Resolution of Indeterminacy of Rational Maps to Proper Tame Stacks

Abstract: We show the resolution of indeterminacy of rational maps from a regular surface to a tame stack locally of finite type over an excellent scheme. The proof uses the valuative criterion for proper tame morphisms, which was proved by Bresciani and Vistoli, together with the resolution of singularities for excellent surfaces and the root stack construction. Using Hironaka's results on the resolution of singularities over fields of characteristic zero, we extend the result to rational maps from a regular scheme of arbitrary dimension to a tame stack locally of finite type over a field of characteristic zero. We also provide a Purity Lemma for higher dimensional tame stacks, generalizing results of Abramovich, Olsson, and Vistoli, which also plays an essential role in the proof.