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

2003/03/15 by Gunther Cornelissen, Fumiharu Kato · 10 citations
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models

paper · doi:10.1215/s0012-7094-03-11632-4

Abstract

We compute the dimension of the tangent space to, and the Krull dimension of, the prorepresentable 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\backslash X. The second one is the "analytic" deformation functor of a fixed embedding of a finitely generated discrete group N in \rm PGL(2, K) over a nonarchimedean-valued field K, and the data are computed in terms of the Bass-Serre representation of N via a graph of groups. Finally, if Γ is a free subgroup of N such that N is contained in the normalizer of Γ in \rm PGL(2, K), then the Mumford curve associated to Γ becomes equipped with an action of N/Γ, and we show that the algebraic functor deforming the latter action coincides with the analytic functor deforming the embedding of N.

Cited by

• Deformation of Curves with Automorphisms and representations on Riemann-Roch spaces
• Relevements des revetements de courbes faiblement ramifies (Lifts of weakly ramified coverings of curves)
• Problems from the workshop on "Automorphisms of Curves" (Leiden, August, 2004)
• Three examples of the relation between rigid-analytic and algebraic deformation parameters
• On the Krull Dimension of the deformation ring of curves with automorphisms
• Size of families of curves with large automorphism group in positive characteristic
• Which weakly ramified group actions admit a universal formal deformation?
• On the tangent space of the deformation functor of curves with automorphisms
• Generalized Artin-Mumford curves over finite fields
• A homological approach for computing the tangent space of the deformation functor of curves with automorphisms
• Deformation functors of local actions
• Equiramified deformations of covers in positive characteristic

Related

• Equivariant deformation of Mumford curves and of ordinary curves in positive characteristic
• Quadratic differentials and equivariant deformation theory of curves
• Equiramified deformations of covers in positive characteristic
• Induction and restriction in formal deformation of coverings
• On the Krull Dimension of the deformation ring of curves with automorphisms