Reduction and deformation of one-point Galois covers

The etale fundamental group of an algebraic curve encodes information about both its finite etale covers and the unramified extensions of its function field. Over an algebraically closed field of characteristic 0, the Riemann existence theorem provides a powerful tool to compute these fundamental groups. The situation is, however, significantly more complicated over fields of positive characteristic p. In this thesis, I extend two techniques for studying these fundamental groups. The first relies on the relationship between the fundamental group of a curve and that of its reduction and involves showing that covers of elliptic curves defined over "small" fields branched at exactly one point have good reduction to positive characteristic. This generalizes results of Raynaud and Obus. The second generalizes techniques that Pries used to show the existence of a cover of P^1 with a single wildly ramified branch point whose conductor is as small as possible. These techniques include studying degenerations of one-point covers in positive characteristic.