Rational points of universal curves in positive characteristics

TL;DR

This paper proves that for generic curves over function fields in positive characteristic, the only rational points are the tautological ones, and confirms Grothendieck's Section Conjecture in certain cases, extending prior characteristic zero results.

Contribution

It extends the understanding of rational points and the Section Conjecture for generic curves from characteristic zero to positive characteristic.

Findings

Only tautological points are rational for g≥3.

Grothendieck's Section Conjecture holds for g≥4, n=0.

Extension of Hain's work to positive characteristic.

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 $g\geq3$, 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.