Fake Liftings of Galois Covers between Smooth Curves

TL;DR

This paper explores the problem of lifting Galois covers of algebraic curves from characteristic p to characteristic 0, proposing a refined conjecture and investigating fake liftings to support the Oort conjecture.

Contribution

It introduces the notion of fake liftings, refines existing results on Galois cover liftings, and develops a smoothening process to potentially prove the Oort conjecture.

Findings

Refined version of Garuti's main result on Galois cover liftings

Definition and analysis of fake liftings and their geometry

Development of a smoothening process to eliminate fake liftings

Abstract

In this paper we investigate the problem of lifting of Galois covers between algebraic curves from characteristic p>0 to characteristic 0. We prove a refined version of the main result of Garuti concerning this problem in [Ga]. 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 on some examples the smoothening process, which ultimately aims to show that fake liftings do not exist. This in turn would imply the Oort conjecture.