  Compactification of the heterotic string on toroidal orbifolds is a promising
set-up for the construction of realistic unified models of particle physics.
The target space dynamics of such models, however, drives them slightly away
from the orbifold point in moduli space. This resolves curvature singularities,
but makes the string computations very difficult. On these smooth manifolds we
have to rely on an effective supergravity approximation in the large volume
limit. By comparing an orbifold example with its blow-up version, we try to
transfer the computational power of the orbifold to the smooth manifold. Using
local properties, we establish a perfect map of the the chiral spectra as well
as the (local) anomalies of these models. A key element in this discussion is
the Green-Schwarz anomaly polynomial. It allows us to identify those
redefinitions of chiral fields and localized axions in the blow-up process
which are relevant for the interactions (such as Yukawa-couplings) in the model
on the smooth space.
