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Rational points of universal curves

Author: Richard Hain
Journal: J. Amer. Math. Soc. 24 (2011), 709-769
MSC (2010): Primary 14G05, 14G27, 14H10, 14H25; Secondary 11G30, 14G32
Published electronically: January 25, 2011
MathSciNet review: 2784328
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Abstract: Suppose that $ k$ is a field of characteristic zero and that $ g+n>2$. The universal curve $ C$ of type $ (g,n)$ is the restriction of the universal curve to the generic point $ \operatorname{Spec} k(\mathcal{M}_{g,n})$ of the moduli stack $ \mathcal{M}_{g,n}$ of $ n$-pointed smooth projective curves of genus $ g$. In this paper we prove that if $ g \ge 3$, then its set of rational points $ C(k(\mathcal{M}_{g,n}))$ consists only of the $ n$ tautological points. We then prove that if $ g\ge 5$ and $ n=0$, then Grothendieck's Section Conjecture holds for $ C$ when, for example, $ k$ is a number field or a non-archimedean local field. When $ n>0$, we consider a modified version of Grothendieck's conjecture in which the geometric fundamental group of $ C$ is replaced by its $ \ell$-adic unipotent completion. We prove that if $ k$ is a number field or a non-archimedean local field, then this modified version of the Section Conjecture holds for all $ g \ge 5$ and $ n \ge 1$.

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Additional Information

Richard Hain
Affiliation: Department of Mathematics, Duke University, Durham, North Carolina 27708-0320

Received by editor(s): January 27, 2010
Received by editor(s) in revised form: September 19, 2010, and December 29, 2010
Published electronically: January 25, 2011
Additional Notes: The author was supported in part by grant DMS-0706955 from the National Science Foundation and by MSRI
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.