  For the moduli stack $\mathcal{M}_{g,n/\mathbb{F}_p}$ of smooth curves over
$\text{Spec}~\mathbb{F}_p$ with the function field $K$, we show that if
$g\geq3$, then the only $K$-rational points of the generic curve over $K$ are
its $n$ tautological points. Furthermore, we show that if $g\geq4$ and $n=0$,
then Grothendieck's Section Conjecture holds for the generic curve over $K$.
This is an extension of Hain's work in characteristic $0$ to positive
characteristics.
