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 \(g\geq3\), 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.