Lifting of curves with automorphisms

Abstract

The lifting problem for curves with automorphisms asks whether we can lift a smooth projective characteristic p curve with a group G of automorphisms to characteristic zero. This was solved by Grothendieck when G acts with prime-to-p stabilizers, and there has been much progress over the last few decades in the wild case. We survey the techniques and obstructions for this lifting problem, aiming at a reader whose background is limited to scheme theory at the level of Hartshorne's book. Throughout, we include numerous examples and clarifying remarks. We also provide a list of open questions.