Positivity Results for spaces of rational curves

Let $X$ be a very general hypersurface of degree $d$ in ${\mathbb{P}}^n$. 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 $n\le d$, there are no rational curves in the locus $Y\subset X$ swept out by lines. And we exhibit differential forms on a smooth compactification of $R_e(X)$ for every $e$ and $n-2\ge d\ge \frac{n+1}{2}$.