Rational points of universal curves

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 $n \ge 1$.