  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 $n \leq d$, 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 $n-2 \geq d \geq \frac{n+1}{2}$.
