The Kodaira dimension of spaces of rational curves on low degree hypersurfaces

For a hypersurface in complex projective space $X\subset {\text{PP}}^n$, we investigate the singularities and Kodaira dimension of the Kontsevich moduli spaces $\text{Kbm}0,0X,e$ parametrizing rational curves of degree $e$ on $X$. If $d+e\le n$ and $X$ is a general hypersurface of degree $d$, we prove that $\text{Kbm}0,0X,e$ has only canonical singularities and we conjecture the same is true for the coarse moduli space $\text{kbm}0,0X,e$.This investigation is motivated by the question of which Fano hypersurfaces are unirational.