Rational points of universal curves in positive characteristics

Watanabe, Tatsunari

doi:10.1090/tran/7842

Rational points of universal curves in positive characteristics
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by Tatsunari Watanabe PDF

Trans. Amer. Math. Soc. 372 (2019), 7639-7676 Request permission

Abstract:

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

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