A generalization of the Oort Conjecture

TL;DR

This paper extends the Oort conjecture to Galois extensions with cyclic p-Sylow subgroups, reducing it to a characteristic p problem and proving it in specific cases, including D_9 as a local Oort group.

Contribution

It generalizes the local Oort conjecture to broader Galois extensions and establishes new cases where the conjecture holds, notably for D_9.

Findings

Reduction of the conjecture to a pure characteristic p statement

Proof of the conjecture for certain Galois extensions

Identification of D_9 as a local Oort group

Abstract

The Oort conjecture (now a theorem of Obus-Wewers and Pop) states that if k is an algebraically closed field of characteristic p, then any cyclic branched cover of smooth projective k-curves lifts to characteristic zero. This is equivalent to the local Oort conjecture, which states that all cyclic extensions of k[[t]] lift to characteristic zero. We generalize the local Oort conjecture to the case of Galois extensions with cyclic p-Sylow subgroups, reduce the conjecture to a pure characteristic p statement, and prove it in several cases. In particular, we show that D_9 is a so-called local Oort group.