  We revisit a statement of Birch that the field of moduli for a marked
three-point ramified cover is a field of definition. Classical criteria due to
D\`ebes and Emsalem can be used to prove this statement in the presence of a
smooth point, and in fact these results imply more generally that a marked
curve descends to its field of moduli. We give a constructive version of their
results, based on an algebraic version of the notion of branches of a morphism
and allowing us to extend the aforementioned results to the wildly ramified
case. Moreover, we give explicit counterexamples for singular curves.
