Rational Points of Universal Curves in Positive Characteristics

Abstract

For the moduli stack $\mathcal{M}_{g,n/\mathbb{F}_p}$ of smooth curves of type $(g,n)$ over 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 $g\geq 3$ and $n=0$, then Grothendieck's Section Conjecture holds for the generic curve over $K$. A primary tool used in this thesis is the theory of weighted completion developed by Richard Hain and Makoto Matsumoto.