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Oort groups and lifting problems

Part of: Ring extensions and related topics Field extensions Curves Families, fibrations Permutation groups

Published online by Cambridge University Press:  01 July 2008

T. Chinburg ,
R. Guralnick  and
D. Harbater
T. Chinburg
Affiliation:
Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395, USA (email: ted@math.upenn.edu)
R. Guralnick
Affiliation:
Department of Mathematics, University of Southern California, Los Angeles, CA 90089-2532, USA (email: guralnic@usc.edu)
D. Harbater
Affiliation:
Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395, USA (email: harbater@math.upenn.edu)
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Abstract

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Let k be an algebraically closed field of positive characteristic p. We consider which finite groups G have the property that every faithful action of G on a connected smooth projective curve over k lifts to characteristic zero. Oort conjectured that cyclic groups have this property. We show that if a cyclic-by-p group G has this property, then G must be either cyclic or dihedral, with the exception of A4 in characteristic two. This proves one direction of a strong form of the Oort conjecture.

Keywords

curvesautomorphismsGalois groupscharacteristic pliftingOort conjecture

MSC classification

Secondary: 12F10: Separable extensions, Galois theory 14H37: Automorphisms 20B25: Finite automorphism groups of algebraic, geometric, or combinatorial structures 13B05: Galois theory 14D15: Formal methods; deformations 14H30: Coverings, fundamental group
Type
Research Article
Information
Compositio Mathematica , Volume 144 , Issue 4 , July 2008 , pp. 849 - 866
Copyright
Copyright © Foundation Compositio Mathematica 2008