Largeness of the wild locus

Abstract: We show that the wild locus of an étale morphism of smooth adic spaces over an affinoid field is either empty or contains a divisorial point. As a consequence, we establish the equivalence of curve tameness and valuation tameness without using resolution of singularities. Moreover, given a finite étale map of regular schemes $f \colon Y \to X$ that is curve tame over a base $S$, we construct a compactification $\bar{f} \colon \overline{Y} \to \overline{X}$ over $S$ that is numerically tame.