  We construct a natural branch divisor for equidimensional projective
morphisms where the domain has lci singularities and the target is nonsingular.
The method involves generalizing a divisor contruction of Mumford from sheaves
to complexes. The construction is valid in flat families. The generalized
branch divisor of a stable map to a nonsingular curve X yields a canonical
morphism from the space of stable maps to a symmetric product of X. This branch
morphism (together with virtual localization) is used to compute the Hurwitz
numbers of covers of P^1 for all genera and degrees in terms of Hodge
integrals.
