We show that for n ≤ d, there are no rational curves other than lines in the locus Y ⊂ X swept out by lines. We exhibit differential forms on a smooth compactification of R_e (X) for every e and n − 2 ≥ d ≥¹₂ (n + 1).