Commentarii Mathematici Helvetici


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Volume 92, Issue 3, 2017, pp. 551–620
DOI: 10.4171/CMH/419

Published online: 2017-07-27

A generalization of the Oort conjecture

Andrew Obus[1]

(1) University of Virginia, Charlottesville, USA

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.

Keywords: Branched cover, lifting, Galois group, metacyclic group, KGB obstruction, Oort conjecture

Obus Andrew: A generalization of the Oort conjecture. Comment. Math. Helv. 92 (2017), 551-620. doi: 10.4171/CMH/419