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

By G. Cornelissen, Fumiharu Kato. 2001

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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 characteristic 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 P GL(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 is a free subgroup of N such that N is contained in the normalizer of in P GL(2, K), then the Mumford curve associated to becomes equipped with an action of N/, and we show that the algebraic functor deforming the latter action coincides with the analytic functor deforming the embedding of N.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/785354768d9c8723f9f38a23aecb6ace80dac005