The lifting problem we will consider roughly asks:
given a smooth, proper,
geometrically connected curve X in characteristic p and a
finite group G of automorphisms of
X, does there exist a smooth, proper curve X' in characteristic
zero and a finite group of
automorphisms G' of X' such that (X', G') lifts (X, G)? It
turns out that solving this
lifting problem reduces to solving a local lifting problem in a
formal neighborhood of each
point of X where G acts with non-trivial inertia. The Oort
conjecture states that this
local lifting problem should be solvable whenever the inertia
group is cyclic. A new result
of Stefan Wewers and the speaker shows that the local lifting
problem is solvable whenever
the inertia group is cyclic of order not divisible by p^{4}, and
in many cases even when the
inertia group is cyclic and arbitrarily large. We will discuss
this result, after giving a
good amount of background on the local lifting problem in
general.