Rational points of universal curves in positive characteristics

Source abstract

Let K K be the function field of the moduli stack M g , n / F p {\mathcal M}_{g,n/{\mathbb F}_p} of curves over Spec F p \text {Spec}{\mathbb F}_p , and let C / K C/K be the restriction of the universal curve to Spec K \text {Spec} K . We show that if g ≥ 3 g\geq 3 , then the only K K -rational points of C C are its n n tautological points. Furthermore, we show that if g ≥ 4 g\geq 4 and n = 0 n=0 , then Grothendieck’s section conjecture holds for C C over K K . This extends Hain’s work in characteristic 0 0 to positive characteristics.