Branched Covers of Curves With Fixed Ramification Locus

We examine conditions under which there exists a non-constant family of finite maps of curves over an algebraically closed field k of fixed degree and fixed ramification locus, under a notion of equivalence derived from considering linear series on a fixed source curve X. If we additionally impose that the maps are Galois, we show such a family exists precisely when the following conditions are satisfied: there is a unique ramification point, char(k) = p > 0, and the Galois group is (Z/pZ)^n for some integer n > 0. In the non-Galois case, we conjecture that a given map occurs in such a family precisely when at least one ramification index is at least p. One direction of this conjecture is proven and the reverse implication is proven in several cases. We also prove a result concerning the smoothness of the moduli space of maps considered up to this notion of equivalence.