Fake Liftings of Galois Covers between Smooth Curves

摘要

In this paper we investigate the problem of lifting of Galois covers betweenalgebraic curves from characteristic p>0 to characteristic 0. We prove arefined version of the main result of Garuti concerning this problem in [Ga].We formulate a refined version of the Oort conjecture on liftings of cyclicGalois covers between curves. We introduce the notion of fake liftings ofcyclic Galois covers between curves, their existence would contradict the Oortconjecture, and we study the geometry of their semi-stable models. Finally, weintroduce and investigate on some examples the smoothening process, whichultimately aims to show that fake liftings do not exist. This in turn wouldimply the Oort conjecture.