Let $\mathcal{M}\_{g,1}\to \mathcal{M}\_g$ be the universal genus $g$ curve, let $K\_g$ denote the funciton field of $\mathcal{M}\_g$. Take the generic fiber $C\to \mathrm{Spec}{(K\_g)}$, after some finite extension $L/K\_g$, $C\_L$ will admit a rational point. Is there a good choice for $L$? If there is, do we know $\mathrm{Gal}(L/K\_g)$?



Section of universal curve
Let me assume $g>2$, so that there is a curve $C$ over $K\_g$ of genus $g$ (the *generic curve*) -- the genus condition assures that the generic curve has no automorphisms. It is known but not obvious that $C$ has no $K\_g$-rational points (this is due to Hain-Matsumoto in characteristic zero and Watanabe in positive characterstic, if I remember correctly). If you choose any closed point of $C$ with residue field $L$, the base change to $L$ will have a rational point.


There is a reasonably canonical choice of $L$. Namely, let $W\_g$ be the moduli space of curves with a Weierstrass point. Then $W\_g$ is connected and $W\_g\to M\_g$ is not Galois, but it is generically etale with monodromy group is $S\_{g(g^2-1)}$ by [this paper](http://link.springer.com/article/10.1007/BF01388708) of Eisenbud-Harris. Thus letting $L\_g$ be the function field of $W\_g$, the generic curve obtains a rational point over $L\_g$ (namely, the generic Weierstrass point). $L\_g/K\_g$ is not Galois, but its Galois closure has Galois group $S\_{g(g^2-1)}$.


Of course this works in genus $<2$ as well, as long as you consider the generic point of $\mathcal{M}\_g$ to be a stacky point rather than Spec of a field.


