  We prove a simultaneous generalization of the classical Riemann-Hurwitz and
Plucker formulas, addressing the total inflection of a morphism from a (smooth,
projective) curve to an arbitrary (smooth, projective) higher-dimensional
variety. Our definition of ramification is relative to an algebraic family of
divisors on the target variety, and our formula is obtained using the theory of
refined top Chern classes. In assigning multiplicities to ramification points,
we frequently have to consider excess degeneracy loci, but we are able to show
nonetheless that the multiplicities are always nonnegative, and are positive
under very mild hypotheses.
