Rational curves on cubic hypersurfaces in positive characteristic

We study the moduli spaces of rational curves on cubic hypersurfaces in characteristic $\mathrm{\ne }2,3$. As a result, we prove that for every integer $d\ge 1$ the Kontsevich moduli space of stable maps on a smooth cubic hypersurface $X$ of degree $d$ is irreducible if the dimension of $X$ is greater than or equal to $4$.