  The classifying space BDiff$(S_{g,n})$ of the orientation-preserving
diffeomorphism group of the surface $S_{g,n}$ of genus $g>1$ with $n$ ordered
marked points has a universal bundle \[ S_g \to
\text{UDiff}(S_{g,n})\xrightarrow{\pi}\text{BDiff}(S_{g,n}). \] The fixed $n$
points provide $n$ sections $s_i$ of $\pi$. In this paper we prove a conjecture
of R. Hain that any section of $\pi$ is homotopic to some $s_i$. Let
$\text{PConf}_n(S_g)$ be the ordered $n$-tuples of distinct points on $S_g$. As
part of the proof, we prove a result of independent interest: any surjective
homomorphism $\pi_1(\text{PConf}_n(S_g))\to \pi_1(S_g)$ is equal to one of the
forgetful maps $\{p_i:\pi_1(\text{PConf}_n(S_g))\to \pi_1(S_g)\}$, possibly
post-composed with an automorphism of $\pi_1(S_g)$. Using similar arguments, we
then show that the universal surface bundle that fixes $n$ points as a set does
not have any section.
