Rational points of universal curves in positive characteristics

Abstract

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 $g\ge 3$, then the only $K$-rational points of the generic curve over $K$ are its $n$ tautological points. Furthermore, we show that if $g\ge 4$ 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.

31 pages. Significantly shortened. Proposition 4.3 from the 1st version had a mistake and is replaced with a weaker proposition. This article draws heavily from arXiv:1001.5008 "Rational points of universal curve." by Richard Hain and is an extension from char. zero to positive characteristic

References in corpus (3)

• Foundations of Topological Stacks I
• The Johnson homomorphism and its kernel
• Relative Weight Filtrations on Completions of Mapping Class Groups

Cited by in corpus (1)

• Surface bundles and the section conjecture