We study the deformation of G-marked stable curves in the case where G is a cyclic group, and construct a parameterizing space for G-marked stable curves of a given numerical type.

This is then used in order to study the components of the locus of stable curves admitting the action of a cyclic group of non prime order d, extending work of F. Catanese in the case where d is prime. The main tools we use here are Teichmuller theory and the deformation of G-marked curves.

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