Tamely ramified morphisms of curves and Belyi's theorem in positive characteristic

Abstract

We show that every smooth projective curve over a finite field k admits a finite tame morphism to the projective line over k. Furthermore, we construct a curve with no such map when k is an infinite perfect field of characteristic two. Our work leads to a refinement of the tame Belyi theorem in positive characteristic, building on results of Sa\"idi, Sugiyama-Yasuda, and Anbar-Tutdere.