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The Oort conjecture on lifting covers of curves. (English) Zbl 1311.12003

This paper completes the proof of the Oort conjecture, which states that any cyclic branched cover of smooth projective curves in characteristic \(p\) lifts to characteristic zero, by reducing the general case to a special case that was proven in [the reviewer and S. Wewers, Ann. Math. (2) 180, No. 1, 233–284 (2014; Zbl 1307.14042)].
In particular, it is well-known that the Oort conjecture is equivalent to its local version, which states that if \(k\) is an algebraically closed field of characteristic \(p\), then for any cyclic \(G\)-extension \(k[[z]]/k[[t]]\), there is a DVR \(R\) with residue field \(k\) such that \(k[[z]]/k[[t]]\) lifts to a \(G\)-extension \(R[[Z]]/R[[T]]\). The result of Obus-Wewers above proves this conjecture under the assumption of no essential ramification, which says that the breaks \((u_1, \ldots, u_n)\) in the higher ramification filtration for the upper numbering of the extension satisfy \(u_i < u_{i-1} + p\) for all \(i\). This paper completes the proof of the (local) Oort conjecture by reducing it to the “no essential ramification” case.
The key result is a so-called “characteristic \(p\) Oort conjecture,” which loosely states that any cyclic extension \(k[[z]]/k[[t]]\) has an equicharacteristic deformation to a cyclic extension of \(k[[t]][[\varpi]]\) where the generic fiber has no essential ramification (although it is generally branched at multiple maximal ideals). This deformation is constructed explicitly using Artin-Schreier-Witt theory. The author then proves a global version of this conjecture, stating that a branched Galois cover of \(\mathbb{P}^1_k\) with cyclic inertia groups deforms to a branched cover of \(\mathbb{P}^1_{k((\varpi))}\) with cyclic inertia groups and no essential ramification. The result of Obus-Wewers shows that this (deformed) cover lifts to characteristic zero. A careful algebraic geometry argument of the author now shows that one can obtain a lift of the original cover from this.

MSC:

12F10 Separable extensions, Galois theory
11G20 Curves over finite and local fields
13B05 Galois theory and commutative ring extensions
13F35 Witt vectors and related rings
14H25 Arithmetic ground fields for curves
14G32 Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory)

Citations:

Zbl 1307.14042
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References:

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