| In this thesis, I investigate wildly ramified Galois covers of curves &phis; : Y → P1k with group G branched at exactly one point over an algebraically closed field k of characteristic p. The questions I consider relate to which inertia groups and which filtrations of higher ramification groups can be realized for such a cover &phis;. The answer to these questions will give information about the minimal possible genus of Y and will solve a conjecture of Abhyankar about inertia. I give partial answers to these questions in the case that the Sylow- p subgroup of G has order p. I also prove a result that proper families of such covers must be constant after an affine linear transformation of the base. My techniques include formal patching, an analysis of deformations of covers of semi-stable curves, and the construction of an affine parameter space for such covers. |
