5. The moduli space of rational curves of hypersurfaces

Problem 5.1.
In char. $p$ if $n\geq d+2$, are the moduli spaces of rational curves on a general hypersurface of degree $d$ in $\mathbb{P}^{n}$ irreducible of the expected dimension? 
Problem 5.2.
What is the dimension of rational curves in hypersurfaces of degree $d$ in $\mathbb{P}^{n}$ in char. $p$? Does char. $0$ proof go through?
Cite this as: AimPL: Rational subvarieties in positive characteristic, available at http://aimpl.org/ratsubvarpos.