Fake liftings of Galois covers between smooth curves

Abstract

In this paper we formulate a refined version of the Oort conjecture on liftings of cyclic Galois covers between curves. We introduce the notion of fake liftings of cyclic Galois covers between curves; their existence would contradict the Oort conjecture, and we study the geometry of their semi-stable models. Finally, we introduce and investigate some examples of the smoothening process, which ultimately aims to show that fake liftings do not exist. This in turn would imply the Oort conjecture.