2001/03/29 by Gunther Cornelissen, Fumiharu Kato, Cornelissen, Gunther +1
Mathematics · #14D15 #14G22 #14H37 #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:14D15 #msc:14G22 #msc:14H37
paper · pdf · doi:10.48550/arxiv.math/0103207
31 pages, version 2 contains major changes in section 4 on versal deformations
arxiv created 2001/08/24 · arxiv updated 2009/11/30
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.