Rational points of universal curves in positive characteristics

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.

Categories: math.AG