In this dissertation, we study deformations of actions of a cyclic group of order p on the formal power series ring k[[u1,...,un]], where k is a field of characteristic p>0. We draw upon work of B. Peskin to reduce, under certain hypotheses, the task of determining the tangent space of the deformation functor D to a problem in invariant theory. When n=2 and p=3, we use these results to explicitly compute the tangent space of D and then generalize results of Mezard and Bertin for smooth curves to smooth surfaces. In particular, we compute the prorepresentable hull of the equicharacteristic local deformation functor D of a smooth surface with finite, cyclic group action at a point of wild ramification in characteristic 3.
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