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Oort groups and lifting problems. (English) Zbl 1158.12003

Let \(k\) be an algebraically closed field of characteristic \(p\) and let \(Y\) be a smooth projective curve over \(k\) equipped with an action of a finite group \(G\). By definition, a lifting of that action to characteristic zero is an action of \(G\) on a smooth projective curve \({\mathfrak Y}\) over a complete discrete valuation ring \(R\) of characteristic zero and residue field \(k\), together with a \(G\)-equivariant isomorphism between \(Y\) and the special fibre \({\mathfrak Y}\times_Rk\). F. Oort [Algebraic geometry, Proc. Summer Res. Inst., Brunswick/Main 1985, part 2, Proc. Symp. Pure Math. 46, 165–195 (1987; Zbl 0645.14017)] conjectured that every faithful action of a cyclic group on a smooth projective curve over \(k\) lifts to characteristic zero. The authors call a finite group \(G\) an Oort group if every faithful action of \(G\) on a smooth projective curve over \(k\) lifts to characteristic zero and prove that a finite group \(G\) is an Oort group if and only if every cyclic-by-\(p\) subgroup of \(G\) is. The authors’ “Strong Oort conjecture” asserts that a cyclic-by-\(p\) group \(G\) is an Oort group if and only if \(G\) is a cyclic group, a dihedral group of order \(2p^n\) for some \(n\), or (if \(p=2)\) the alternating group \(A_4\). In this work the forward direction of the strong Oort conjecture is proved, that is to say it is shown that a cyclic-by-\(p\) Oort group is cyclic, isomorphic to a dihedral \(p\)-group, or to \(A_4\) (if \(p=2)\).
Reviewer: B. Z. Moroz (Bonn)

MSC:

12F10 Separable extensions, Galois theory
14H37 Automorphisms of curves
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
13B05 Galois theory and commutative ring extensions
14D15 Formal methods and deformations in algebraic geometry
14H30 Coverings of curves, fundamental group
13F30 Valuation rings
14L30 Group actions on varieties or schemes (quotients)

Citations:

Zbl 0645.14017
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