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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Rational points of universal curves in positive characteristics


Author: Tatsunari Watanabe
Journal: Trans. Amer. Math. Soc. 372 (2019), 7639-7676
MSC (2010): Primary 14D23, 14D10, 14H30, 14H10
DOI: https://doi.org/10.1090/tran/7842
Published electronically: July 29, 2019
MathSciNet review: 4029677
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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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Additional Information

Tatsunari Watanabe
Affiliation: Mathematics Department, Embry-Riddle Aeronautical University, Prescott, Arizona 86301
MR Author ID: 1256428
Email: watanabt@erau.edu

Received by editor(s): November 30, 2016
Received by editor(s) in revised form: January 4, 2019
Published electronically: July 29, 2019
Article copyright: © Copyright 2019 American Mathematical Society