Equivariant deformation of Mumford curves and of ordinary curves in positive characteristic

Details Equivariant deformation of Mumford curves and of ordinary curves in positive characteristic Equivariant deformation of Mumford curves and of ordinary curves in positive characteristic

2001-03-29

arxiv.org/abs/math/0103207v2

Mathematics

Algebraic Geometry

31 pages, version 2 contains major changes in section 4 on versal deformations

Scientific paper

We compute the dimension of the tangent space to, and the Krull dimension of the pro-representable hull of two deformation functors. The first one is the ``algebraic'' deformation functor of an ordinary curve X over a field of positive charateristic with prescribed action of a finite group G, and the data are computed in terms of the ramification behaviour of X -> G\X. The second one is the ``analytic'' deformation functor of a fixed embedding of a finitely generated discrete group N in PGL(2,K) over a non-archimedean valued field K, and the data are computed in terms of the Bass-Serre representation of N via a graph of groups. Finally, if F is a free subgroup of N such that N is contained in the normalizer of F in PGL(2,K), then the Mumford curve associated to F becomes equipped with an action of N/F, and we show that the algebraic functor deforming the latter action coincides with the analytic functor deforming the embedding of N.

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