Kontsevich spaces of rational curves on Fano hypersurfaces

We investigate the spaces of rational curves on a general hypersurface. In particular, we show that for a general degree $d$ hypersurface in ${\mathbb{P}}^n$ with $n\ge d+2$, the space $\overline{{\mathcal{M}}_{0,0}}(X,e)$ of degree $e$ Kontsevich stable maps from a rational curve to $X$ is an irreducible local complete intersection stack of dimension $e(n-d+1)+n-4$.