The mapping class group and special loci in moduli of curves

The general problem this thesis is concerned with is that of studying the subvarieties of the moduli space [special characters omitted] corresponding to curves with extra automorphisms. A typical curve with marked points has no automorphisms; but some do, depending upon the choice of curves and position of marked points. This gives us certain subvarieties in the moduli space. For Riemann surfaces, these subvarieties are characterized by specifying a finite group of mapping-classes whose action on a curve is fixed topologically. This thesis builds upon previous work by González-Díez, Harvey and Schneps. González-Díez and Harvey [GH92] considered these irreducible subvarieties for genus g ≥ 2 curves without marked points over the complex numbers, where they have studied the coarse moduli space for surfaces with a specified automorphism group. Cornalba [Cor87] gives a complete classification of the irreducible subvarieties corresponding to the curves whose automorphism group contains a fixed cyclic subgroup of prime order, in the case g ≥ 1, n = 0 over the complex numbers. Later Schneps [Sch02a] considered the situation of genus 0 with n marked points, and genus 1 with n = 1 or 2 marked points, corresponding to the curves having a cyclic group in its automorphism group, over the complex numbers. In this thesis, we consider more general cases in higher genus and in characteristic p, with n marked points, corresponding to the marked curves whose automorphism group contains a fixed cyclic subgroup acting in a fixed way