Wild ramification kinks

TL;DR

This paper investigates conditions under which the inverse image of an open disk under a branched cover of smooth projective curves remains an open disk, especially within families of covers over non-archimedean fields.

Contribution

It provides a criterion determining when the inverse image of an open disk under a branched cover is again an open disk, including in analytic families of covers.

Findings

Criteria for inverse image of open disks to be open disks

Results on stability of disk preimages in families of covers

Insights into ramification behavior in non-archimedean geometry

Abstract

Given a branched cover $f:Y\to X$ between smooth projective curves over a non-archimedian mixed-characteristic local field and an open rigid disk $D\subset X$, we study the question under which conditions the inverse image $f^{-1}(D)$ is again an open disk. More generally, if the cover $f$ varies in an analytic family, is this true at least for some member of the family? Our main result gives a criterion for this to happen.