Boundary of Hurwitz Spaces and Explicit Patching

We describe a general method for the study and computation of Hurwitz spaces of curves of any genus. It is based on a careful combinatorial study of the associated Jacobian. The key tool is an adapted cell decomposition of the cohomology of a graph (used here for the intersection graphs of special curves). We illustrate this method in the context of modular curves to produce modular units. We also give a detailed simple example and show how the algebraic difficulty of Hurwitz spaces computation can be reduced to its minimum.