For a curve over a field of characteristic not equal to 2, it is easy to produce finite separable morphisms to the projective line which have only simple branching, and hence are everywhere tamely ramified. In characteristic 2, the existence of a tamely ramified morphism is a much subtler question.
Building on work of Schroeer, Sugiyama-Yasuda, and Anbar-Tutdere, we identify a Brauer obstruction to the existence of such a morphism on one hand, and the other hand show that such morphisms always exist when the base field is algebraically closed (by Sugiyama-Yasuda) or finite (a new result). We also describe consequences for the tame Belyi theorem in characteristic 2.
This is Joint work with Daniel Litt and Jakub Witaszek.