Oort groups and lifting problems

TL;DR

This paper investigates which finite groups acting on smooth projective curves over algebraically closed fields of positive characteristic can be lifted to characteristic zero, providing partial confirmation of the Oort Conjecture.

Contribution

It proves that cyclic-by-p groups with the lifting property are either cyclic or dihedral, except A_4 in characteristic 2, advancing the understanding of the Oort Conjecture.

Findings

Cyclic-by-p groups with the property are cyclic or dihedral

A_4 in characteristic 2 is an exception

Partial proof of the Oort Conjecture for certain groups

Abstract

Let k be an algebraically closed field of positive characteristic p. We consider which finite groups G have the property that every faithful action of G on a connected smooth projective curve over k lifts to characteristic zero. Oort conjectured that cyclic groups have this property. We show that if a cyclic-by-p group G has this property, then G must be either cyclic or dihedral, with the exception of A_4 in characteristic 2. This proves one direction of a strong form of the Oort Conjecture.